Limiting Conditional Distributions for Sums of Random Variables

Kudlaev, E. M.

doi:10.1137/1129102

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…|E(e isX n Y n )| 2 = |E((e isX n − e iθ )Y n )| 2 13) and thus, by the arithmetic-geometric inequality,…”

Section: Proofsmentioning

confidence: 97%

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Moment convergence in conditional limit theorems

Janson

2001

Journal of Applied Probability

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Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

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“…|E(e isX n Y n )| 2 = |E((e isX n − e iθ )Y n )| 2 13) and thus, by the arithmetic-geometric inequality,…”

Section: Proofsmentioning

confidence: 97%

“…General limit theorems yielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5], [7] and Kudlaev [13]. (See also results on asymptotic expansions in [2], [14], [4], [8] and on rates of convergence in [15].…”

Section: Introductionmentioning

confidence: 99%

Moment convergence in conditional limit theorems

Janson

2001

Journal of Applied Probability

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“…When the distribution of (X (n) j , Y (n) j ) does not depends on n, the Gibbs conditioning principle ( [22,5,6]) states that L n converges weakly to the degenerated distribution concentrated on a point χ depending on the conditioning value (see Corollary 2.6). Around the Gibbs conditioning principle, general limit theorems yielding the asymptotic behavior of the conditioned sum are given in [21,13,17]. Asymptotic expansions for the distribution of the conditioned sum are proved in [11,18].…”

Section: Introductionmentioning

confidence: 99%

Conditional large and moderate deviations for sums of discrete random variables. Combinatoric applications

Gamboa¹,

Klein²,

Prieur³

2012

Bernoulli

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“…The scheme of decomposable statistics (DSs) constitutes a rather wide class of dependent random variables, which appear both in various branches of mathematics and some other sciences (see, e.g., [3,6,7,8,14,15]). This means the existence of independent random variables whose joint conditional distribution (under the condition that their sum is fixed at some "point") coincides with the joint distribution of the terms of that sample.…”

Section: Introductionmentioning

confidence: 99%

“…) -(112)15(n#,',_)-(Ai -#tl )'-'] + ie, E{[I +it, .-I(h,,(.\" l )-ul(n.))](112)~.,((,,~,~.) Further, according to Condition A' aitd formula(7).with {,, and 11,~ replaced by ~o and 0 ,l, ~, respectively, we obtain ~f ~o~ t~_)exp{-it2a ~ dr2 ~eOlo.O(tt I,i ~ = a 15) determines(according to (8)) the regular conditional distribution Q~.l~176 I ") of the variable ~o. Further, according to Condition A' aitd formula(7).with {,, and 11,~ replaced by ~o and 0 ,l, ~, respectively, we obtain ~f ~o~ t~_)exp{-it2a ~ dr2 ~eOlo.O(tt I,i ~ = a 15) determines(according to (8)) the regular conditional distribution Q~.l~176 I ") of the variable ~o.…”

mentioning

confidence: 99%