ON THE STRANGE DOMAIN OF ATTRACTION TO GENERALIZED DICKMAN DISTRIBUTIONS FOR SUMS OF INDEPENDENT RANDOM VARIABLES

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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 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 = c, for some c [, 1] 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 ROSS G PINSKY 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 at the end of this section 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 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 The ribution decays very rapidly; indeed, it is not hard to show that p 1 x) In fact, for all θ >, expθ 1 probability ribution e γ Γx+1), x [6] 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 These ribution decays very rapidly; indeed, it is not hard to show that

3 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 3 p θ x) C θ Γx+1), x 1, for an appropriate constant C θ A fundamental fact about these ributions is that 12) 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 From 12) 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] It will follow from the proof of Theorem 1 below that expθ 1 e λx 1 x dx) is the Laplace transform of a probability ribution In section 5 we will prove that a random variable with such a ribution satisfies 12), and that if a random variable satisfies 12), then it has a density of the form c θ ρ θ, where ρ θ satisfies 11) Thus, this paper is self-contained with regard to all the above noted facts, with the exception of the rate of decay and the value e θγ Γθ) of the normalizing constant c θ in p θ For more on these ributions, including a derivation of the normalizing constant, see, for example, [1] and [8] 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 Mimicing the proof of 12) that we give in section 5 shows that 13) D X ) θ = U 1 θ D X ) θ + X ), where U is ributed according to the uniform ribution on [, 1], and U, D X ) θ and X on the right hand side above are independent From 13)

4 4 ROSS G PINSKY it is immediate that D θ = X 1 U 1 θ 1 + X 2 U 1 U 2 ) 1 θ + X3 U 1 U 2 U 3 ) 1 θ +, where {U n } n=1 and {X n} n=1 are mutually independent sequences of IID random variables, with U 1 ributed according to the uniform ribution on [, 1] and X 1 ributed according to the ribution of X 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 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 In light of the above discussion, we will consider the following setting Let {B }, {X } random variables be mutually independent sequences of independent Assume that {B } are Bernoulli random variables

5 satisfying: STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 5 14) P B = 1) = 1 P B = ) = p [, 1), and assume that {X } satisfy: 15) X >, EX < Let 16) = and define 17) W n = 1 p, B X We will be interested in the limiting behavior of W n trivialities, we will assume that 18) 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 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

6 6 ROSS G PINSKY Theorem 1 Let W n be as in 17), where {B }, {X } and are as in 14)-16) and 18) i Assume that { X } is uniformly integrable which occurs automatically X if lim = 1) a Assume also that max 1 n 19) lim = n Then lim W n = 1 n b Assume also that there exists a sequence {K n } n=1 11) lim and n =K n+1 p =, such that 111) lim n If max 1 Kn = M Kn 112) c lim n exists, then lim W n = c n If 112) 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 113) lim = X Assume also that { } is increasing, that lim p = and that there exist θ, L, ) such that 114) lim p +1 = θ, lim M = L

7 Then STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 7 lim W n = LD X ) θ), n where D X ) θ) is a random variable with the GD X ) θ) ribution Remar 1 In 114), necessarily L 1 θ Indeed, if {p } and {θ } 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 Cµ 2, 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 17), where {B }, {X } and are as in 14)-16) Assume also that lim X = 1 Let J µ, J p be nonnegative integers, let c µ, c p > and define 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

8 8 ROSS G PINSKY with b Jp Assume that 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 18) 115) ii b j = 1, j J p ; iii a j =, j J p 1, and a Jp >, then 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 116) 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 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

9 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 9 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 18) 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 18) 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 [7], where the GD1) ribution arises, one has b = b 1 = 1, a =, a 1 = 1, c p = c µ = 1 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 Finally, in section 5 we prove the basic facts about the Dicman ribution and its density, as was promised earlier in this section

10 1 ROSS G PINSKY As mentioned above, we end this section with a little bacground concerning the Dicman function ρ ρ 1 The Dicman function arises in probabilistic number theory in the context of so-called smooth numbers; that is, numbers all of whose prime divisors are small 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] = {1,, 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)

11 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 11 ribution arises as a normalized limit in the context of certain natural probability measures that one can place on N; see [3], [7] 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 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

12 12 ROSS G PINSKY 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) 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 18) holds Recalling that = when E =, and that 1 otherwise, note that = EI n = E E = p

13 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 13 Thus, in the present context the requirement 18) is E 21) =, 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 { µ b lim n n n Then In EI n 1 = µ 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 j +1 j 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 Proof Assume first that the condition in part i-a) holds We claim that since { µ 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 { µ n n =γn+1 } n=1 is also bounded Indeed, assume to the contrary Then, in particular, {µ n } n=1 is unbounded Also, since <, we have n < γ n + n =γ n+1, and it would follow that { µn γ n } n=1 is bounded for all sequences {γ n} n=1 satisfying lim n γ n =, which is a contradiction Let {γ n } n=1 be such a sequence Then = E min >γ n E ) =γ n

14 14 ROSS G PINSKY Thus, the condition in i-a) guarantees that 19) holds Now assume that the condition in part i-b) holds Since n, it follows again that 19) 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 conclude from part ii) of Theorem 1 that N a I n EI n 3 Proof of Theorem 1 1 θ GDX ) θ), where θ = 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 112) 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 ) ))

