Large Deviation Theorem Research Articles

Convergence results on multitype, multivariate branching random walks

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L−⊂L, where L is an open set in ℂd, and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

Convergence results on multitype, multivariate branching random walks

We consider a multi-type branching random walk on d-dimensional Euclidian space. The~uniform convergence, as n goes to infinity, of a scaled version of the Laplace transform of the point process given by the nth generation particles of each type is obtained. Similar results in the one-type case, where the transform gives a martingale, were obtained in Biggins (1992) and Barral (2001). This uniform convergence of transforms is then used to obtain limit results for numbers in the underlying point processes. Supporting results, which are of interest in their own right, are obtained on (i) ‘Perron-Frobenius theory’ for matrices that are smooth functions of a variable λ∈L and are nonnegative when λ∈L −⊂L, where L is an open set in ℂ d , and (ii) saddlepoint approximations of multivariate distributions. The saddlepoint approximations developed are strong enough to give a refined large deviation theorem of Chaganty and Sethuraman (1993) as a by-product.

On complexity of multistage stochastic programs

In this paper we derive estimates of the sample sizes required to solve a multistage stochastic programming problem with a given accuracy by the (conditional sampling) sample average approximation method. The presented analysis is self-contained and is based on a relatively elementary, one-dimensional, Cramér's Large Deviations Theorem.

Sample-path large deviations for tandem and priority queues with Gaussian inputs

This paper considers Gaussian flows multiplexed in a queueing network. A single node being a useful but often incomplete setting, we examine more advanced models. We focus on a (two-node) tandem queue, fed by a large number of Gaussian inputs. With service rates and buffer sizes at both nodes scaled appropriately, Schilder’s sample-path large-deviations theorem can be applied to calculate the asymptotics of the overflow probability of the second queue. More specifically, we derive a lower bound on the exponential decay rate of this overflow probability and present an explicit condition for the lower bound to match the exact decay rate. Examples show that this condition holds for a broad range of frequently used Gaussian inputs. The last part of the paper concentrates on a model for a single node, equipped with a priority scheduling policy. We show that the analysis of the tandem queue directly carries over to this priority queueing system.

Approximation by the Compound Poisson Law

We consider a specification of the convergence of distributions of sums of independent nonnegative integer random variables to the compound Poisson distribution. The theorems of large deviations and asymptotic expansion under some weak conditions are proved.

A remark on a large deviation theorem for Markov chain with a finite number of states

A remark on a large deviation theorem for Markov chain with a finite number of states

On Bounds for Moderate Deviations for Student's Statistic

Let $X_1,X_2,\dots$ be independent random variables with zero means and finite variances. In this paper we prove lower bounds for a Cramer-type large deviation theorem for self-normalized sums which imply that the bounds obtained by Jing, Shao, and Wang [{\em Ann. Probab.}, 31 (2003), pp. 2167--2215] are sharp.

Large deviations in Axiom A endomorphisms

By applying a general large-deviation theorem of Kifer and Ruelle's Smale space technique, some large-deviation estimates are proved for Axiom A endomorphisms.

Self-normalized Cramér-type large deviations for independent random variables

Let $X_1, X_2, \ldots $ be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramér-type large deviation result for the standardized partial sums. In this paper, we show that a Cramér-type large deviation theorem holds for self-normalized sums only under a finite $(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show $P(S_n /V_n \geq x)=\break (1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$ for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} = (\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.

Extremal slabs in the cube and the Laplace transform

We study the volume of symmetric slabs in the unit cube. We show that, for t< 3 4 , the slab parallel to a face has the minimal volume among all symmetric slabs with width t. For large width, we prove the asymptotic extremality of the slab orthogonal to the main diagonal. The proof is based on certain concavity properties of the Laplace transform and on several limit theorems from probability: the central limit theorem and classical principles of moderate and large deviations. Finally, we extend some of the results to more general classes of bodies.

Theorems of Large Deviations in the Approximation by the Compound Poisson Distribution

The paper is devoted to the research of large deviation probabilities in the approximation by compound Poisson law.

Integral Limit Theorems on Large Deviations for Multidimensional Hypergeometric Distribution

Integral large deviation theorems are obtained for multidimensional hypergeometric distribution. These theorems allow us to evaluate the probabilities of large deviations with the remainder term of order~$O(1/N)$. The corresponding hypergeometric distribution of a random vector $(\mu_1\ldots\mu_s)$ has the form {\bf{P}}\{(\mu_1\ldots\mu_s)=(k_1\ldots k_s)\}=\frac{C_{M_1}^{k_1}\cdots C_{M_s}^{k_s}}{C_N^n}\,, and $k_j\le M_j$, $j=1\ldots s$; 0~in the remaining cases.

On Large Deviations in Testing Ornstein-Uhlenbeck Type Models with Delay

We obtain an explicit form of fine large deviation theorems for the log-likelihood ratio in testing models with observed Ornstein-Uhlenbeck processes and get explicit rates of decrease for error probabilities of Neyman-Pearson, Bayes, and minimax tests. We also give expressions for the rates of decrease of error probabilities of Neyman-Pearson tests in models with observed processes solving affine stochastic delay differential equations.

Asymtotical expansions in local limit theorem of large deviations

We continue investigation of the local distribution of multiplicative arithmetic functions from the class M(G). In the present paper the local limit theorem of large deviations with asymptotical expansion of the principal term is obtained.

Large deviation theorems for weighted summation of Gamma-distribution random variables

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Exact rates in Vapnik–Chervonenkis bounds

Vapnik–Chervonenkis bounds on rates of uniform convergence of empirical means to their expectations have been continuously improved over the years since the precursory work in [26]. The result obtained by Talagrand in 1994 [21] seems to provide the final word as far as universal bounds are concerned. However, in the case where there are some additional assumptions on the underlying probability distribution, the exponential rate of convergence can be fairly improved. Alexander [1] and Massart [15] have found better exponential rates (similar to those in Bennett–Bernstein inequalities) under the assumption of a control on the variance of the empirical process. In this paper, the case of a particular distribution is considered for the empirical process indexed by a family of sets, and we provide the exact exponential rate based on large deviations theorems, as predicted by Azencott [2].

On the relationship between the sample path and moment lyapunov exponents for jump linear systems

In this paper, we study the relationship between the sample and moment Lyapunov exponents for jump linear systems. Using a large deviation theorem, a modified version of Arnold's formula for connecting sample path and moment Lyapunov exponents for continuous-time linear stochastic systems is extended to discrete-time jump linear systems. Sample path stability properties of linear stochastic systems are determined by the top Lyapunov exponent and relating sample and moment Lyapunov exponents may be useful for developing computationally efficient methods for determining the almost-sure (sample path) stability of linear stochastic systems.

Large deviation theorems for weighted sums applied to a geographical problem

A large deviation expansion is used to evaluate the impact of errors in a geographical database on the computation of travel times. We work in the framework of discrete random variables and improve a theorem by Book to solve this problem. Simulations are provided to illustrate the methodology.

Large deviation theorems for weighted sums applied to a geographical problem

A large deviation expansion is used to evaluate the impact of errors in a geographical database on the computation of travel times. We work in the framework of discrete random variables and improve a theorem by Book to solve this problem. Simulations are provided to illustrate the methodology.

The local limit theorem of large deviations for multiplicative functions

In the present paper we continue investigation of the local distribution of multiplicative ari­thmetic functions from the class M(G). A local theorem of large deviations is obtained.