Random Bernstein-Markov factors

Size: px
Start display at page:

Download "Random Bernstein-Markov factors"

Transcription

1 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 circle, 0 < p, with equality for P n (z = cz n. We study this inequality for random polynomials, and show that the expected (average and almost sure value of P n / P n is often different from the classical deterministic upper bound n. In particular, for circles of radii less than one, the ratio P n / P n is almost surely bounded as n tends to infinity, and its expected value is uniformly bounded for all degrees under mild assumptions on the random coefficients. For norms on the unit circle, Borwein and Lockhart mentioned that the asymptotic value of P n / P n in probability is n/ 3, and we strengthen this to almost sure it for p = 2. If the radius R of the circle is larger than one, then the asymptotic value of P n / P n in probability is n/r, matching the sharp upper bound for the deterministic case. We also obtain bounds for the case p = on the unit circle. Introduction Let D C denote the unit disc centered at the origin. Let P n (z = n a kz k be a polynomial of degree n. The classical Bernstein s inequality for the derivative of a polynomial states that P n n P n, (. where f denotes the sup norm of f over D. Polynomials P n (z = cz n are extremal for the inequality (.. It is also well known that inequality (. holds when is replaced by L p norms p taken over D (with respect to dθ/(2π, for any p > 0. More information about Bernstein s inequality and its history can be found in [8]. Bernstein s inequality is a direct analogue of Markov s inequality for the sup norm of the derivative of a polynomial on the interval [, ], see [8]. Since Markov s inequality is apparently the first sharp estimate for the derivative of a polynomial, the ratio of norms M n (P n := P n P n 200 Mathematics Subject Classification: Primary 4A7 Secondary 30E0, 60F05

2 is often referred to as the Markov factor of P n, while in our case the Bernstein-Markov factor is a more appropriate term. If P n (z 0 then we set M n (P n := 0, which holds for any constant polynomial, and this convention is carried through to all other norms used in our paper. In this article, we consider random polynomials of the form P n (z = A k z k, where the {A k } i.i.d. complex random variables. Throughout this article we will assume that our random variables are non trivial, meaning that P(A k = 0 <. Our goal is to study the asymptotics of the average Bernstein-Markov factors, namely P E[M n (P n ] = E n P n for various norms, as well as probabilistic its of these factors. Define the L p norm on the unit circle D by ( 2π /p P n p = P n (e it dt p, p > 0. 2π 0 The Bernstein-Markov factors of random polynomials for the L p norms on the unit circle were considered by Borwein and Lockhart, who stated an asymptotic for the case of random Littlewood polynomials. But Theorems and 2 in [] give the following more general result. Theorem. Let 0 < p <. Consider random polynomials P n (z = n A kz k where the A k are i.i.d. real valued random variables with mean 0, variance and additionally if p > 2, E ( A 0 2p <. Then P n p P as n, n P n p 3 where the convergence holds in probability. As a consequence, the average Bernstein-Markov factors satisfy P n E n p =. P n p 3 We first show that when p = 2, the convergence of Bernstein-Markov factors holds in the stronger almost sure sense. Theorem 2. If {A k } are i.i.d. complex valued random variables with E[ A 0 2 ] <, then P n 2 = n P n 2 3 a.s. 2

3 Hence the average Bernstein-Markov factors satisfy P n E n 2 =. P n 2 3 We next consider the average Bernstein-Markov factors for the L p norm on D r = {z : z = r}, for r > 0. This norm is defined by P n p,r = ( 2π /p P n (re it dt p, p > 0. 2π 0 Using a simple change of variable z rz, we obtain from the Bernstein inequality on the unit circle that P n p,r n r P n p,r, 0 < p, where equality holds for polynomials of the form P n (z = cz n. However, we show that in the randomized model of this problem, the Bernstein-Markov factors remain bounded as n with probability one, for all r (0,. At a heuristic level, this is because for z such that z = r, we can view P n (z as a partial sum of a random power series F (z. Hence one expects the it of M n to be the corresponding ratio of norms for F and F. Theorem 3. Suppose that 0 < p and r (0,. If {A k } are i.i.d. complex random variables with E[log + A 0 ] <, then there exists a positive random variable X p,r such that 0 < P n p,r P n p,r = X p,r < a.s. If E[ A 0 ] <, and for some ε > 0 the distribution of A 0 is absolutely continuous on [0, ε], with the density bounded on [0, ε], then P n p,r sup E <. n N P n p,r The second statement above admits a stronger result when p = 2. This we collect as a proposition. Proposition 4. Let r (0,. Let {A k } be i.i.d. complex valued random variables with E( A 0 2 <. Then there exists a positive random variable X 2,r such that 0 < P n 2,r P n 2,r = X 2,r < a.s. Furthermore, E [ P n 2,r P n 2,r ] = E[X 2,r ] <. 3

4 While the Bernstein-Markov factors are typically bounded on circles of radii less than one, we now show that they are asymptotic to n/r (have maximal growth for the L 2 -norms on circles of radii R >. Theorem 5. Assume that {A k } are i.i.d. complex valued random variables with E[ A 0 2 ] <. If for some ɛ > 0, the distribution of A 0 2 is absolutely continuous on [0, ε], and its density is bounded on [0, ε], then for any R > we have P n 2,R P n P n 2,R R as n. As a consequence, we obtain that P n E n 2,R = P n 2,R R, R >. We return to the supremum norm on the unit circle in the last two results. Results on the asymptotic behaviour of the norm of random Littlewood polynomials are proved in [5], see also [6]. Theorem 6. If {A k } are i.i.d. complex valued random variables with common distribution invariant under complex conjugation, then the expected Bernstein-Markov factors in the sup norm on the unit circle satisfy P E n n P n 2. It is clear that the result applies to all real i.i.d. random coefficients, as well as many standard complex ones. Before we state our next theorem, we recall that a random variable X is said to have the standard complex normal distribution, denoted by N C (0,, if it has density π exp( z 2 in the complex plane C. We note for future reference here that if X N C (0,, then X 2 E(, the exponential distribution with parameter. Theorem 7. If {A k } are i.i.d. complex valued random variables with N C (0, distribution, then the expected Bernstein-Markov factors in the sup norm on the unit circle satisfy P inf 2 n E n P sup P n n E n 2 P n 3. Remarks: The L 2 norm of polynomials on the unit circle has an explicit formula in terms of the coeffficients of the polynomial and this helps to derive almost sure convergence of the Bernstein-Markov factors as in Theorem 2. For general p (0,, p 2, one does not have such a formula. However by proving convergence in distribution of certain related 4