15 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 15 Since n =K n+1 1 p ) n =K n+1 1 p 1 E exp λ X ) )) 1, it follows from assumption 11) 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 In light of 111) and the assumption that { X } is uniformly integrable, it follows that for all ɛ >, there exists 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 111) 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 37) 1 + ɛ)λ 1 ɛ)λ Kn = ɛ p log K n Kn = ɛ p, n n ɛ 1 p 1 E exp λ X ) ))

16 16 ROSS G PINSKY If M Kn 38) c lim n 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 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 114) 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) exp λ X )dp P A c ;δ M ), n A c ;δ

17 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 17 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) 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 ɛ as n Let n) j n = max ɛ = +1, is a partition of µ = and lim n n = L We n 1 n), of the partition converges to ɛ n 1 n), where ɛ j n n Without loss of generality, assume either that {j n } is bounded or that lim n j n = In the former case it is clear that max ɛ n 1 n) = n) j n = µ jn+1 µ jn n

18 18 ROSS G PINSKY Now consider the latter case From assumption 114) 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 114) we have lim p +1 = θ In light of these facts, along with 315), 316) and the fact that ɛ > is arbitrary, 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 n 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 115) 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 x log j) ) 1 ; j=1 +1 c µ a JP log J P ) ) a Jp 1 = Consequently, 41) lim x 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

19 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 19 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 115) does not hold We need to show that {K n } n=1 can be defined so that 11) and 111) hold, and so that 112) holds with c {, 1} We also have to show when c = and when c = 1 Recall the definitions in 116) If { j J µ : a j } is empty, or if it is not empty and a κµ <, then { } is bounded Therefore, 11) and 111) hold with 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 from 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 115) is fulfilled and we have J p 44) p c p x log j) ) 1 j=1 Since we are assuming that 115) does not hold, at least one of the other two conditions in 115) 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 > )

20 2 ROSS G PINSKY 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 11) and 111) hold by choosing K n = n Thus, from part i-a) of Theorem 1, lim n W n = 1 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 11) and 111) hold, and so that 112) 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 ) 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;

21 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 21 thus, 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 We now consider the case that { j J p : b j 1} is not empty Then in order to fulfill the second condition in 18), we have b κp < 1 We write κ p 1 411) p c p x j=1 log j) ) 1 log κ p) ) b κp J p j=κ p+1 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 b κp +1 L κp n), κ µ = κ p ; log κp) n) 1 bκp L κp n), κ µ > κ p, log j) ) b j 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 From 43) and 412) we have 414) µ Kn log κµ) K n log κµ) n log j) n ) b j log κp) K n ) 1 bκp ) 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 J p j=κ p+1 It is immediate 43) and 414) that if κ µ κ p, then 11) and 111) 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 log j) K n ) bj ]

22 22 ROSS G PINSKY 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 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 M Kn log κµ) K n ) aκµ log κµ) ; n 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) 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, where n K n, n) 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 11) and 111) hold, and that 112) 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 in fact the Laplace transform of a probability ribution, which we have denoted by GDθ) In particular, if we let X = = and p = θ, in which case = n p = θn, then it follows from Theorem 1 that 51) Ŵ n θw n = 1 n where D θ GDθ) We now prove 12) Let B D θ, J + n = max{ n : B },

23 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 23 with max We write 52) 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 53) P J + n n x) = 1 We have n =[xn+1] Also, by the independence of {B }, we have 1 θ ) xθ, x, 1) 54) J 1 n + 1 J n + B {J n + 1 = } = B = Ŵ 1, 2 Letting n in 52) and using 51), 53) and 54), we conclude that 12) holds, where U is a ributed according to the uniform ribution on [, 1], D θ GDθ) and U and D θ on the right hand side are independent We now use 12) to show that 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) Let F θ x) = P D θ ribution Then from 12) we have 55) x) denote the ribution function for the GDθ) F θ x) = P D θ x) = P U 1 θ Dθ + 1) x) = 1 F θ xy 1 θ 1)dy 1 For x >, maing the change of variables, v = xy 1 θ 55) as 56) 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 56) 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

24 24 ROSS G PINSKY Consequently, it follows from 56) 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 56) it follows that F is differentiable on, 1) and on 1, ), and that, letting p θ = F θ, 57) p θ = c θ x θ 1, < x < 1, c θ = θ 2 F Dθ v)1 + v) 1 θ dv, and 58) 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 58), 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, 59) xp θ x) + 1 θ)p θx) + θp θ x 1) =, x > 1 From 57) and 59) 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 57) shows that c θ = θ 1 + v) θ p θ v)dv = θe1 + D θ ) θ 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)

25 STRANGE DOMAIN OF ATTRACTION TO DICKMAN DISTRIBUTIONS 25 [7] Pinsy R, A Natural Probabilistic Model on the Integers and its Relation to Dicman- Type Distributions and Buchstab s Function, preprint, 216) [8] Penrose, M and Wade, A, Random minimal directed spanning trees and Dicmantype ributions, Adv in Appl Probab 36 24), [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: