Hsien-Kuei Hwang. Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab., 6(1):297-319, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....estimate for each of these distances. Our results are thus approximation theorems rather than limit theorems . The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]) Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall discuss some of these, including Halasz ....

Hwang, H.-K. (1996) Large deviations for combinatorial distributions, I. central limit theorems. Annals of Applied Probability 6, 297--319.

....Ch. 6] and yields a better error term. We complete the asymptotic normality of n by its strong concentration property (2) using a simple technique (cf. 27, Ch. III] amended from the usual Chernoff bound. We can also derive a local limit theorem in the form of Cram er type large deviations (cf. [19, 27]) It suffices to replace condition (M3) by the following stronger one. M3 ) There exists a fixed constant c 3 0 such that g(r) Gamma e e g(r iy) c 3 (log(1=r) uniformly for =2 jyj and Gamma , as r 0 . Let Y (u; s) be the Mellin transform of the function log(1 ....

....for k = 3; 4; 5; 5) U(w) ff 1)ff Gamma Y (1; ff) Here the symbol [z ]f(z) denotes the coefficient of z in the Taylor expansion of f(z) Note that U is convex due to the same property of Y and that n = U . The first two terms of k are given by (cf. [19]) 3 = 6 U (0) and 4 = 4) 0) Gamma As an interesting consequence, we state the following Corollary 1 If m = n xoe n 2 Z , where x = o jxj jxj ; 6) uniformly in x. The proof of this theorem utilizes essentially the two dimensional saddle point ....

H.-K. Hwang, Large deviations for combinatorial distributions. I: central limit theorems, Annals of Applied Probability, 6 (1996), 297--319.

....Ch. 6] and yields a better error term. We complete the asymptotic normality of # n by its strong concentration property (3) using a simple technique (see [27, Ch. III] amended from the usual Cherno# bound. We can also derive a local limit theorem in the form of Cramer type large deviations (see [20] [27] It su#ces to replace condition (M3) by the following stronger one. M3 ) There exists a fixed constant c 3 0 such that g(r) Let Y (u, s) be the Mellin transform of the function log(1 ue x ) log(1 ue x ) dx for s 0. 4) As we will see, Y is essentially ....

....(w) for k = 3, 4, 5, 6) U(w) # 1)# # (# 1) A . Here the symbol [z ]f(z) denotes the coe#cient of z in the Taylor expansion of f(z) Note that U is convex due to the same property of Y and that n = U # (0)n . The first two terms of # k are given by (see [20]) # 3 = U ### (0) and # 4 = 4) 0) U ### (0) As an interesting consequence, we state the following Corollary 1. If m = n x# n , where x = o , 7) uniformly in x. The proof of this theorem utilizes essentially the two dimensional saddle point method and ....

H.-K. Hwang, Large deviations of combinatorial distributions. II. Local limit theorems, Annals of Applied Probability, 8 (1998), 163--181.

.... feature of integer partitions is that the limiting distribution of the number of summands is non Gaussian in almost all cases if the multiplicity of each summand is taken into account (see [15, 23, 31] in contrast to the ubiquitous normal law in a large class of combinatorial structures (see [12, 18]) Intuitively, the former phenomenon may be ascribed to the predominance of small summands when the number of summands becomes large, say, larger than the mean value. However, Gaussian limiting distribution appears if the parts are counted without multiplicity, this being intuitively clear since ....

H.-K. Hwang, Large deviations for combinatorial distributions. I: central limit theorems, Annals of Applied Probability, 6 (1996), 297--319. 32

....estimate for each of these distances. Our results are thus approximation theorems rather than limit theorems . The common form of the underlying structure of these distributions suggests the study of an analytic scheme as we did previously for normal approximation and large deviations (cf. [53, 54]) Many concrete examples from probabilistic number theory and combinatorial structures will justify the study of this scheme. Our treatment being completely general, many extensions can be further pursued with essentially the same line of methods. We shall discuss some of these, including Hal asz ....

Hwang, H.-K. (1996) Large deviations for combinatorial distributions, I. central limit theorems. Annals of Applied Probability 6, 297--319.

....any ball of the rst type. The property then results immediately from the fact that (z) S(z) in the notations of Lemma 1) is analytic in jzj 2 . We next turn to large deviations, for which the book of den Hollander [15] can serve as a smooth introduction. It is known from the works of Hwang [28] that a quasi power approximation (in the sense of Lemma 3) for a family of PGFs leads to very precise moderate deviation estimates valid in some range not too far from the center of the distribution. We state: Corollary 5 (Large deviations) Let be a number of the open interval (0; 7 ) ....