5 random variables, Borwein and Lockhart [] derive their result. We believe that almost sure convergence holds for all p > 0 and it would be interesting to see a proof of such a result. In Theorem 5, we once again expect almost sure convergence although we have not been able to prove it so far. Finally for Theorem 7, we believe that the iting value of the scaled average Bernstein-Markov factors exist, and that this it is / 3. 2 Proofs Proof of Theorem. Under the conditions stated, Theorems and 2 in [] give P n p n P Γ( + p/2 p and P n p n 3 P Γ( + p/2 p 3 where in both cases convegence holds in probability. Combining the two displays gives the first result. Taking expectation and applying the dominated convergence theorem yields the result about convergence of the expected Bernstein-Markov factors. Proof of Theorem 2. It is an easy computation to see that if P n (z = n A kz k P n 2 = ( n A k 2 /2 and P n 2 = ( n k= k2 A k 2 /2. Therefore we obtain that then ( P n n 2 k= = A k 2 /2 P n n 2 A. k 2 (2. By scaling, we can assume without loss of generality that E[ A k 2 ] =. With this assumption, since A k 2 are i.i.d., the Law of Large Numbers gives n + A k 2 a.s. (2.2 To deal with the numerator in (2. we use the following lemma which is a strengthening of the Law of Large Numbers, cf. Section 5.2, Theorem 3 in [2]. Lemma 8. Let {X n } be i.i.d. random variables with finite mean. If {a n } are positive constants with S n = n k= a k, such that b n = S n /a n and b 2 n b 2 k=n k = O(n, then S n a k X k E[X ] a.s. k= 5

6 Continuing with the proof of Theorem 2, we apply Lemma 8 with X k = A k 2, a k = k 2, and S n = n k= k2 to obtain Next, we write S n k 2 A k 2 a.s. (2.3 k= ( n k= k2 A k 2 n A k 2 /2 = Sn n + n k= k2 A k 2 S n n A k 2 n + /2. (2.4 Dividing equation (2.4 by n and then letting n, we obtain from (2.2 and (2.3 that P n 2 = a.s. n P n 2 3 A simple use of the Bounded Convergence Theorem yields P n E n 2 = P n 2 3 and finishes the proof of Theorem 2. Proof of Theorem 3. Our assumption E[log + A 0 ] < implies that sup A n /n a.s., see, e.g., [7]. Therefore, with probability one, both series A k z k and ka k z k converge uniformly on compact subsets of the unit disk to the random analytic functions F 0 and F 0 respectively. This means P n converge uniformly on z = r to F, and P n converge uniformly on z = r to F. As a consequence, we obtain that P n p,r = F p,r 0 and P n p,r = F p,r 0 Hence the first part of this theorem follows. For the second part, we need to show that the expected Bernstein-Markov factors are uniformly bounded for all n N. We start with deterministic estimates for the norms of P n and P n. It is immediate that P n p,r P n,r k A k r k, 0 < p. k= 6 k= a.s.

7 If p then we use the elementary estimate P n A k to give the lower bound P n p,r P n,r = A k r k z k A k r k, k = 0,,..., n. Hence we have for all P n that P n p,r r 2 ( A 0 + A, p. If p (0, then we use a theorem of Hardy and Littlewood (cf. Theorem 6.2 in [3, p. 95] to estimate P n p,r = A k r k z k c p r( A 0 p + A p /p c p r( A 0 + A, 0 < p <, p where c p > 0 depends only on p. It follows that for all p (0, ] P n n p,r k= k A ( k r k P n p,r r min(c p, /2( A 0 + A = A n r min(c p, /2 A 0 + A + k=2 k A k r k A 0 + A ( n k=2 + k A k r k. r min(c p, /2 A 0 + A Taking expectation, and using independence of the A k, we obtain that ( [ P ] E n p,r + E E k A k r k P n p,r r min(c p, /2 A 0 + A k=2 ( = + E[ A 0 ] E kr k. r min(c p, /2 A 0 + A Since n k=2 krk k=2 krk < for 0 < r <, it remains to show that E [( A 0 + A ] is finite under the additional assumption on the density f of A 0. Let Z = A 0 + A, and denote the density function of this random variable on [0, ε] by g. It is known that g is given by the convolution f f. We finish the proof with the following estimate: ε ( x P(Z > ε E [/Z] f(x tf(t dt dx + x ε 0 ε sup x [0,ε] x 0 x 0 f(x tf(t dt + /ε ε sup f 2 (x + /ε <. x [0,ε] k=2 7

