ON THE STRANGE DOMAIN OF ATTRACTION TO GENERALIZED DICKMAN DISTRIBUTIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES
|
|
- Maximilian Nicholas Richard
- 5 years ago
- Views:
Transcription
1 ON THE STRANGE DOMAIN OF ATTRACTION TO GENERALIZED DICKMAN DISTRIBUTIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES ROSS G PINSKY Abstract Let {B }, {X } all be independent random variables Assume that {B } are {, 1}-valued Bernoulli random variables satisfying B = Berp ), with p =, and assume that {X } satisfy X > and EX < Let = n p, assume that and define the normalized sum of independent random variables W n = 1 n B X We give a general condition under which W n c, for some c [, 1], and a general condition under which W n converges wealy to a ribution from a family of ributions that includes the generalized Dicman ributions GDθ), θ > In particular, we obtain the following result, which reveals a strange domain of attraction to generalized Dicman ributions Assume that lim X = 1 Let J µ, J p be nonnegative integers, let c µ, c p > and let µ n c µn a J µ j=1 logj) n) a j, p n c p n b Jp j=1 logj) n) b j ) 1, b Jp, where log j) denotes the jth iterate of the logarithm If i J p J µ; ii b j = 1, j J p; iii a j =, j J p 1, and a Jp >, then lim n W n = 1 cp GDθ), where θ = θ a Jp Otherwise, lim n W n = δ c, where c {, 1} depends on the above parameters We also give an application to the statistics of the number of inversions in certain random shuffling schemes 2 Mathematics Subject Classification 6F5 Key words and phrases Dicman function, generalized Dicman ribution, domain of attraction, normalized sums of independent random variables, inversions, permutations 1
2 2 1 Introduction and Statement of Results The Dicman function ρ 1 is the unique function, continuous on, ), and satisfying the differential-delay equation ρ 1 x) =, x ; ρ 1 x) = 1, x, 1]; xρ 1x) + ρ 1 x 1) =, x > 1 This function has an interesting role in number theory and probability, which we describe briefly in the final section of the paper With a little wor, one can show that the Laplace transform of ρ 1 is given by ρ 1 x)e λx dx = expγ + 1 e λx 1 x dx), where γ is Euler s constant See, for example, [6] or [9]) From this it follows that ρ 1 x)dx = e γ, and consequently, that e γ ρ 1 is a probability density on [, ) We will call this probability ribution the Dicman ribution We denote its density by p 1 x) = e γ ρ 1 x), and we denote by D 1 a random variable ributed according to the Dicman ribution Differentiating the Laplace transform E exp λd 1 ) = exp 1 e λx 1 x dx) of D 1 at λ = shows that ED 1 = 1 These ributions decay very rapidly; indeed, it is not hard to show that p 1 x) e γ Γx+1), x [6] In fact, for all θ >, expθ 1 probability ribution e λx 1 x dx) is the Laplace transform of a We will prove this directly; however, this fact follows from the theory of infinitely divisible ributions, and shows that the ribution in question is infinitely divisible) This ribution has density p θ = e θγ Γθ) ρ θ, where ρ θ satisfies the differential-delay equation ρ θ x) =, x ; 11) ρ θ x) = x θ 1, < x 1; xρ θ x) + 1 θ)ρ θx) + θρ θ x 1) =, x > 1 We will call such ributions generalized Dicman ributions and denote them by GDθ) We denote by D θ a random variable with the GDθ) ribution Differentiating its Laplace transform at λ = shows that ED θ = θ
3 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 3 These ribution decays very rapidly; indeed, it is not hard to show that p θ x) C θ Γx+1), x 1, for an appropriate constant C θ In fact, the scope of this paper leads us to consider a more general family of ributions than the generalized Dicman ributions Let X be a random variable satisfying EX 1 Then, as we shall see, for θ >, there exists a ribution whose Laplace transform is exp θ 1 E exp λxx ) 1 x dx ) We will denote this ribution by GD X ) θ) and we denote a random variable with this ribution by D X ) θ When X 1, we revert to the previous notation for generalized Dicman ributions) Differentiating the Laplace transform at λ = shows that ED X ) θ = θex It is nown that the generalized Dicman ribution GDθ) arises as the limiting ribution of 1 n n Y, where the {Y } are independent random variables with Y ributed according to the Poisson ribution with parameter θ [1] It is also nown that the Dicman ribution GD1) arises as the limiting ribution of 1 n n Y as n, where the {Y } are independent Bernoulli random variables satisfying P Y = 1) = 1 P Y = ) = 1 Such behavior is in inct contrast to the law of large numbers behavior of a well-behaved sequence of independent random variables {Z } with finite first moments; namely, that 1 n Z converges in ribution to 1 as n, where = n EZ The purpose of this paper is to understand when the law of large numbers fails and a ribution from the family GD X ) θ) arises in its stead From the above examples, we see that generalized Dicman ributions sometimes arise as limits of normalized sums from a sequence {V } of independent random variables which are non-negative and satisfy the following three V conditions: i) lim P V = ) = 1, ii) lim V > EV V >) = 1 and iii) EV = In the above examples, Y plays the role of V ) It turns out that these three conditions are very far from sufficient for a generalized Dicman ribution to arise In fact, as we shall see in Theorem 2 below, such ributions arise only in a strange sequence of very narrow windows of opportunity
4 4 ROSS G PINSKY In light of the above discussion, we will consider the following setting Let {B }, {X } random variables satisfying: be mutually independent sequences of independent Assume that {B } 12) P B = 1) = 1 P B = ) = p [, 1), and assume that {X } satisfy: 13) X >, EX < Let 14) = and define 15) W n = 1 p, are Bernoulli random variables B X We will be interested in the limiting behavior of W n trivialities, we will assume that 16) lim n = and p =, since otherwise n=1 B X is almost surely finite In order to avoid Note that for the example brought with the Pois θ )-ribution, we have p = 1 e θ, X is ributed according to Y {Y > }, where Y has the Pois θ ) ribution, = θ 1 e θ and = nθ And for the example with the Ber 1 )-ribution, we have p = 1, X = deterministically, = and = n In the first of these two examples, X 1, and in the second one, X = 1 for all Our first theorem gives a general condition for W n c which is the law of large numbers if c = 1), and a general condition for convergence to a limiting ribution from the family of ributions GD X ) θ) Using this theorem, we can prove our second theorem, which reveals the strange domain of attraction to generalized Dicman ributions Of course, we
5 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 5 are using the term domain of attraction not in its classical sense, since our sequence of random variables, although independent, are not identically ributed) Let δ c denote the degenerate ribution at c Theorem 1 Let W n be as in 15), where {B }, {X } and are as in 12)-14) and 16) i Assume that { X } is uniformly integrable which occurs automatically X if lim = 1) a Assume also that max 1 n 17) lim = n Then lim W n = 1 n b Assume also that there exists a sequence {K n } n=1 such that 18) lim and 19) lim n If n =K n+1 max 1 Kn p =, = M Kn 11) c lim n exists, then lim W n = c n If 11) does not hold, then the ributions of {W n } n=1 form a tight sequence whose set of accumulation points is {δ c : c A}, where A denotes the set of accumulation points of the sequence { M Kn } n=1 ii Assume that there exists a random variable X such that X 111) lim = X