....from the mean value is exponentially small, the rate of exponential decay being explicitly related to the already encountered Abelian integrals. Proof. Notice rst that E(Xn ) 7 (n 1) so that (51) quanti es the left part of the distribution as approximately given by e . Theorem 2. 1 in [28] asserts that quasi powers approximations near u = 1 systematically entail moderate large deviation estimates expressed in terms of large values of the Gaussian error function. The proof below recycles most of the technology of [28] though the range is di erent. For completeness, we outline the ....

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Hwang, H.-K. Large deviations for combinatorial distributions. I. Central limit theorems. The Annals of Applied Probability 6, 1 (1996), 297-319.

....representation for our problem, a fact related to the technique of transfer matrices [6] see Section 5.1) Then Perron Frobenius properties and their perturbed versions apply, as detailed in Section 5.2; see especially Lemmas 2 and 3. A quasi powers approximation (in the sense of Bender and Hwang [4, 23, 24]) for the probability generating function of is then inferred, see Eq. 45) As developed in Section 5.3, this suces to establish the central limit law (32) by a well known process that parallels the usual proof of the central limit theorem for sums of independent random variables [4, 17, 23, ....

....[4, 23, 24] for the probability generating function of is then inferred, see Eq. 45) As developed in Section 5. 3, this suces to establish the central limit law (32) by a well known process that parallels the usual proof of the central limit theorem for sums of independent random variables [4, 17, 23, 24]. Speed of convergence estimates expressed by (32) arise in this context from the Berry Esseen inequalities. A similar analysis provides large deviation estimates as represented in a simpli ed form by (33) Additional strong positivity properties that are available when the pattern is primitive ....

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Hsien-Kuei Hwang, Large deviations for combinatorial distributions: I. Central limit theorems, The Annals of Applied Probability, 6, 297-319, 1996.

.... ]f(z, y) p(y) 1 2 #(y) n 1 1 O(n K ) 30) for any K 0 and y near the unity, where #(y) 31) From this result, we can deduce several results on the underlying random variable, say # n having probability generating function [z ]f(z, y) by the results in [17, 18, 19]; we give only the result for convergence rate, details as well as other finer results being omitted here. Theorem 3. The random variable # n is asymptotically normally distributed sup # x # # # # # # # n # 2# e u 2 # # n 1 2 provided that # # # : q # (1) 2p # ....

H.-K. Hwang, Large deviations of combinatorial distributions. II. Local limit theorems, Annals of Applied Probability, 8, 163--181 (1998). 39

.... ]f(z, y) p(y) 1 2 #(y) n 1 1 O(n K ) 30) for any K 0 and y near the unity, where #(y) 31) From this result, we can deduce several results on the underlying random variable, say # n having probability generating function [z ]f(z, y) by the results in [17, 18, 19]; we give only the result for convergence rate, details as well as other finer results being omitted here. Theorem 3. The random variable # n is asymptotically normally distributed sup # x # # # # # # # n # 2# e u 2 # # n 1 2 provided that # # # : q # (1) 2p # ....

H.-K. Hwang, Large deviations for combinatorial distributions. I. Central limit theorems, Annals of Applied Probability, 6, 297--319 (1996).

.... y) p(y) i(y) Gamman Gamma1 Gamma 1 O(n GammaK ; 30) for any K 0 and y near the unity, where i(y) 31) From this result, we can deduce several results on the underlying random variable, say n having probability generating function [z ]f(z; y) by the results in [17, 18, 19]; we give only the result for convergence rate, details as well as other finer results being omitted here. Theorem 3. The random variable n is asymptotically normally distributed sup Gamma1 x 1 fi fi Gammau =2 fi fi provided that oe 6= 0, where : ....

H.-K. Hwang, Large deviations of combinatorial distributions. II. Local limit theorems, Annals of Applied Probability, 8, 163--181 (1998). 39

.... y) p(y) i(y) Gamman Gamma1 Gamma 1 O(n GammaK ; 30) for any K 0 and y near the unity, where i(y) 31) From this result, we can deduce several results on the underlying random variable, say n having probability generating function [z ]f(z; y) by the results in [17, 18, 19]; we give only the result for convergence rate, details as well as other finer results being omitted here. Theorem 3. The random variable n is asymptotically normally distributed sup Gamma1 x 1 fi fi Gammau =2 fi fi provided that oe 6= 0, where : ....

H.-K. Hwang, Large deviations for combinatorial distributions. I. Central limit theorems, Annals of Applied Probability, 6, 297--319 (1996).

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H.-K. Hwang (1996). Large deviations for combinatorial distributions. I. Central limit theorems. Annals of Applied Probability, 6, 297--319.

No context found.

H.-K. Hwang (1996). Large deviations for combinatorial distributions. I. Central limit theorems. Annals of Applied Probability, 6, 297--319.

No context found.

Hsien-Kuei Hwang. Large deviations for combinatorial distributions. I. Central limit theorems. Ann. Appl. Probab., 6(1):297-319, 1996.

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