8 Proof of Proposition 4. A direct computation shows that P n 2,r = ( n A k 2 r 2k /2, and similarly P n 2,r = ( n k= k2 A k 2 r 2k 2 /2. Recall that the Bernstein-Markov factor is Mn = P n 2,r / P n 2,r. It is straightforward to see by considering the differences M n+ M n, that M n increases with n and that the it M n = X 2,r exists with probability one, where k= X 2,r = k2 A k 2 r 2k 2 A. k 2 r 2k After applying the monotone convergence theorem, it remains to show that E[X 2,r ] <. For this, we first observe that since A 0 is a non-trivial random variable, there exists α > 0 and 0 < c < such that P ( A 0 2 < α = c. Applying the Cauchy-Schwarz inequality, we obtain that ( ( /2 ( /2 E(X 2,r E k 2 A k 2 r (E 2k 2 A. k 2 r 2k k= Since E( A k 2 <, the first term has finite expectation. Hence it is enough to show that ( E <, Z 2,r where Z 2,r = A k 2 r 2k. For simplicity of notation, we henceforth just write Z for this random variable. As a preparation we claim that P ( αr 2n+2 Z < αr 2n c n+, n 0. (2.5 Assume for the moment that the estimate (2.5 is true. Then if c < r <, we have ( ( E = E Z Z ; Z > α α + E n=0 n=0 ( + E Z ; 0 < Z < α ( Z ; αr2n+2 Z < αr 2n α + αr P ( αr 2n+2 Z < αr 2n 2n+2 α + α n=0 ( c r 2 n+ <, where the last line is true because of our choice of r. So E[X 2,r ] < for c < r <. Observe that for a fixed n, the quantity rp n 2,r P n 2,r is monotonically increasing with respect to r. This 8

9 property can be shown by simply computing the derivative with respect to r and checking that it is nonnegative. It follows that for s c < r < sx 2,s = sp n 2,s P n 2,s rp n 2,r P n 2,r = rx 2,r almost surely. The latter shows E[X 2,r ] < for all r (0,. It now remains to prove (2.5, which is easily done as follows: P ( ( αr 2n+2 Z < αr 2n P(Z < αr 2n P A k 2 r 2k < αr 2n n P ( A k 2 < αr 2n 2k c n+. n P ( A k 2 < α Proof of Theorem 5. As before, we have R P n 2,R P n 2,R = n k= k2 A k 2 R 2k n A k 2 R 2k. Denote a k = A k 2. After a scaling, we may assume without loss of generality that E[a k ] =. For j 0, let Y j = j a kr 2k. By the Abel summation formula, we obtain that k 2 a k R 2k = n 2 k= n a k R 2k a 0 (2j + Y j. j= Therefore R P n 2,R n P n 2,R = n k= k2 a k R 2k n 2 n a kr = a n 0 j= (2j + Y j. (2.6 2k n 2 Y n n 2 Y n a 0 Since a 0, we see that 0 a.s., and hence in probability. We now show that the Y n n 2 Y n random variables n j= U n = (2j + Y j n 2 Y n also converge to 0 in probability, which proves this theorem. Convergence in probability in turn follows from E[Un /2 ] 0 as n. Indeed, applying the Cauchy-Schwarz inequality we obtain 9

10 E[Un /2 ] ( n /2 ( /2 (2j + E[Y j ] E (2.7 n 2 Y n j= Since E[a k ] =, we have that E[Y j ] = j R2k. It is a simple matter to now see that n j= (2j + E[Y j] = O (nr 2n. To estimate the other term we use the fact that Y n R 2n 2 (a n + a n to obtain ( E n 2 Y n n 2 R E = O, (2.8 2n 2 a n + a n n 2 R 2n where we used the same idea as in the end of proof for Theorem 3 to conclude that E [/(a n + a n ] <. Combining the two estimates and using (2.7, we obtain that E[Un /2 ] 0 as n and finish the proof. Proof of Theorem 6. For any polynomial p n (z = n a kz k with complex coefficients, we associate its conjugate reciprocal polynomial q n (z := z n p n (/ z = a n k z k. Obviously, p n (z = q n (z holds for z =. It is also clear that the polynomial p n q n is self reciprocal, i.e. coincides with its conjugate reciprocal. For such polynomials, the Bernstein- Markov factor in the sup norm on the unit circle is known to be equal to the degree divided by 2, see [8, p. 527]. Hence we obtain that It follows that n p n q n = n p n q n = p nq n + p n q n p n q n + p n q n. p n p n + q n q n n. (2.9 Applying (2.9 with p n = P n being our random polynomial, and taking into account that the expected values of both additive terms on the left of (2.9 are identical under our assumptions on random coefficients, we arrive at the required result. Proof of Theorem 7. The inequality involving the inf of the Bernstein-Markov factors follows immediately from Theorem 6. Thus we need to prove the inequality involving the sup. For this we will give asymptotic bounds for P n and P n separately. We start with a bound for P n. 0

11 Proposition 9. inf P n n log(n a.s. Proof. We first observe that for z with z =, P n (z is complex Gaussian, distributed as N C (0, n+. This means that for such z, Pn(z 2 E(, where E( denotes the exponential n+ random variable with parameter. For 0 l n, let z l = exp( i2πl. Note that for j, l n n+ with j l we have ( E P n (z j P n (z l = (z j z l k = 0. (2.0 Hence P n (z j are uncorrelated for different values of j. But since they are Gaussian, this means that P n (z j, 0 j n, are independent. This in turn implies the independence of X n,j = Pn(z j 2 n+, 0 j n. Therefore P n 2 (n + log(n + max 0jn X n,j. (2. log(n + We need the following lemma, whose proof follows by a simple modification of the more well known result involving extremes of i.i.d. exponential random variables, see [4], page 07. Lemma 0. Let Y n,j, n N, j n, be a triangular array of random variables each of which is distributed as E(, and in which every row is jointly independent. Let T n = max jn Y n,j. Then T n log(n = a.s. Applying Lemma 0 with X n,j in place of Y n.j and making use of (2., we obtain inf This conculdes the proof of Proposition 9. P n 2 (n + log(n + a.s. Our next objective is to bound P n which we do by means of Proposition. sup Proof. Note that for each z with z =, claim that for every c > 2 P n= P n n3 log(n 2 3 a.s. P n(z 2 n(n + (2n + /6 ( P n 2 + (2n + > cn(n log(n <. 6 is distributed as E(. We