6 6 ROSS G PINSKY Assume also that { } is increasing, that lim p = and that there exist θ, L, ) such that 112) lim Then p +1 = θ, lim lim W n = LD X ) θ), n M = L where D X ) θ) is a random variable with the GD X ) θ) ribution Remar 1 In 112), necessarily L 1 θ Indeed, if {p } and {} satisfy the conditions of part ii), and we choose X =, then W n LD θ Since EW n = 1 and ED θ = θ, it follows from Fatou s lemma that L 1 θ In most cases of interest, one has L = 1 θ Remar 2 By Fatou s lemma, the random variable X in part ii) must satisfy EX 1 Remar 3 The uniform integrability of { X } in part i) occurs automatically if lim X = 1, because if a sequence {Y } of random variables satisfies Y Y, and E Y <, then E Y E Y is equivalent to uniform integrability Remar 4 In the case that X =, or more generally, if EX 2 Cµ2, for all and some C >, then V arw n ) C N p µ 2 M 2 n n = C p µ 2 n p ) 2 C sup 1 n Thus, in this case part i-a) follows directly from the second moment method Using Theorem 1, we can prove the following theorem that exhibits the strange domain of attraction to generalized Dicman ributions Let log j) denote the jth iterate of the logarithm, and mae the convention j=1 = 1 Theorem 2 Let W n be as in 15), where {B }, {X } and are as in 12)-14) Assume also that lim X = 1 Let J µ, J p be nonnegative
7 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 7 integers, let c µ, c p > and define with b Jp Assume that µx) = c µ x a J µ j=1 J p log j) x) a j, px) = c p x b log j) x) b ) j 1, j=1 µ), p p); +1 µ ) Assume that the exponents {a j } Jµ j=, {b j} Jp j= holds If i J p J µ ; have been chosen so that 16) 113) ii b j = 1, j J p ; then iii a j =, j J p 1, and a Jp >, lim W n = 1 n θ D θ, with θ = c p, a Jp where D θ is a random variable with the GDθ) ribution Otherwise, lim n W n = c, where c {, 1} To determine c, let 114) κ µ = min{ j J µ : a j } and κ p = min{ j J p : b j 1} If { j J µ : a j } is not empty, a κµ > and either { j J p : b j 1} is empty and κ µ < J p, or { j J p : b j 1} is not empty and κ µ < κ p, then c = ; otherwise, c = 1 Remar 1 Note that if one chooses = µ) and p = p), then the condition +1 µ ) is always satisfied Remar 2 Theorem 2 shows that to obtain a generalized Dicman ribution, {p } in particular must be set in a very restricted fashion For some intuition regarding this phenomenon, tae the situation where X =, and consider the sequence {σ 2 W n )} n=1 of variances This sequence converges
8 8 ROSS G PINSKY to in the cases where W n converges to 1, converges to in the cases where W n converges to, and converges to a positive number in the cases where W n converges to a generalized Dicman ribution We now state explicitly what Theorem 2 yields in the cases J p =, 1 J p = We have p n c p n b, b >, µ n c µ n a J µ j=1 log j) n) a j In order that 16) hold, we require b 1 We also require either: a b > 1; or a b = 1 and a 1 > 1; or a b = a 1 = 1 and a 2 > 1; etc If b = 1 and a >, then Otherwise, lim n W n = 1 lim W n = 1 n θ D θ, where θ = c p a J p = 1 We have p n c p n b log n) b 1, b 1, µ n c µ n a J µ j=1 log j) n) a j In order that 16) hold, we require either b = and b 1 >, or < b < 1, or b = 1 and b 1 1 We also require either: a b > 1; or a b = 1 and a 1 b 1 > 1; or a b = a 1 b 1 = 1 and a 2 > 1; etc If J µ 1, b = b 1 = 1, a = and a 1 >, then lim W n = 1 n θ D θ, where θ = c p a 1 If b = 1 and a >, then lim n W n = Otherwise, lim n W n = 1 Remar In [3] and [8], where the GD1) ribution arises, one has J p = 1 with b = b 1 = 1, a =, a 1 = 1, c p = c µ = 1
9 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 9 The organization of the rest of the paper is as follows In section 2 we use Theorems 1 and 2 to investigate a question raised in [5] concerning the statistics of the number of inversions in certain random shuffling schemes In sections 3 and 4 respectively we prove Theorems 1 and 2 In section 5 we prove a couple basic facts about generalized Dicman ributions In particular, we provide a rather probabilistic proof that the ribution whose Laplace transform is given by expθ 1 e λx 1 x dx) possesses a density p θ of the form p θ = c θ ρ θ, where ρ θ satisfies 11) We also give a reference for the formula c θ = e θγ Γθ) Finally, in section 6, we offer a little historical bacground concerning the Dicman function ρ 1 and its connection to number theory and probability 2 An application to random permutations We consider a setup that appeared in [5], and which in the terminology of this paper can be described as follows For each N, let E {1,, 1} Let X be uniformly ributed on E, and let B = Ber E ) So Define = 1 E I n = l, p = E l E B X We allow E =, in which case B = and X is not defined In such a case, we define B X = and = We always have E 1 = Consider first the case that E = {1,, 1} Then B 1 X 1 = and for 2 n, B X is uniformly ributed over {, 1, 1} In this case, I n has the ribution of the number of inversions in a uniformly random permutation from S n The authors in [5] have a typo and wrote E = {1,, } instead) To see this, consider the following shuffling procedure for n cards, numbered from 1 to n The cards are to be inserted in a row, one by one, in order of their numbers At step one, card number 1 is set down The number of inversions created by this step is zero, which is given by B 1 X 1 At step, for {2,, n}, card number is randomly
10 1 ROSS G PINSKY inserted in the current row of cards, numbered 1 to 1 Thus, for any j {, 1,, 1}, card number has probability 1 of being placed in the position with j cards to its right and 1 j cards to its left), in which case this step will have created j new inversions, and this is represented by B X It is clear from the construction that the random variables {B X } n are independent Thus, I n indeed gives the number of inversions in a uniformly random permutation from S n It is well-nown that the law of large numbers and the central limit theorem hold for I n in this case Indeed, using the above representation, a direction calculation shows that EI n = nn 1) 4 and that VarI n ) = On 3 ); thus the central limit theorem follows from the second moment method Consider now the general case that E {1,, 1} Then I n gives the number of inversions in a random permutation created by a shuffling procedure in the same spirit as the above one At step, with probability 1 E, card number is inserted at the right end of the row, thereby creating no new inversions, and for each j E, with probability 1 it is inserted in the position with j cards to its right, thereby creating j new inversions In particular, as a warmup consider the cases E = {1} and E = { 1}, 2 n In each of these two cases, at step, 2 n, card number is inserted at the right end of the row with probability 1 1 In the first case, with probability 1 card number is inserted immediately to the left of the right most card, thereby creating one new inversion, while in the second case, with probability 1 card number is inserted at the left end of the row, thereby creating 1 new inversions In both cases Xn µ n = 1 for all n, and in both cases, p = 1 In the first case, = 1 while in the second case, = 1 Thus, in the first case, = n p log n, and in the second case, n Therefore, it follows from Theorem 1 or 2 that in the first case In log n converge in ribution to 1, while in the second case, In n converges in ribution to GD1)
