Structure Theory of Set Addition and Local Limit Theorems for Independent Lattice Random Variables

Moskvin, D. A.; Frejman, G. A.; Yudin, A. A.

doi:10.1137/1119005

Cited by 16 publications

(7 citation statements)

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“…Then (see [5]), g(x) = e S(x) , x ∈ D is the generating function for the sequence {c n } ∞ 0 defined by (1.3). Namely, 13) and moreover, the series (2.12) and (2.13) converge in the same domain D.…”

mentioning

confidence: 89%

“…For this case, general necessary and sufficient conditions for validity of the local limit theorem are not known. For some cases results in this direction were obtained in [13] and [3]. In the first of these two papers a sufficient condition was established (see ([13], Theorem 2, condition III) in the case of an array of general lattice random variables.…”

Section: )mentioning

confidence: 99%

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Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

Freiman

Granovsky

2002

Isr. J. Math.

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We construct a probability model seemingly unrelated to the considered stochastic process of coagulation and fragmentation. By proving for this model the local limit theorem, we establish the asymptotic formula for the partition function of the equilibrium measure for a wide class of parameter functions of the process. This formula proves the conjecture stated in [5] for the above class of processes. The method used goes back to A.Khintchine.

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confidence: 89%

Section: )mentioning

confidence: 99%

Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

Freiman

Granovsky

2002

Isr. J. Math.

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“…Constructing now the set K 1 (u) = × d j=1 K 1 (u (j) ) as described in the formulation of Theorem 5, we see that inequalities (24) and (25) follow from (51), ( 52) and (54). Theorem 5 is proved.…”

Section: Proofs Of Theorems 5-8mentioning

confidence: 99%

“…These ideas were used by Nguyen and Vu [27] and [28] as well. It should also be mentioned that Freiman himself has used his theory to obtain local limit theorems and bounds for concentration functions (see, e.g., [8], [17] and [25]).…”

Section: Introductionmentioning

confidence: 99%

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Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem

Götze¹,

Eliseeva²,

Зайцев³

2018

Theory Probab. Appl.

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Let X, X 1 , . . . , X n be independent identically distributed random variables. In this paper we study the behavior of concentration functions of weighted sums n k=1 X k a k depending on the arithmetic structure of coefficients a k . The results obtained for the last ten years for the concentration functions of weighted sums play an important role in the study of singular numbers of random matrices. Recently, Tao and Vu proposed a so-called inverse principle in the Littlewood-Offord problem. We discuss the relations between this Inverse Principle and a similar principle for sums of arbitrarily distributed independent random variables formulated by Arak in the 1980's. be independent identically distributed (i.i.d.) random variables. Let a = (a 1 , . . . , a n ) = 0, where a k = (a k1 , . . . , a kd ) ∈ R d , k = 1, . . . , n. Starting with seminal papers of Littlewood and Offord [24] and Erdös [13], the behavior of the concentration functions of the weighted sums S a = n k=1 X k a k . is studied intensively. In the sequel, let F a denote the distribution of the sum S a . The first results were obtained for the case τ = 0 and d = 1, that is, here the maximal probability max x∈R P{S a = x}. was investigated. For a detailed history of this part of the problem we refer to a recent review of Nguyen and Vu [28].In the last ten years, refined concentration results for the weighted sums S a play an important role in the study of singular values of random matrices (see, for instance, Nguyen and Vu [27], Rudelson and Vershynin [31], [32], Tao and Vu [35], [36] Vershynin [38]). Recently, the authors of the present paper (see [9], [10], and [12]) improved some of concentration bounds of the papers [18], [31], [32], [38]. These results reflect the dependence of the bounds on the arithmetic structure of coefficients a k under various conditions on the vector a ∈ (R d ) n and on the distribution L(X). Several years ago, Tao and Vu [35] (see also [27]) proposed the so-called inverse principle in the Littlewood-Offord problem (see § 2). In the present paper, we discuss the relations between this inverse principle and similar principles formulated by Arak (see [1] and [2]) in his papers from the 1980's. In the one-dimensional case, Arak has found a connection of the concentration function of the sum with the arithmetic structure of supports of distributions of independent random variables for arbitrary distributions of summands.Apparently the authors of the publications mentioned above were not aware of the results from the papers of Arak [1] and [2]. Although Arak himself did not use the concept of "inverse principle" in his works, in essence such a principle was there formulated. It is related to general bounds for concentration functions of distributions of sums of independent one-dimensional random variables. The results were used for the estimation of the rate of approximation of n-fold convolutions of probability distributions by infinitely divisible ones. Later, the methods based on Arak's inverse principle admitted to pro...

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Dolgopyat,

Sarig

2023

Lecture Notes in Mathematics

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Structure Theory of Set Addition and Local Limit Theorems for Independent Lattice Random Variables

Cited by 16 publications

References 3 publications

Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

Asymptotic formula for a partition function of reversible coagulation-fragmentation processes

Arak Inequalities for Concentration Functions and the Littlewood--Offord Problem

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