12 By the Borel-Cantelli lemma this will prove Proposition. To show convergence, let us take ɛ > 0 so small that c = ( ɛ 2 c > 2 and then cover the unit circle by a net of size ɛ/(n. Let z,ɛ, z 2,ɛ...z k,ɛ be these points, where k = (n. We will need the following ɛ lemma whose proof is well known, but for the reader s benefit we present it below. Lemma 2. Let Q be a polynomial of degree n. Let t D be such that Q(t = Q. Let 0 < ɛ <. Then for all z such that z t < ɛ/n, we have Q(z ( ɛ Q. Proof of Lemma 2. From the integral representation, we have Q(z Q(t z t Q z t n Q, where we used Bernstein s inequality (. in the latter estimate. Using z t ɛ/n, and combining with the previous line, the result now follows from Q(z Q(t Q(z Q(t Q ɛ Q = ( ɛ Q. Using the lemma along with the fact that for z with z =, as E(, we obtain P n(z 2 is distributed n(n + (2n + /6 ( ( k P P n 2 + (2n + P > cn(n log(n P n(z j,ɛ 2 6 n(n + (2n + /6 > c log(n. j= k n C c ɛ n c and since c > 2, the corresponding series converges and this proves Proposition. Combining Propositions 9 and yields sup An application of Fatou s Lemma gives sup P n n P n n E which is the upper bound in Theorem 7. ( P n P n 2 3 a.s. 2 3, Acknowledgment: We thank Manjunath Krishnapur for valuable discussions and references. 2

13 References [] P. Borwein and R. Lockhart, The expected L p norm of random polynomials, Proc. Amer. Math. Soc. 29 (200, [2] Y. S. Chow and H. Teicher, Probability theory: Independence, Interchangeability, Martingales, Springer-Verlag, 978. [3] P. L. Duren, Theory of H p Spaces, Dover, New York, [4] A. Gut, Probability: A graduate course, Springer Texts in Statistics, [5] G. Halász, On a result of Salem and Zygmund concerning random polynomials, Studia Sci. Math. Hungar. 8 (973, [6] J. P. Kahane, Some random series of functions, vol.5, Cambridge studies in Advanced Mathematics, Cambrdige, 2nd edition (985. [7] I. E. Pritsker, Asymptotic zero distribution of random polynomials spanned by general bases, Contemp. Math. 66 (206, [8] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Clarendon Press, Oxford, I. P.: Department of Mathematics, Oklahoma State University Stillwater, OK 74078, USA igor@math.okstate.edu K. R.: Tata Institute of Fundamental Research, Centre for Applicable Mathematics Bengaluru, India koushik@tifrbng.res.in 3

Natural boundary and Zero distribution of random polynomials in smooth domains arxiv: v1 [math.pr] 2 Oct 2017

Natural boundary and Zero distribution of random polynomials in smooth domains arxiv: v1 [math.pr] 2 Oct 2017 Natural boundary and Zero distribution of random polynomials in smooth domains arxiv:1710.00937v1 [math.pr] 2 Oct 2017 Igor Pritsker and Koushik Ramachandran Abstract We consider the zero distribution

More information

Zeros of lacunary random polynomials

Zeros of lacunary random polynomials Zeros of lacunary random polynomials Igor E. Pritsker Dedicated to Norm Levenberg on his 60th birthday Abstract We study the asymptotic distribution of zeros for the lacunary random polynomials. It is

More information

Zeros of Polynomials with Random Coefficients

Zeros of Polynomials with Random Coefficients Zeros of Polynomials with Random Coefficients Igor E. Pritsker Department of Mathematics, Oklahoma State University Stilwater, OK 74078 USA igor@math.okstate.edu Aaron M. Yeager Department of Mathematics,

More information

Complete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions

Complete moment convergence of weighted sums for processes under asymptotically almost negatively associated assumptions Proc. Indian Acad. Sci. Math. Sci.) Vol. 124, No. 2, May 214, pp. 267 279. c Indian Academy of Sciences Complete moment convergence of weighted sums for processes under asymptotically almost negatively

More information

STAT 7032 Probability Spring Wlodek Bryc

STAT 7032 Probability Spring Wlodek Bryc STAT 7032 Probability Spring 2018 Wlodek Bryc Created: Friday, Jan 2, 2014 Revised for Spring 2018 Printed: January 9, 2018 File: Grad-Prob-2018.TEX Department of Mathematical Sciences, University of Cincinnati,

More information

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS

NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS NEWMAN S INEQUALITY FOR INCREASING EXPONENTIAL SUMS Tamás Erdélyi Dedicated to the memory of George G Lorentz Abstract Let Λ n := {λ 0 < λ < < λ n } be a set of real numbers The collection of all linear

More information

Estimates for probabilities of independent events and infinite series

Estimates 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 information

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let

Let D be the open unit disk of the complex plane. Its boundary, the unit circle of the complex plane, is denoted by D. Let HOW FAR IS AN ULTRAFLAT SEQUENCE OF UNIMODULAR POLYNOMIALS FROM BEING CONJUGATE-RECIPROCAL? Tamás Erdélyi Abstract. In this paper we study ultraflat sequences (P n) of unimodular polynomials P n K n in

More information

AN INEQUALITY FOR THE NORM OF A POLYNOMIAL FACTOR IGOR E. PRITSKER. (Communicated by Albert Baernstein II)

AN INEQUALITY FOR THE NORM OF A POLYNOMIAL FACTOR IGOR E. PRITSKER. (Communicated by Albert Baernstein II) PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 9, Number 8, Pages 83{9 S -9939()588-4 Article electronically published on November 3, AN INQUALITY FOR TH NORM OF A POLYNOMIAL FACTOR IGOR. PRITSKR (Communicated

More information

Inequalities for sums of Green potentials and Blaschke products

Inequalities for sums of Green potentials and Blaschke products Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Inequalities for sums of reen potentials and Blaschke products Igor. Pritsker Abstract We study inequalities for the ima

More information

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms

More information

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011

LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011 LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic

More information

L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by

L p Functions. Given a measure space (X, µ) and a real number p [1, ), recall that the L p -norm of a measurable function f : X R is defined by L p Functions Given a measure space (, µ) and a real number p [, ), recall that the L p -norm of a measurable function f : R is defined by f p = ( ) /p f p dµ Note that the L p -norm of a function f may

More information

The Hilbert Transform and Fine Continuity

The Hilbert Transform and Fine Continuity Irish Math. Soc. Bulletin 58 (2006), 8 9 8 The Hilbert Transform and Fine Continuity J. B. TWOMEY Abstract. It is shown that the Hilbert transform of a function having bounded variation in a finite interval

More information

BALANCING GAUSSIAN VECTORS. 1. Introduction

BALANCING 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 information

A class of trees and its Wiener index.

A class of trees and its Wiener index. A class of trees and its Wiener index. Stephan G. Wagner Department of Mathematics Graz University of Technology Steyrergasse 3, A-81 Graz, Austria wagner@finanz.math.tu-graz.ac.at Abstract In this paper,

More information

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n

GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. 2π) n GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS OF CONVEX BODIES. A. ZVAVITCH Abstract. In this paper we give a solution for the Gaussian version of the Busemann-Petty problem with additional

More information

ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA

ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA ON THE BOUNDEDNESS BEHAVIOR OF THE SPECTRAL FACTORIZATION IN THE WIENER ALGEBRA FOR FIR DATA Holger Boche and Volker Pohl Technische Universität Berlin, Heinrich Hertz Chair for Mobile Communications Werner-von-Siemens

More information

A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin

A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN Dedicated to the memory of Mikhail Gordin Abstract. We prove a central limit theorem for a square-integrable ergodic

More information

Complete Moment Convergence for Weighted Sums of Negatively Orthant Dependent Random Variables

Complete Moment Convergence for Weighted Sums of Negatively Orthant Dependent Random Variables Filomat 31:5 217, 1195 126 DOI 1.2298/FIL175195W Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Complete Moment Convergence for

More information

Random Process Lecture 1. Fundamentals of Probability

Random Process Lecture 1. Fundamentals of Probability Random Process Lecture 1. Fundamentals of Probability Husheng Li Min Kao Department of Electrical Engineering and Computer Science University of Tennessee, Knoxville Spring, 2016 1/43 Outline 2/43 1 Syllabus

More information

Wald for non-stopping times: The rewards of impatient prophets

Wald for non-stopping times: The rewards of impatient prophets Electron. Commun. Probab. 19 (2014), no. 78, 1 9. DOI: 10.1214/ECP.v19-3609 ISSN: 1083-589X ELECTRONIC COMMUNICATIONS in PROBABILITY Wald for non-stopping times: The rewards of impatient prophets Alexander

More information

Continued Fraction Digit Averages and Maclaurin s Inequalities

Continued 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 information

ON THE STRONG LIMIT THEOREMS FOR DOUBLE ARRAYS OF BLOCKWISE M-DEPENDENT RANDOM VARIABLES

ON THE STRONG LIMIT THEOREMS FOR DOUBLE ARRAYS OF BLOCKWISE M-DEPENDENT RANDOM VARIABLES Available at: http://publications.ictp.it IC/28/89 United Nations Educational, Scientific Cultural Organization International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

More information

The Codimension of the Zeros of a Stable Process in Random Scenery

The Codimension of the Zeros of a Stable Process in Random Scenery The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar

More information

If Y and Y 0 satisfy (1-2), then Y = Y 0 a.s.

If Y and Y 0 satisfy (1-2), then Y = Y 0 a.s. 20 6. CONDITIONAL EXPECTATION Having discussed at length the limit theory for sums of independent random variables we will now move on to deal with dependent random variables. An important tool in this

More information

A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series

A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series Publ. Math. Debrecen 73/1-2 2008), 1 10 A note on the growth rate in the Fazekas Klesov general law of large numbers and on the weak law of large numbers for tail series By SOO HAK SUNG Taejon), TIEN-CHUNG

More information

SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS

SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint

More information

PCA with random noise. Van Ha Vu. Department of Mathematics Yale University

PCA with random noise. Van Ha Vu. Department of Mathematics Yale University PCA with random noise Van Ha Vu Department of Mathematics Yale University An important problem that appears in various areas of applied mathematics (in particular statistics, computer science and numerical

More information

COMPLETE QTH MOMENT CONVERGENCE OF WEIGHTED SUMS FOR ARRAYS OF ROW-WISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES

COMPLETE QTH MOMENT CONVERGENCE OF WEIGHTED SUMS FOR ARRAYS OF ROW-WISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES Hacettepe Journal of Mathematics and Statistics Volume 43 2 204, 245 87 COMPLETE QTH MOMENT CONVERGENCE OF WEIGHTED SUMS FOR ARRAYS OF ROW-WISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES M. L. Guo

More information

On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables

On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables On the Set of Limit Points of Normed Sums of Geometrically Weighted I.I.D. Bounded Random Variables Deli Li 1, Yongcheng Qi, and Andrew Rosalsky 3 1 Department of Mathematical Sciences, Lakehead University,

More information

Part II Probability and Measure

Part II Probability and Measure Part II Probability and Measure Theorems Based on lectures by J. Miller Notes taken by Dexter Chua Michaelmas 2016 These notes are not endorsed by the lecturers, and I have modified them (often significantly)

More information

1 Sequences of events and their limits

1 Sequences of events and their limits O.H. Probability II (MATH 2647 M15 1 Sequences of events and their limits 1.1 Monotone sequences of events Sequences of events arise naturally when a probabilistic experiment is repeated many times. For

More information

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE

CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator

More information

9. Series representation for analytic functions

9. Series representation for analytic functions 9. Series representation for analytic functions 9.. Power series. Definition: A power series is the formal expression S(z) := c n (z a) n, a, c i, i =,,, fixed, z C. () The n.th partial sum S n (z) is