11 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 11 The authors of [5] as which choices of {E } lead to the Dicman ribution and which choices lead to the central limit theorem Of course, the law of large numbers is a prerequisite for the central limit theorem The following theorem gives sufficient conditions for the law of large numbers to hold and sufficient conditions for convergence to a ribution from the family GD X ) θ) In order to avoid trivialities, we need to assume that 16) holds Recalling that = when E =, and that 1 otherwise, note that = EI n = E E Thus, in the present context the requirement 16) is 21) E =, = p which holds in particular if E for all sufficiently large Theorem 3 Assume that 21) holds i Assume that at least one of the following conditions holds: a lim E = and { max 1 n max b lim 1 n n n = Then 1 In EI n n } n=1 is bounded; ii Assume that E = N 1, for all large, and that X X Also assume that µ) and +1 µ ), where µx) = c µ x a J µ j=1 logj) x) a j, with a > Then In EI n 1 θ DX ) θ, with θ = N a, where D X ) θ is a random variable with the GD X ) θ) ribution Remar 1 The condition on { } in part i-a) is just a very wea regularity requirement on its growth rate recall that 1 < 1) The condition in part i-b) is fulfilled if µ) and +1 µ ), where µx) = c µ Jµ j=1 logj) x) a j with J µ Remar 2 Note that the random variable X in part ii) taes on no more than N inct values
12 12 ROSS G PINSKY Proof Assume first that the condition in part i-a) holds We claim that since { max 1 n n } n=1 is bounded, there exists a sequence of positive integers {γ n } n=1 satisfying lim n γ n = and such that { max 1 n is also bounded n } n=1 =γn+1 Indeed, assume to the contrary that the above sum is unbounded for all such choices of {γ n } n=1 Then necessarily, {µ n} n=1 is unbounded Indeed, since by assumption E 1 for sufficiently large, the same is true for, and thus, choosing, for example, γ n = [n 1 2 ], it follows that for sufficiently large n, n =γ n+1 must be unbounded if { max 1 n we have n of { max 1 n n n =γn+1 < γ n + n =γ n+1 n =γ n log n Thus {µ n} n=1 } n=1 is unbounded) Also, since <,, and it follows from the boundedness } n=1 and the unboundedness of { max 1 n n =γn } n=1 that the unbounded sequence {max 1 n } n=1 has the property that { max 1 n γ n } n=1 is bounded for all sequences {γ n } n=1 satisfying lim n γ n =, which is impossible Now let {γ n } n=1 be a sequence such that lim n γ n = and { max 1 n is bounded Since and { max 1 n = n } n=1 =γn+1 that 17) holds E µ min E ) >γ n =γ n n } n=1 =γn+1 is bounded, it follows from the first condition in i-a) Now assume that the condition in part i-b) holds Since n, it follows again that 17) holds Thus, assuming either i-a) or i-b), it follows from part i-a) of Theorem 1 that In EI n 1 Now assume that the condition in part ii) holds Then p = N, for large, and c a J µ j=1 logj) ) a j, with a > Thus, and lim = M = a N E Nc µ a n a J µ j=1 log j) n) a j, Also, if the condition in part ii) holds, then +1 a c a 1 J µ j=1 logj) ) a j Thus, lim p +1 = N a We
13 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 13 conclude from part ii) of Theorem 1 that N a I n EI n 1 θ GDX ) θ), where θ = 3 Proof of Theorem 1 Since EW n = 1, for all n, the ributions of {W n } n=1 are tight Thus, since the random variables are nonnegative, it suffices to show that their Laplace transforms E exp λw n ) converge under the conditions of part i) to exp λc), for the specified value of c, and under the conditions of part ii) to expθ 1 Ee LλxX 1 x dx), which is the Laplace transform of LD X ) θ) Proof of part i) Note that part i-a) is the particular case of part i-b) in which one can choose K n = n, and then 11) holds with c = 1 Thus, it suffices to consider part i-b) We have for λ >, 31) E exp λw n ) = = K n n E exp λ B X ) = n 1 p 1 E exp λ X ) )) 1 p 1 E exp λ X ) )) = n =K n+1 1 p 1 E exp λ X ) )) Since n =K n+1 1 p ) n =K n+1 1 p 1 E exp λ X ) )) 1, it follows from assumption 18) that 32) lim n n =K n+1 1 p 1 E exp λ X ) )) = 1 Applying the mean value theorem to E exp λ X ) as a function of λ, and recalling that = EX, we have 33) λ EX exp λ X ) 1 E exp λ X ) λ
14 14 ROSS G PINSKY The assumption that { X } is uniformly integrable means that lim N sup 1 < E X 1 X ) = Thus, in light of 19) and the uniform integrability assumption, it follows that for all ɛ >, there exists >N an n ɛ such that 34) λ EX exp λ X ) = λ E X exp λ X ) 1 ɛ)λ, 1 K n, n n ɛ Thus, 33) and 34) yield 35) 1 ɛ)λ 1 E exp λ X ) λ, 1 K n, n n ɛ Since for any ɛ >, there exists an x ɛ > such that 1+ɛ)x log1 x) x, for < x < x ɛ, it follows from 35) and 19) that there exists an n ɛ such that 36) µ λ 1 + ɛ)λp log 1 p 1 E exp X ) )) 1 ɛ)λp, 1 K n, n n ɛ From 36) we have Kn = 1 + ɛ)λ ɛ p log 37) If 1 ɛ)λ K n Kn = ɛ p, n n ɛ M Kn 38) c lim n 1 p 1 E exp λ X ) )) Kn = lim p n exists, then from 31), 32), 37) and 38), along with the fact that ɛ > is arbitrary, we conclude that lim E exp λw n) = exp λc), n which proves that lim n W n = c The rest of the results in part i-b), concerning accumulation points, follow in the same manner
15 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 15 Proof of part ii) From 31), we have 39) log E exp λw n ) = λ log 1 p 1 E exp X ) )) Since by assumption lim p =, for any ɛ > there exists a ɛ such that 31) λ 1 + ɛ)p 1 E exp X ) ) λ log 1 p 1 E exp X ) )) p 1 E exp λ X ) ), ɛ We now show that for any ɛ > there exists a ɛ such that 311) 1 ɛ)e exp λ X ) E exp λ X ) 1+ɛ)E exp λ X ), ɛ By assumption 112) and the assumption that {µ n } n=1 is increasing, there exists a C such that C, for 1 n and n 1 By assumption, X X Without loss of generality, we assume that all of these random variables are defined on the same space and that X X as For δ >, let A ;δ = {sup X l X δ} l µ l Then A ;δ is increasing in and lim P A ;δ ) = 1 We have 312) A c ;δ exp λ X )dp P A c ;δ ), and 313) exp λcδ) A ;δ exp λ X )dp expλcδ) exp λ X )dp A ;δ A ;δ exp λ X )dp Now 311) follows from 312) and 313)
16 16 ROSS G PINSKY Letting ɛ = max ɛ, ɛ), it follows from 31) and 311) that 314) 1 + ɛ)p 1 1 ɛ)e exp λ µ ) λ X ) log 1 p 1 E exp X ) )) p ɛ)e exp λ µ ) X ), ɛ From 39) and 314) we have 315) = ɛ = ɛ p 1 + ɛ) 1 1 ɛ)e exp λ µ ) X ) + o1) log E exp λw n ) p ɛ)e exp λ µ ) X ), as n Define x n) =, ɛ ɛ n 1 Then we have 316) = ɛ = ɛ By assumption, { } [ ɛ, µn ] n, and n) p 1 1 ± ɛ)e exp λ µ ) X ) = 1 1 ± ɛ)e exp λx n) X ) x n) = x n) +1 xn) n) ) p +1 is increasing; thus {xn) }n = µ By assumption, lim ɛ n now show that the mesh, max as n Let n) j n = max ɛ ɛ n 1 n) n 1 n) ɛ = +1, is a partition of µ = and lim n n = L We, of the partition converges to, where ɛ j n n Without loss of generality, assume either that {j n } is bounded or that lim n j n = In n the former case it is clear that max ɛ n 1 n) = n) j n = µ jn+1 µ jn Now consider the latter case From assumption 112) and the assump- µ tion that lim p =, it follows that lim n+1 µ n n = Then we have max n) ɛ n 1 = n) j n = µ j n+1 µ jn = µ j n+1 µ jn M jn M jn µ j n+1 µ jn M jn Finally, we note that from 112) we have lim p +1 = θ In light of these facts, along with 315), 316) and the fact that ɛ > is arbitrary, n
17 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 17 it follows that 317) lim log E exp λw n) = θ n L E exp λxx ) 1 1 dx = θ x 4 Proof of Theorem 2 E exp λlxx ) 1 dx, x We will assume that J p, J µ 1 so that we can use a uniform notation, leaving it to the reader to verify that the proof also goes through if J p or J µ is equal to zero First assume that 113) holds Then by the assumptions in the theorem, Thus, 1 J p J µ ; c µ J µ j=j p log j) ) aj, a Jp > ; J p p c p j log j) ) 1 ; j=1 +1 c µ a JP log J P ) ) a Jp 1 = Consequently, 41) lim j J p 1 j=1 logj) p c µ c p log Jp) n) a Jp a Jp M = a J p c p and lim J µ j=j p+1 J µ j=j p+1 log j) ) a j log j) n) a j p +1 = c p a Jp Thus, from part ii) of Theorem 1 it follows that lim n W n = 1 θ D θ, where θ = cp a Jp Now assume that 113) does not hold We need to show that {K n } n=1 can be defined so that 18) and 19) hold, and so that 11) holds with c {, 1} We also have to show when c = and when c = 1 Recall the definitions in 114) If { j J µ : a j } is empty, or if it is not empty and a κµ <, then { } is bounded Therefore, 18) and 19) hold with
18 18 ROSS G PINSKY K n = n and it follows from part i-a) of Theorem 1 that lim n W n = 1 Thus, from now on we assume that { j J µ : a j } is not empty and that a κµ > In order to use uniform notation, we will assume that κ µ >, leaving the reader to verify that the proof goes through if κ µ = Thus, we have 42) J µ j=κ µ log j) ) aj, κ µ 1, a κµ > In order to simplify notation, for the rest of this proof, we will let L l ) denote a positive constant multiplied by a product of powers possibly of varying sign) of iterated logarithms log j), where the smallest j is strictly larger than l The exact form of this expression may vary from line to line Sometimes we will need to inguish between two such expressions in the same formula, in which case we will use the notation L 1) l ), L 2) l ) Thus, we rewrite 42) as 43) log κµ) ) aκµ L κ), κ µ 1, a κµ > If { j J p : b j 1} is empty, then the second condition in 113) is fulfilled and we have J p 44) p c p j log j) ) 1 j=1 Since we are assuming that 113) does not hold, at least one of the other two conditions in 113) must fail This forces κ µ J p Recall that we are assuming that { j J µ : a j } is not empty and that a κµ > ) 45) Consider first the case that κ µ > J p Then from 43) and 44) we have = p log Jp+1) n)log κµ) n) aκµ L κµ n), where κ µ J p + 1 From 43) and 45) it follows that 18) and 19) hold by choosing K n = n Thus, from part i-a) of Theorem 1, lim n W n = 1
19 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 19 Now consider the case κ µ < J p Then from 43) and 44) we have 46) = p log κµ) n) aκµ L κµ n), where κ µ J p 1, and for any K n satisfying K n and K n n, we have 47) p c p log J p+1) n log Jp+1) ) K n = cp log logjp) n =K n log Jp) K n From 43) and 46) we have 48) µ Kn M Kn log κ µ) K ) n aκµ L 1) κ µ K n ) log κµ) n L 2) κ µ n), κ µ J p 1, a κµ > ; log κ µ) K ) n aκµ L 1) κ µ K n ) log κµ) n L 2) κ µ n), κ µ J p 1, a κµ > ; As we explain in some detail below, since κ µ < J p, we can choose {K n } n=1 so that log Jp) K n 49) lim n log Jp) n log κ µ) K ) n aκµ L 1) κ = 1 and lim µ K n ) n log κµ) n L 2) κ µ n) = From 43) and 47)-49), we conclude that {K n } can be defined so that 18) and 19) hold, and so that 11) holds with c = This proves that lim n W n = To explain 49), note that L1) κµ Kn) L 2) κµ n) log κµ+1) n) A, for some A > and all large n Recall that the powers of the iterated logarithms in L 2) µ can be negative) Thus, in place of the second limit in 49), it suffices to show that δ n log κµ) K n log κµ) n thus, ) aκµ log κµ+1) n) A n We have log κµ) K n = δ n ) 1 aκµ log κµ+1) n) A aκµ log κµ) n; 41) log κµ+1) K n log κµ+1) n = log δ n a κµ log κµ+1) n A logκµ+2) n a κµ log κµ+1) n + 1 Defining K n by choosing δ n = log κµ+1) n) 1, it follows from 41) and the fact that J p κ µ + 1 that the two equalities in 49) hold
20 2 ROSS G PINSKY We now consider the case that { j J p : b j 1} is not empty Then in order to fulfill the second condition in 16), we have b κp < 1 We write κ p 1 411) p c p j j=1 log j) ) 1 log κ p) ) b κp J p j=κ p+1 log j) ) b j From 43) and 411) it follows that = n p satisfies log κµ) n) aκµ L κµ n), κ µ < κ p ; 412) log κp) n) aκp bκp+1 L κp n), κ µ = κ p ; log κp) n) 1 bκp L κp n), κ µ > κ p, and from 411) it follows that for any K n satisfying K n and K n n, 413) =K n p c [ p log κ p) n ) 1 b κp 1 b κp J p j=κ p+1 log j) n ) b j log κp) K n ) 1 bκp J p j=κ p+1 log j) K n ) bj ] From 43) and 412) we have 414) µ Kn log κµ) K n log κµ) n ) aκµ L 1) κµ Kn) L 2) κµ n), κ µ < κ p ; log κp) K n) aκp L 1) κp ) Kn) aκp log κp) bκp n +1 L ) 2) aκµ log κµ) K n L 1) κµ ) Kn) 1 bκp log κp) n L 2) κp n), κ µ = κ p ; κp n), κ µ > κ p It is immediate 43) and 414) that if κ µ κ p, then 18) and 19) hold by choosing K n = n For the case κ µ = κ p, recall that b κp, 1)) Thus, from part i-a) of Theorem 1, lim n W n = 1 Now consider the case κ µ < κ p For simplicity, we will assume that the higher order iterated logarithmic terms do not appear; that is, we will
21 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 21 assume from 412)-414) that =K n p c [ p log κ p) n ) 1 b κp log κp) ) ] 1 bκp K n ; 1 b κp 415) µ Kn log κµ) K n ) aκµ log κµ) ; n M Kn log κµ) K n ) aκµ log κµ) n The additional logarithmic terms can be dealt with similarly to the way they were dealt with for 49), as explained in the paragraph following 49) Applying the mean value theorem to the function x 1 bκp, we obtain 416) where n K n, n) log κ p) n ) 1 b κp log κp) K n ) 1 bκp = 1 b κ p ) log κp) n K n log κp) n ) bκp, Since κ µ < κ p, we can choose K n such that log lim κµ) K n n =, but lim κp) Kn log κµ) n n log n = 1 For such a choice of {K n }, it follows from 43), 415) and 416) that 18) and 19) hold, and that 11) holds with c = ; thus, lim n W n = 5 Basic Facts Concerning Generalized Dicman Distributions We proved in Theorem 1 that expθ 1 e λx 1 x dx) is the Laplace transform of a probability ribution, which we have denoted by GDθ) Proposition 1 Let D θ GDθ) Then 51) D θ = U 1 θ Dθ + 1), where U is ributed according to the uniform ribution on [, 1], and U and D θ on the right hand side above are independent Remar 1 From 51) it is immediate that D θ = U 1 θ 1 + U 1 U 2 ) 1 θ + U1 U 2 U 3 ) 1 θ +, where {U n } n=1 are IID random variables ributed according to the uniform ribution on [, 1]
22 22 ROSS G PINSKY Remar 2 Our proof of the proposition is rather probabilistic; a more analytic proof can be found in [7] Proof The proof of Theorem 1 showed in particular that if we let X = = and p = θ, in which case = n p = θn, then 52) Ŵ n θw n = 1 n where D θ GDθ) Let B D θ, J + n = max{ n : B }, with max We write 53) Ŵ n 1 n n=1 B = J n + 1 n 1 J + n 1 J n + 1 B ) + J + n n, where the first of the two summands on the right hand side above is interpreted as equal to if J + n 54) P J + n n x) = 1 We have n =[xn+1] Also, by the independence of {B }, we have 1 θ ) xθ, x, 1) 55) J 1 n + 1 J n + B {J n + 1 = } = B = Ŵ 1, 2 Letting n in 53) and using 52), 54) and 55), we conclude that 51) holds, where U is ributed according to the uniform ribution on [, 1], D θ GDθ) and U and D θ on the right hand side are independent Proposition 2 The GDθ) ribution has a density function p θ satisfying p θ = c θ ρ θ, for some c θ >, where ρ θ satisfies 11) Remar For a derivation of the formula c θ = e θγ Γθ), see [1]
23 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 23 Proof Let F θ x) = P D θ GDθ) ribution Then from 51) we have 56) F θ x) = P D θ x) = P U 1 θ Dθ + 1) x) = 1 F θ xy 1 θ 1)dy x) denote the ribution function for the 1 For x >, maing the change of variables, v = xy 1 θ 56) as 57) F θ x) = θx θ x 1 F Dθ v)1 + v) 1 θ dv, x > P D θ + 1 xy 1 θ )dy = 1, we can rewrite From 57) and the fact that F θ x) =, for x, it follows that F θ is continuous on R Also, since F θ x) =, for x, we have x 1 F Dθ v)1 + v) 1 θ dv = F Dθ v)1 + v) 1 θ dv, x 1 Consequently, it follows from 57) that F θ x) = C θ x θ, for x [, 1], where C θ = θ F Dθ v)1 + v) 1 θ dv From this and 57) it follows that F is differentiable on, 1) and on 1, ), and that, letting p θ = F θ, 58) p θ = c θ x θ 1, < x < 1, c θ = θ 2 F Dθ v)1 + v) 1 θ dv, and 59) p θ x) = θ 2 x θ 1 x 1 θ x F θx) F θ x 1)), x > 1 F Dθ v)1 + v) 1 θ dv θx 1 F θ x 1) = From 59), it follows that p θ is differentiable on x > 1, and that xp θ x)) = θ p θ x) p θ x 1) ), for x > 1, or equivalently, 51) xp θ x) + 1 θ)p θx) + θp θ x 1) =, x > 1 From 58) and 51) we conclude that p θ x) = c θ ρ θ, where ρ θ satisfies 11) Integrating by parts in the formula for c θ in 58) shows that c θ = θ 1 + v) θ p θ v)dv = θe1 + D θ ) θ
24 24 ROSS G PINSKY 6 The Dicman function in number theory and probability The Dicman function ρ ρ 1 arises in probabilistic number theory in the context of so-called smooth divisors are small numbers; that is, numbers all of whose prime Let Ψx, y) denote the number of positive integers less than or equal to x with no prime divisors greater than y Numbers with no prime divisors greater than y are called y-smooth numbers Then for s 1, ΨN, N 1 s ) Nρs), as N This result was first proved by Dicman in 193 [4], whence the name of the function, with later refinements by de Bruijn [2] See also [6] or [9] Let [n] = {1,, n} and let p + n) denote the largest prime divisor of n Then Dicman s result states that the random variable log p+ j) log n, j [n], on the probability space [n] with the uniform ribution converges in ribution as n to the ribution whose ribution function is ρ 1 x ), x [, 1], and whose density is ρ 1 x ) x 2 = ρ 1 x 1) x, x [, 1] It is easy to see that an equivalent statement of Dicman s result is that the random variable log p+ j) log j, j [n], on the probability space [n] with the uniform ribution converges in ribution as n to the ribution whose ribution function is ρ 1 x ), x [, 1], We note that the length of the longest cycle of a uniformly random permutation of [n], normalized by dividing by n, also converges to a limiting ribution whose ribution function is ρ 1 x ) If instead of using the uniform measure on S n, the set of permutations of [n], one uses the Ewens sampling ribution on S n, obtained by giving each permutation σ S n the probability proportional to θ Cσ), where Cσ) denotes the number of cycles in σ, then the length of the longest cycle of such a random permutation of [n], normalized by dividing by n, converges to a limiting ribution whose ribution function is ρ θ 1 x ), x [, 1] This ribution is also the ribution of the first coordinate of the Poisson-Dirichlet ribution PDθ) see [1]) The examples in the above paragraph lead to limiting ributions where the Dicman function arises as a ribution function, not as a density as is the case with the GDθ) ributions discussed in this paper The GDθ)
25 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 25 ribution arises as a normalized limit in the context of certain natural probability measures that one can place on N; see [3], [8] References [1] Arratia, R, Barbour, A and Tavaré, S, Logarithmic Combinatorial Structures: A Probabilistic Approach, EMS Monographs in Mathematics, European Mathematical Society EMS), Zurich, 23) [2] de Bruijn, N, On the number of positive integers x and free of prime factors > y, Nederl Acad Wetensch Proc Ser A 54, 1951), 5 6 [3] Cellarosi, F and Sinai, Y, Non-standard limit theorems in number theory, Prohorov and Contemporary Probability Theory, , Springer Proc Math Stat, 33, Springer, Heidelberg, 213) [4] Dicman, K, On the frequency of numbers containing prime factors of a certain relative magnitude, Ar Math Astr Fys 22, ) [5] Hwang, H-K and Tsai, T-H Quicselect and the Dicman function, Combin Probab Comput 11 22), [6] Montgomery, H and Vaughan, R, Multiplicative Number Theory I Classical Theory, Cambridge Studies in Advanced Mathematics, 97, Cambridge University Press, Cambridge, 27) [7] Penrose, M and Wade, A, Random minimal directed spanning trees and Dicmantype ributions, Adv in Appl Probab 36 24), [8] Pinsy R, A Natural Probabilistic Model on the Integers and its Relation to Dicman- Type Distributions and Buchstab s Function, preprint, 216) [9] Tenenbaum, G, Introduction to Analytic and Probabilistic Number Theory, Cambridge Studies in Advanced Mathematics, 46, Cambridge University Press, Cambridge, 1995) Department of Mathematics, Technion Israel Institute of Technology, Haifa, 32, Israel address: pinsy@mathtechnionacil URL:
ON THE STRANGE DOMAIN OF ATTRACTION TO GENERALIZED DICKMAN DISTRIBUTIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES
ON THE STRANGE DOMAIN OF ATTRACTION TO GENERALIZED DICKMAN DISTRIBUTIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES ROSS G PINSKY Abstract Let {B }, {X } all be independent random variables Assume that {B
More informationA Note on the Approximation of Perpetuities
Discrete Mathematics and Theoretical Computer Science (subm.), by the authors, rev A Note on the Approximation of Perpetuities Margarete Knape and Ralph Neininger Department for Mathematics and Computer
More informationKrzysztof Burdzy University of Washington. = X(Y (t)), t 0}
VARIATION OF ITERATED BROWNIAN MOTION Krzysztof Burdzy University of Washington 1. Introduction and main results. Suppose that X 1, X 2 and Y are independent standard Brownian motions starting from 0 and
More informationNotes 6 : First and second moment methods
Notes 6 : First and second moment methods Math 733-734: Theory of Probability Lecturer: Sebastien Roch References: [Roc, Sections 2.1-2.3]. Recall: THM 6.1 (Markov s inequality) Let X be a non-negative
More informationExercises and Answers to Chapter 1
Exercises and Answers to Chapter The continuous type of random variable X has the following density function: a x, if < x < a, f (x), otherwise. Answer the following questions. () Find a. () Obtain mean
More informationTHE POISSON DIRICHLET DISTRIBUTION AND ITS RELATIVES REVISITED
THE POISSON DIRICHLET DISTRIBUTION AND ITS RELATIVES REVISITED LARS HOLST Department of Mathematics, Royal Institute of Technology SE 1 44 Stockholm, Sweden E-mail: lholst@math.kth.se December 17, 21 Abstract
More informationk-protected VERTICES IN BINARY SEARCH TREES
k-protected VERTICES IN BINARY SEARCH TREES MIKLÓS BÓNA Abstract. We show that for every k, the probability that a randomly selected vertex of a random binary search tree on n nodes is at distance k from
More informationON COMPOUND POISSON POPULATION MODELS
ON COMPOUND POISSON POPULATION MODELS Martin Möhle, University of Tübingen (joint work with Thierry Huillet, Université de Cergy-Pontoise) Workshop on Probability, Population Genetics and Evolution Centre
More informationA Generalization of Wigner s Law
A Generalization of Wigner s Law Inna Zakharevich June 2, 2005 Abstract We present a generalization of Wigner s semicircle law: we consider a sequence of probability distributions (p, p 2,... ), with mean
More informationUpper Bounds for Partitions into k-th Powers Elementary Methods
Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 9, 433-438 Upper Bounds for Partitions into -th Powers Elementary Methods Rafael Jaimczu División Matemática, Universidad Nacional de Luján Buenos Aires,
More informationDepartment of Mathematics
Department of Mathematics Ma 3/103 KC Border Introduction to Probability and Statistics Winter 2017 Lecture 8: Expectation in Action Relevant textboo passages: Pitman [6]: Chapters 3 and 5; Section 6.4
More informationλ(x + 1)f g (x) > θ 0
Stat 8111 Final Exam December 16 Eleven students took the exam, the scores were 92, 78, 4 in the 5 s, 1 in the 4 s, 1 in the 3 s and 3 in the 2 s. 1. i) Let X 1, X 2,..., X n be iid each Bernoulli(θ) where
More informationDecomposing oriented graphs into transitive tournaments
Decomposing oriented graphs into transitive tournaments Raphael Yuster Department of Mathematics University of Haifa Haifa 39105, Israel Abstract For an oriented graph G with n vertices, let f(g) denote
More informationNotes 9 : Infinitely divisible and stable laws
Notes 9 : Infinitely divisible and stable laws Math 733 - Fall 203 Lecturer: Sebastien Roch References: [Dur0, Section 3.7, 3.8], [Shi96, Section III.6]. Infinitely divisible distributions Recall: EX 9.
More informationAPPROXIMATING CONTINUOUS FUNCTIONS: WEIERSTRASS, BERNSTEIN, AND RUNGE
APPROXIMATING CONTINUOUS FUNCTIONS: WEIERSTRASS, BERNSTEIN, AND RUNGE WILLIE WAI-YEUNG WONG. Introduction This set of notes is meant to describe some aspects of polynomial approximations to continuous
More informationADVANCED PROBABILITY: SOLUTIONS TO SHEET 1
ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1 Last compiled: November 6, 213 1. Conditional expectation Exercise 1.1. To start with, note that P(X Y = P( c R : X > c, Y c or X c, Y > c = P( c Q : X > c, Y
More informationCycle Lengths of θ-biased Random Permutations
Cycle Lengths of θ-biased Random Permutations Tongjia Shi Nicholas Pippenger, Advisor Michael Orrison, Reader Department of Mathematics May, 204 Copyright 204 Tongjia Shi. The author grants Harvey Mudd
More informationEquidivisible consecutive integers
& Equidivisible consecutive integers Ivo Düntsch Department of Computer Science Brock University St Catherines, Ontario, L2S 3A1, Canada duentsch@cosc.brocku.ca Roger B. Eggleton Department of Mathematics
More informationFirst Order Convergence and Roots
Article First Order Convergence and Roots Christofides, Demetres and Kral, Daniel Available at http://clok.uclan.ac.uk/17855/ Christofides, Demetres and Kral, Daniel (2016) First Order Convergence and
More informationMOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES
J. Korean Math. Soc. 47 1, No., pp. 63 75 DOI 1.4134/JKMS.1.47..63 MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES Ke-Ang Fu Li-Hua Hu Abstract. Let X n ; n 1 be a strictly stationary
More informationWelsh s problem on the number of bases of matroids
Welsh s problem on the number of bases of matroids Edward S. T. Fan 1 and Tony W. H. Wong 2 1 Department of Mathematics, California Institute of Technology 2 Department of Mathematics, Kutztown University
More informationMath 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative
Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance
More information3 Integration and Expectation
3 Integration and Expectation 3.1 Construction of the Lebesgue Integral Let (, F, µ) be a measure space (not necessarily a probability space). Our objective will be to define the Lebesgue integral R fdµ
More informationarxiv: v3 [math.pr] 7 Aug 2016
arxiv:60602965v3 [mathpr] 7 Aug 206 A ATURAL PROBABILISTIC MODEL O THE ITEGERS AD ITS RELATIO TO DICKMA-TYPE DISTRIBUTIOS AD BUCHSTAB S FUCTIO ROSS G PISKY Abstract Let {p } = denote the set of prime numbers
More informationHOW OFTEN IS EULER S TOTIENT A PERFECT POWER? 1. Introduction
HOW OFTEN IS EULER S TOTIENT A PERFECT POWER? PAUL POLLACK Abstract. Fix an integer k 2. We investigate the number of n x for which ϕn) is a perfect kth power. If we assume plausible conjectures on the
More informationHints/Solutions for Homework 3
Hints/Solutions for Homework 3 MATH 865 Fall 25 Q Let g : and h : be bounded and non-decreasing functions Prove that, for any rv X, [Hint: consider an independent copy Y of X] ov(g(x), h(x)) Solution:
More informationLaplace s Equation. Chapter Mean Value Formulas
Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationA LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS
Applicable Analysis and Discrete Mathematics, 1 (27, 371 385. Available electronically at http://pefmath.etf.bg.ac.yu A LINEAR BINOMIAL RECURRENCE AND THE BELL NUMBERS AND POLYNOMIALS H. W. Gould, Jocelyn
More informationA prolific construction of strongly regular graphs with the n-e.c. property
A prolific construction of strongly regular graphs with the n-e.c. property Peter J. Cameron and Dudley Stark School of Mathematical Sciences Queen Mary, University of London London E1 4NS, U.K. Abstract
More informationThe number of Euler tours of random directed graphs
The number of Euler tours of random directed graphs Páidí Creed School of Mathematical Sciences Queen Mary, University of London United Kingdom P.Creed@qmul.ac.uk Mary Cryan School of Informatics University
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationAsymptotic Statistics-III. Changliang Zou
Asymptotic Statistics-III Changliang Zou The multivariate central limit theorem Theorem (Multivariate CLT for iid case) Let X i be iid random p-vectors with mean µ and and covariance matrix Σ. Then n (
More informationWeighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes
More informationLecture 5: Expectation
Lecture 5: Expectation 1. Expectations for random variables 1.1 Expectations for simple random variables 1.2 Expectations for bounded random variables 1.3 Expectations for general random variables 1.4
More informationLogarithmic scaling of planar random walk s local times
Logarithmic scaling of planar random walk s local times Péter Nándori * and Zeyu Shen ** * Department of Mathematics, University of Maryland ** Courant Institute, New York University October 9, 2015 Abstract
More informationOn bounded redundancy of universal codes
On bounded redundancy of universal codes Łukasz Dębowski Institute of omputer Science, Polish Academy of Sciences ul. Jana Kazimierza 5, 01-248 Warszawa, Poland Abstract onsider stationary ergodic measures
More informationExchangeable Bernoulli Random Variables And Bayes Postulate
Exchangeable Bernoulli Random Variables And Bayes Postulate Moulinath Banerjee and Thomas Richardson September 30, 20 Abstract We discuss the implications of Bayes postulate in the setting of exchangeable
More informationEcon 508B: Lecture 5
Econ 508B: Lecture 5 Expectation, MGF and CGF Hongyi Liu Washington University in St. Louis July 31, 2017 Hongyi Liu (Washington University in St. Louis) Math Camp 2017 Stats July 31, 2017 1 / 23 Outline
More informationMath 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative
Math 259: Introduction to Analytic Number Theory Functions of finite order: product formula and logarithmic derivative This chapter is another review of standard material in complex analysis. See for instance
More information1 Basic Combinatorics
1 Basic Combinatorics 1.1 Sets and sequences Sets. A set is an unordered collection of distinct objects. The objects are called elements of the set. We use braces to denote a set, for example, the set
More informationn! (k 1)!(n k)! = F (X) U(0, 1). (x, y) = n(n 1) ( F (y) F (x) ) n 2
Order statistics Ex. 4. (*. Let independent variables X,..., X n have U(0, distribution. Show that for every x (0,, we have P ( X ( < x and P ( X (n > x as n. Ex. 4.2 (**. By using induction or otherwise,
More informationProbability and Measure
Chapter 4 Probability and Measure 4.1 Introduction In this chapter we will examine probability theory from the measure theoretic perspective. The realisation that measure theory is the foundation of probability
More informationProbabilistic number theory and random permutations: Functional limit theory
The Riemann Zeta Function and Related Themes 2006, pp. 9 27 Probabilistic number theory and random permutations: Functional limit theory Gutti Jogesh Babu Dedicated to Professor K. Ramachandra on his 70th
More informationEstimates for probabilities of independent events and infinite series
Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences
More informationRandom Bernstein-Markov factors
Random Bernstein-Markov factors Igor Pritsker and Koushik Ramachandran October 20, 208 Abstract For a polynomial P n of degree n, Bernstein s inequality states that P n n P n for all L p norms on the unit
More informationTheory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk
Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments
More informationSelected Exercises on Expectations and Some Probability Inequalities
Selected Exercises on Expectations and Some Probability Inequalities # If E(X 2 ) = and E X a > 0, then P( X λa) ( λ) 2 a 2 for 0 < λ
More informationLectures 2 3 : Wigner s semicircle law
Fall 009 MATH 833 Random Matrices B. Való Lectures 3 : Wigner s semicircle law Notes prepared by: M. Koyama As we set up last wee, let M n = [X ij ] n i,j=1 be a symmetric n n matrix with Random entries
More informationNonparametric hypothesis tests and permutation tests
Nonparametric hypothesis tests and permutation tests 1.7 & 2.3. Probability Generating Functions 3.8.3. Wilcoxon Signed Rank Test 3.8.2. Mann-Whitney Test Prof. Tesler Math 283 Fall 2018 Prof. Tesler Wilcoxon
More information0.1 Uniform integrability
Copyright c 2009 by Karl Sigman 0.1 Uniform integrability Given a sequence of rvs {X n } for which it is known apriori that X n X, n, wp1. for some r.v. X, it is of great importance in many applications
More informationSOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES
Communications on Stochastic Analysis Vol. 4, No. 3 010) 45-431 Serials Publications www.serialspublications.com SOLUTIONS OF SEMILINEAR WAVE EQUATION VIA STOCHASTIC CASCADES YURI BAKHTIN* AND CARL MUELLER
More informationAsymptotic Behavior of a Controlled Branching Process with Continuous State Space
Asymptotic Behavior of a Controlled Branching Process with Continuous State Space I. Rahimov Department of Mathematical Sciences, KFUPM, Dhahran, Saudi Arabia ABSTRACT In the paper a modification of the
More informationMath 564 Homework 1. Solutions.
Math 564 Homework 1. Solutions. Problem 1. Prove Proposition 0.2.2. A guide to this problem: start with the open set S = (a, b), for example. First assume that a >, and show that the number a has the properties
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More informationMath 141: Lecture 19
Math 141: Lecture 19 Convergence of infinite series Bob Hough November 16, 2016 Bob Hough Math 141: Lecture 19 November 16, 2016 1 / 44 Series of positive terms Recall that, given a sequence {a n } n=1,
More informationWAITING FOR A BAT TO FLY BY (IN POLYNOMIAL TIME)
WAITING FOR A BAT TO FLY BY (IN POLYNOMIAL TIME ITAI BENJAMINI, GADY KOZMA, LÁSZLÓ LOVÁSZ, DAN ROMIK, AND GÁBOR TARDOS Abstract. We observe returns of a simple random wal on a finite graph to a fixed node,
More information4 Expectation & the Lebesgue Theorems
STA 205: Probability & Measure Theory Robert L. Wolpert 4 Expectation & the Lebesgue Theorems Let X and {X n : n N} be random variables on a probability space (Ω,F,P). If X n (ω) X(ω) for each ω Ω, does
More informationCONVERGENCE OF RANDOM SERIES AND MARTINGALES
CONVERGENCE OF RANDOM SERIES AND MARTINGALES WESLEY LEE Abstract. This paper is an introduction to probability from a measuretheoretic standpoint. After covering probability spaces, it delves into the
More informationCONSECUTIVE INTEGERS IN HIGH-MULTIPLICITY SUMSETS
CONSECUTIVE INTEGERS IN HIGH-MULTIPLICITY SUMSETS VSEVOLOD F. LEV Abstract. Sharpening (a particular case of) a result of Szemerédi and Vu [4] and extending earlier results of Sárközy [3] and ourselves
More informationDepartment of Applied Mathematics Faculty of EEMCS. University of Twente. Memorandum No Birth-death processes with killing
Department of Applied Mathematics Faculty of EEMCS t University of Twente The Netherlands P.O. Box 27 75 AE Enschede The Netherlands Phone: +3-53-48934 Fax: +3-53-48934 Email: memo@math.utwente.nl www.math.utwente.nl/publications
More informationBALANCING GAUSSIAN VECTORS. 1. Introduction
BALANCING GAUSSIAN VECTORS KEVIN P. COSTELLO Abstract. Let x 1,... x n be independent normally distributed vectors on R d. We determine the distribution function of the minimum norm of the 2 n vectors
More informationAsymptotic behavior for sums of non-identically distributed random variables
Appl. Math. J. Chinese Univ. 2019, 34(1: 45-54 Asymptotic behavior for sums of non-identically distributed random variables YU Chang-jun 1 CHENG Dong-ya 2,3 Abstract. For any given positive integer m,
More informationRiemann integral and volume are generalized to unbounded functions and sets. is an admissible set, and its volume is a Riemann integral, 1l E,
Tel Aviv University, 26 Analysis-III 9 9 Improper integral 9a Introduction....................... 9 9b Positive integrands................... 9c Special functions gamma and beta......... 4 9d Change of
More informationAdditive functionals of infinite-variance moving averages. Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535
Additive functionals of infinite-variance moving averages Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535 Departments of Statistics The University of Chicago Chicago, Illinois 60637 June
More informationMath Bootcamp 2012 Miscellaneous
Math Bootcamp 202 Miscellaneous Factorial, combination and permutation The factorial of a positive integer n denoted by n!, is the product of all positive integers less than or equal to n. Define 0! =.
More informationUPPER DEVIATIONS FOR SPLIT TIMES OF BRANCHING PROCESSES
Applied Probability Trust 7 May 22 UPPER DEVIATIONS FOR SPLIT TIMES OF BRANCHING PROCESSES HAMED AMINI, AND MARC LELARGE, ENS-INRIA Abstract Upper deviation results are obtained for the split time of a
More informationChapter 7. Basic Probability Theory
Chapter 7. Basic Probability Theory I-Liang Chern October 20, 2016 1 / 49 What s kind of matrices satisfying RIP Random matrices with iid Gaussian entries iid Bernoulli entries (+/ 1) iid subgaussian entries
More informationCONNECTIONS BETWEEN BERNOULLI STRINGS AND RANDOM PERMUTATIONS
CONNECTIONS BETWEEN BERNOULLI STRINGS AND RANDOM PERMUTATIONS JAYARAM SETHURAMAN, AND SUNDER SETHURAMAN Abstract. A sequence of random variables, each taing only two values 0 or 1, is called a Bernoulli
More informationOn the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method
On the Entropy of Sums of Bernoulli Random Variables via the Chen-Stein Method Igal Sason Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, Israel ETH, Zurich,
More information6.1 Moment Generating and Characteristic Functions
Chapter 6 Limit Theorems The power statistics can mostly be seen when there is a large collection of data points and we are interested in understanding the macro state of the system, e.g., the average,
More informationQualifying Exam in Probability and Statistics. https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf
Part 1: Sample Problems for the Elementary Section of Qualifying Exam in Probability and Statistics https://www.soa.org/files/edu/edu-exam-p-sample-quest.pdf Part 2: Sample Problems for the Advanced Section
More informationThis document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.