More information

A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES

A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES Bull. Korean Math. Soc. 52 (205), No. 3, pp. 825 836 http://dx.doi.org/0.434/bkms.205.52.3.825 A NOTE ON THE COMPLETE MOMENT CONVERGENCE FOR ARRAYS OF B-VALUED RANDOM VARIABLES Yongfeng Wu and Mingzhu

More information

ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY

ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY J. Korean Math. Soc. 45 (2008), No. 4, pp. 1101 1111 ON THE COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF DEPENDENT RANDOM VARIABLES UNDER CONDITION OF WEIGHTED INTEGRABILITY Jong-Il Baek, Mi-Hwa Ko, and Tae-Sung

More information

Theorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension. n=1

Theorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension. n=1 Chapter 2 Probability measures 1. Existence Theorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension to the generated σ-field Proof of Theorem 2.1. Let F 0 be

More information

Limit Theorems for Exchangeable Random Variables via Martingales

Limit Theorems for Exchangeable Random Variables via Martingales Limit Theorems for Exchangeable Random Variables via Martingales Neville Weber, University of Sydney. May 15, 2006 Probabilistic Symmetries and Their Applications A sequence of random variables {X 1, X

More information

Weighted norm inequalities for singular integral operators

Weighted norm inequalities for singular integral operators Weighted norm inequalities for singular integral operators C. Pérez Journal of the London mathematical society 49 (994), 296 308. Departmento de Matemáticas Universidad Autónoma de Madrid 28049 Madrid,

More information

Probability and Measure

Probability 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 information

The main results about probability measures are the following two facts:

The main results about probability measures are the following two facts: Chapter 2 Probability measures The main results about probability measures are the following two facts: Theorem 2.1 (extension). If P is a (continuous) probability measure on a field F 0 then it has a

More information

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET ao DOI: 1.478/s1175-14-8-y Math. Slovaca 64 (14), No. 6, 1397 148 POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET Javad Baradaran* Mohsen Taghavi** (Communicated by Stanis lawa Kanas ) ABSTRACT. This paper

More information

Chapter 6. Convergence. Probability Theory. Four different convergence concepts. Four different convergence concepts. Convergence in probability

Chapter 6. Convergence. Probability Theory. Four different convergence concepts. Four different convergence concepts. Convergence in probability Probability Theory Chapter 6 Convergence Four different convergence concepts Let X 1, X 2, be a sequence of (usually dependent) random variables Definition 1.1. X n converges almost surely (a.s.), or with

More information

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART

SZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics

More information

Why Bohr got interested in his radius and what it has led to

Why Bohr got interested in his radius and what it has led to Why Bohr got interested in his radius and what it has led to Kristian Seip Norwegian University of Science and Technology (NTNU) Universidad Autónoma de Madrid, December 11, 2009 Definition of Bohr radius

More information

Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION

Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces ABSTRACT 1. INTRODUCTION Malaysian Journal of Mathematical Sciences 6(2): 25-36 (202) Bernstein-Szegö Inequalities in Reproducing Kernel Hilbert Spaces Noli N. Reyes and Rosalio G. Artes Institute of Mathematics, University of

More information

Functional Analysis, Stein-Shakarchi Chapter 1

Functional Analysis, Stein-Shakarchi Chapter 1 Functional Analysis, Stein-Shakarchi Chapter 1 L p spaces and Banach Spaces Yung-Hsiang Huang 018.05.1 Abstract Many problems are cited to my solution files for Folland [4] and Rudin [6] post here. 1 Exercises

More information

An operator preserving inequalities between polynomials

An operator preserving inequalities between polynomials An operator preserving inequalities between polynomials NISAR A. RATHER Kashmir University P.G. Department of Mathematics Hazratbal-190006, Srinagar INDIA dr.narather@gmail.com MUSHTAQ A. SHAH Kashmir

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journal of Inequalities in Pure and Applied Mathematics NORMS OF CERTAIN OPERATORS ON WEIGHTED l p SPACES AND LORENTZ SEQUENCE SPACES G.J.O. JAMESON AND R. LASHKARIPOUR Department of Mathematics and Statistics,

More information

SCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS

SCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS C O L L O Q U I U M M A T H E M A T I C U M VOL. LXIII 1992 FASC. 2 SCHWARTZ SPACES ASSOCIATED WITH SOME NON-DIFFERENTIAL CONVOLUTION OPERATORS ON HOMOGENEOUS GROUPS BY JACEK D Z I U B A Ń S K I (WROC

More information

A Note on the Central Limit Theorem for a Class of Linear Systems 1

A Note on the Central Limit Theorem for a Class of Linear Systems 1 A Note on the Central Limit Theorem for a Class of Linear Systems 1 Contents Yukio Nagahata Department of Mathematics, Graduate School of Engineering Science Osaka University, Toyonaka 560-8531, Japan.

More information

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS Tamás Erdélyi Abstract. Let Λ n := {λ,λ,...,λ n } be a set of n distinct positive numbers. The span of {e λ t,e λ t,...,e λ nt } over R

More information

Mathematical Institute, University of Utrecht. The problem of estimating the mean of an observed Gaussian innite-dimensional vector

Mathematical Institute, University of Utrecht. The problem of estimating the mean of an observed Gaussian innite-dimensional vector On Minimax Filtering over Ellipsoids Eduard N. Belitser and Boris Y. Levit Mathematical Institute, University of Utrecht Budapestlaan 6, 3584 CD Utrecht, The Netherlands The problem of estimating the mean

More information

A note on a construction of J. F. Feinstein

A note on a construction of J. F. Feinstein STUDIA MATHEMATICA 169 (1) (2005) A note on a construction of J. F. Feinstein by M. J. Heath (Nottingham) Abstract. In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform

More information

Convergence Rates for Renewal Sequences

Convergence Rates for Renewal Sequences Convergence Rates for Renewal Sequences M. C. Spruill School of Mathematics Georgia Institute of Technology Atlanta, Ga. USA January 2002 ABSTRACT The precise rate of geometric convergence of nonhomogeneous

More information

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET

POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9947(XX)0000-0 POLYNOMIALS WITH COEFFICIENTS FROM A FINITE SET PETER BORWEIN, TAMÁS ERDÉLYI, FRIEDRICH LITTMANN

More information

Lecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.

Lecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal

More information

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Jan Wehr and Jack Xin Abstract We study waves in convex scalar conservation laws under noisy initial perturbations.

More information

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure?

3 (Due ). Let A X consist of points (x, y) such that either x or y is a rational number. Is A measurable? What is its Lebesgue measure? MA 645-4A (Real Analysis), Dr. Chernov Homework assignment 1 (Due ). Show that the open disk x 2 + y 2 < 1 is a countable union of planar elementary sets. Show that the closed disk x 2 + y 2 1 is a countable

More information

Introduction and Preliminaries

Introduction and Preliminaries Chapter 1 Introduction and Preliminaries This chapter serves two purposes. The first purpose is to prepare the readers for the more systematic development in later chapters of methods of real analysis

More information

The small ball property in Banach spaces (quantitative results)

The small ball property in Banach spaces (quantitative results) The small ball property in Banach spaces (quantitative results) Ehrhard Behrends Abstract A metric space (M, d) is said to have the small ball property (sbp) if for every ε 0 > 0 there exists a sequence

More information

INTRODUCTION TO PADÉ APPROXIMANTS. E. B. Saff Center for Constructive Approximation

INTRODUCTION TO PADÉ APPROXIMANTS. E. B. Saff Center for Constructive Approximation INTRODUCTION TO PADÉ APPROXIMANTS E. B. Saff Center for Constructive Approximation H. Padé (1863-1953) Student of Hermite Histhesiswon French Academy of Sciences Prize C. Hermite (1822-1901) Used Padé

More information

Math 328 Course Notes

Math 328 Course Notes Math 328 Course Notes Ian Robertson March 3, 2006 3 Properties of C[0, 1]: Sup-norm and Completeness In this chapter we are going to examine the vector space of all continuous functions defined on the

More information

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS. Lecture 3

ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS. Lecture 3 ENTIRE FUNCTIONS AND COMPLETENESS PROBLEMS A. POLTORATSKI Lecture 3 A version of the Heisenberg Uncertainty Principle formulated in terms of Harmonic Analysis claims that a non-zero measure (distribution)

More information

Analytic Theory of Polynomials

Analytic Theory of Polynomials Analytic Theory of Polynomials Q. I. Rahman Universite de Montreal and G. Schmeisser Universitat Erlangen -Niirnberg CLARENDON PRESS-OXFORD 2002 1 Introduction 1 1.1 The fundamental theorem of algebra

More information

Polarization constant K(n, X) = 1 for entire functions of exponential type

Polarization constant K(n, X) = 1 for entire functions of exponential type Int. J. Nonlinear Anal. Appl. 6 (2015) No. 2, 35-45 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.252 Polarization constant K(n, X) = 1 for entire functions of exponential type A.

More information

APPROXIMATING CONTINUOUS FUNCTIONS: WEIERSTRASS, BERNSTEIN, AND RUNGE

APPROXIMATING 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 information

Lecture I: Asymptotics for large GUE random matrices

Lecture I: Asymptotics for large GUE random matrices Lecture I: Asymptotics for large GUE random matrices Steen Thorbjørnsen, University of Aarhus andom Matrices Definition. Let (Ω, F, P) be a probability space and let n be a positive integer. Then a random

More information

MATHS 730 FC Lecture Notes March 5, Introduction

MATHS 730 FC Lecture Notes March 5, Introduction 1 INTRODUCTION MATHS 730 FC Lecture Notes March 5, 2014 1 Introduction Definition. If A, B are sets and there exists a bijection A B, they have the same cardinality, which we write as A, #A. If there exists

More information

ADVANCED PROBABILITY: SOLUTIONS TO SHEET 1

ADVANCED 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 information

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES

NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)

More information

ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS

ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS ESTIMATES FOR MAXIMAL SINGULAR INTEGRALS LOUKAS GRAFAKOS Abstract. It is shown that maximal truncations of nonconvolution L -bounded singular integral operators with kernels satisfying Hörmander s condition

More information

Math 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 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 information

Expected zeros of random orthogonal polynomials on the real line

Expected zeros of random orthogonal polynomials on the real line ISSN: 889-3066 c 207 Universidad de Jaén Web site: jja.ujaen.es Jaen J. Approx. 9) 207), 24 Jaen Journal on Approximation Expected zeros of random orthogonal polynomials on the real line Igor E. Pritsker

More information

3 Integration and Expectation

3 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 information

Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal Czechoslovak Mathematical Journal Oktay Duman; Cihan Orhan µ-statistically convergent function sequences Czechoslovak Mathematical Journal, Vol. 54 (2004), No. 2, 413 422 Persistent URL: http://dml.cz/dmlcz/127899

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory

More information

CARLESON MEASURES AND DOUGLAS QUESTION ON THE BERGMAN SPACE. Department of Mathematics, University of Toledo, Toledo, OH ANTHONY VASATURO

CARLESON MEASURES AND DOUGLAS QUESTION ON THE BERGMAN SPACE. Department of Mathematics, University of Toledo, Toledo, OH ANTHONY VASATURO CARLESON MEASURES AN OUGLAS QUESTION ON THE BERGMAN SPACE ŽELJKO ČUČKOVIĆ epartment of Mathematics, University of Toledo, Toledo, OH 43606 ANTHONY VASATURO epartment of Mathematics, University of Toledo,

More information

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM

More information

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS

ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS Bendikov, A. and Saloff-Coste, L. Osaka J. Math. 4 (5), 677 7 ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS ALEXANDER BENDIKOV and LAURENT SALOFF-COSTE (Received March 4, 4)

More information

Lecture 21 Representations of Martingales

Lecture 21 Representations of Martingales Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let

More information

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS

UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS J. Korean Math. Soc. 45 (008), No. 3, pp. 835 840 UNIMODULAR ROOTS OF RECIPROCAL LITTLEWOOD POLYNOMIALS Paulius Drungilas Reprinted from the Journal of the Korean Mathematical Society Vol. 45, No. 3, May

More information

RECENT RESULTS FOR SUPERCRITICAL CONTROLLED BRANCHING PROCESSES WITH CONTROL RANDOM FUNCTIONS

RECENT RESULTS FOR SUPERCRITICAL CONTROLLED BRANCHING PROCESSES WITH CONTROL RANDOM FUNCTIONS Pliska Stud. Math. Bulgar. 16 (2004), 43-54 STUDIA MATHEMATICA BULGARICA RECENT RESULTS FOR SUPERCRITICAL CONTROLLED BRANCHING PROCESSES WITH CONTROL RANDOM FUNCTIONS Miguel González, Manuel Molina, Inés

More information

On complete convergence and complete moment convergence for weighted sums of ρ -mixing random variables

On complete convergence and complete moment convergence for weighted sums of ρ -mixing random variables Chen and Sung Journal of Inequalities and Applications 2018) 2018:121 https://doi.org/10.1186/s13660-018-1710-2 RESEARCH Open Access On complete convergence and complete moment convergence for weighted

More information

Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales

Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales Fundamental Inequalities, Convergence and the Optional Stopping Theorem for Continuous-Time Martingales Prakash Balachandran Department of Mathematics Duke University April 2, 2008 1 Review of Discrete-Time

More information

Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett/

Hartogs Theorem: separate analyticity implies joint Paul Garrett  garrett/ (February 9, 25) Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ (The present proof of this old result roughly follows the proof

More information

Notes 15 : UI Martingales

Notes 15 : UI Martingales Notes 15 : UI Martingales Math 733 - Fall 2013 Lecturer: Sebastien Roch References: [Wil91, Chapter 13, 14], [Dur10, Section 5.5, 5.6, 5.7]. 1 Uniform Integrability We give a characterization of L 1 convergence.

More information

Remarks on the Rademacher-Menshov Theorem

Remarks on the Rademacher-Menshov Theorem Remarks on the Rademacher-Menshov Theorem Christopher Meaney Abstract We describe Salem s proof of the Rademacher-Menshov Theorem, which shows that one constant works for all orthogonal expansions in all

More information

Lecture 8: February 8

Lecture 8: February 8 CS71 Randomness & Computation Spring 018 Instructor: Alistair Sinclair Lecture 8: February 8 Disclaimer: These notes have not been subjected to the usual scrutiny accorded to formal publications. They

More information

arxiv: v1 [math.pr] 22 Dec 2018

arxiv: v1 [math.pr] 22 Dec 2018 arxiv:1812.09618v1 [math.pr] 22 Dec 2018 Operator norm upper bound for sub-gaussian tailed random matrices Eric Benhamou Jamal Atif Rida Laraki December 27, 2018 Abstract This paper investigates an upper

More information

Lectures 2 3 : Wigner s semicircle law

Lectures 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 information

On the convergence of sequences of random variables: A primer

On the convergence of sequences of random variables: A primer BCAM May 2012 1 On the convergence of sequences of random variables: A primer Armand M. Makowski ECE & ISR/HyNet University of Maryland at College Park armand@isr.umd.edu BCAM May 2012 2 A sequence a :

More information

An idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim

An idea how to solve some of the problems. diverges the same must hold for the original series. T 1 p T 1 p + 1 p 1 = 1. dt = lim An idea how to solve some of the problems 5.2-2. (a) Does not converge: By multiplying across we get Hence 2k 2k 2 /2 k 2k2 k 2 /2 k 2 /2 2k 2k 2 /2 k. As the series diverges the same must hold for the

More information

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series

C.7. Numerical series. Pag. 147 Proof of the converging criteria for series. Theorem 5.29 (Comparison test) Let a k and b k be positive-term series C.7 Numerical series Pag. 147 Proof of the converging criteria for series Theorem 5.29 (Comparison test) Let and be positive-term series such that 0, for any k 0. i) If the series converges, then also

More information

Chapter One. The Calderón-Zygmund Theory I: Ellipticity

Chapter One. The Calderón-Zygmund Theory I: Ellipticity Chapter One The Calderón-Zygmund Theory I: Ellipticity Our story begins with a classical situation: convolution with homogeneous, Calderón- Zygmund ( kernels on R n. Let S n 1 R n denote the unit sphere

More information

Lecture 21: Expectation of CRVs, Fatou s Lemma and DCT Integration of Continuous Random Variables

Lecture 21: Expectation of CRVs, Fatou s Lemma and DCT Integration of Continuous Random Variables EE50: Probability Foundations for Electrical Engineers July-November 205 Lecture 2: Expectation of CRVs, Fatou s Lemma and DCT Lecturer: Krishna Jagannathan Scribe: Jainam Doshi In the present lecture,

More information

Useful Probability Theorems

Useful Probability Theorems Useful Probability Theorems Shiu-Tang Li Finished: March 23, 2013 Last updated: November 2, 2013 1 Convergence in distribution Theorem 1.1. TFAE: (i) µ n µ, µ n, µ are probability measures. (ii) F n (x)

More information

MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS

MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS MARKOV-NIKOLSKII TYPE INEQUALITIES FOR EXPONENTIAL SUMS ON FINITE INTERVALS TAMÁS ERDÉLYI Abstract Let Λ n := {λ 0 < λ 1 < < λ n} be a set of real numbers The collection of all linear combinations of of

More information

P -adic root separation for quadratic and cubic polynomials

P -adic root separation for quadratic and cubic polynomials P -adic root separation for quadratic and cubic polynomials Tomislav Pejković Abstract We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible

More information