This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Composition operators on Hilbert spaces of entire functions Author(s) Doan, Minh Luan; Khoi, Le Hai Citation
More informationCOMPOSITIONS WITH A FIXED NUMBER OF INVERSIONS
COMPOSITIONS WITH A FIXED NUMBER OF INVERSIONS A. KNOPFMACHER, M. E. MAYS, AND S. WAGNER Abstract. A composition of the positive integer n is a representation of n as an ordered sum of positive integers
More informationSTOCHASTIC GEOMETRY BIOIMAGING
CENTRE FOR STOCHASTIC GEOMETRY AND ADVANCED BIOIMAGING 2018 www.csgb.dk RESEARCH REPORT Anders Rønn-Nielsen and Eva B. Vedel Jensen Central limit theorem for mean and variogram estimators in Lévy based
More informationSOME INEQUALITIES FOR THE q-digamma FUNCTION
Volume 10 (009), Issue 1, Article 1, 8 pp SOME INEQUALITIES FOR THE -DIGAMMA FUNCTION TOUFIK MANSOUR AND ARMEND SH SHABANI DEPARTMENT OF MATHEMATICS UNIVERSITY OF HAIFA 31905 HAIFA, ISRAEL toufik@mathhaifaacil
More informationTHE LINDEBERG-FELLER CENTRAL LIMIT THEOREM VIA ZERO BIAS TRANSFORMATION
THE LINDEBERG-FELLER CENTRAL LIMIT THEOREM VIA ZERO BIAS TRANSFORMATION JAINUL VAGHASIA Contents. Introduction. Notations 3. Background in Probability Theory 3.. Expectation and Variance 3.. Convergence
More informationWeak convergence in Probability Theory A summer excursion! Day 3
BCAM June 2013 1 Weak convergence in Probability Theory A summer excursion! Day 3 Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu BCAM June 2013 2 Day 1: Basic
More informationRINGS ISOMORPHIC TO THEIR NONTRIVIAL SUBRINGS
RINGS ISOMORPHIC TO THEIR NONTRIVIAL SUBRINGS JACOB LOJEWSKI AND GREG OMAN Abstract. Let G be a nontrivial group, and assume that G = H for every nontrivial subgroup H of G. It is a simple matter to prove
More informationErgodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.
Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions
More informationON ALLEE EFFECTS IN STRUCTURED POPULATIONS
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 ON ALLEE EFFECTS IN STRUCTURED POPULATIONS SEBASTIAN J. SCHREIBER Abstract. Maps f(x) = A(x)x of
More informationHow many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected?
How many randomly colored edges make a randomly colored dense graph rainbow hamiltonian or rainbow connected? Michael Anastos and Alan Frieze February 1, 2018 Abstract In this paper we study the randomly
More informationIntroduction to Probability
LECTURE NOTES Course 6.041-6.431 M.I.T. FALL 2000 Introduction to Probability Dimitri P. Bertsekas and John N. Tsitsiklis Professors of Electrical Engineering and Computer Science Massachusetts Institute
More informationLecture 17: The Exponential and Some Related Distributions
Lecture 7: The Exponential and Some Related Distributions. Definition Definition: A continuous random variable X is said to have the exponential distribution with parameter if the density of X is e x if
More informationContinued Fraction Digit Averages and Maclaurin s Inequalities
Continued Fraction Digit Averages and Maclaurin s Inequalities Steven J. Miller, Williams College sjm1@williams.edu, Steven.Miller.MC.96@aya.yale.edu Joint with Francesco Cellarosi, Doug Hensley and Jake
More informationCOMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL. Ross G. Pinsky
COMPARISON THEOREMS FOR THE SPECTRAL GAP OF DIFFUSIONS PROCESSES AND SCHRÖDINGER OPERATORS ON AN INTERVAL Ross G. Pinsky Department of Mathematics Technion-Israel Institute of Technology Haifa, 32000 Israel
More informationJónsson posets and unary Jónsson algebras
Jónsson posets and unary Jónsson algebras Keith A. Kearnes and Greg Oman Abstract. We show that if P is an infinite poset whose proper order ideals have cardinality strictly less than P, and κ is a cardinal
More informationSample Spaces, Random Variables
Sample Spaces, Random Variables Moulinath Banerjee University of Michigan August 3, 22 Probabilities In talking about probabilities, the fundamental object is Ω, the sample space. (elements) in Ω are denoted
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationList Decomposition of Graphs
List Decomposition of Graphs Yair Caro Raphael Yuster Abstract A family of graphs possesses the common gcd property if the greatest common divisor of the degree sequence of each graph in the family is
More informationPROBABILITY AND STATISTICS IN COMPUTING. III. Discrete Random Variables Expectation and Deviations From: [5][7][6] German Hernandez
Conditional PROBABILITY AND STATISTICS IN COMPUTING III. Discrete Random Variables and Deviations From: [5][7][6] Page of 46 German Hernandez Conditional. Random variables.. Measurable function Let (Ω,
More informationUC Berkeley Department of Electrical Engineering and Computer Sciences. EECS 126: Probability and Random Processes
UC Berkeley Department of Electrical Engineering and Computer Sciences EECS 6: Probability and Random Processes Problem Set 3 Spring 9 Self-Graded Scores Due: February 8, 9 Submit your self-graded scores
More informationA NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES
Sankhyā : The Indian Journal of Statistics 1998, Volume 6, Series A, Pt. 2, pp. 171-175 A NOTE ON A DISTRIBUTION OF WEIGHTED SUMS OF I.I.D. RAYLEIGH RANDOM VARIABLES By P. HITCZENKO North Carolina State
More informationON THE EQUIVALENCE OF CONGLOMERABILITY AND DISINTEGRABILITY FOR UNBOUNDED RANDOM VARIABLES
Submitted to the Annals of Probability ON THE EQUIVALENCE OF CONGLOMERABILITY AND DISINTEGRABILITY FOR UNBOUNDED RANDOM VARIABLES By Mark J. Schervish, Teddy Seidenfeld, and Joseph B. Kadane, Carnegie
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca October 22nd, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationarxiv: v1 [math.pr] 11 Dec 2017
Local limits of spatial Gibbs random graphs Eric Ossami Endo Department of Applied Mathematics eric@ime.usp.br Institute of Mathematics and Statistics - IME USP - University of São Paulo Johann Bernoulli
More informationThe Moran Process as a Markov Chain on Leaf-labeled Trees
The Moran Process as a Markov Chain on Leaf-labeled Trees David J. Aldous University of California Department of Statistics 367 Evans Hall # 3860 Berkeley CA 94720-3860 aldous@stat.berkeley.edu http://www.stat.berkeley.edu/users/aldous
More informationTHE EXPONENTIAL DISTRIBUTION ANALOG OF THE GRUBBS WEAVER METHOD
THE EXPONENTIAL DISTRIBUTION ANALOG OF THE GRUBBS WEAVER METHOD ANDREW V. SILLS AND CHARLES W. CHAMP Abstract. In Grubbs and Weaver (1947), the authors suggest a minimumvariance unbiased estimator for
More informationTaylor and Laurent Series
Chapter 4 Taylor and Laurent Series 4.. Taylor Series 4... Taylor Series for Holomorphic Functions. In Real Analysis, the Taylor series of a given function f : R R is given by: f (x + f (x (x x + f (x
More informationPermutations with Inversions
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 4 2001, Article 01.2.4 Permutations with Inversions Barbara H. Margolius Cleveland State University Cleveland, Ohio 44115 Email address: b.margolius@csuohio.edu
More information