scholarly journals Large deviations for exchangeable observations with applications

2004 ◽  
Vol 2004 (55) ◽  
pp. 2947-2958
Author(s):  
Jinwen Chen
Keyword(s):  
Large Deviations ◽  
Large Deviation ◽  
Random Variables ◽  
Moment Conditions

We first prove some large deviation results for a mixture of i.i.d. random variables. Compared with most of the known results in the literature, our results are built on relaxing some restrictive conditions that may not be easy to be checked in certain typical cases. The main feature in our main results is that we require little knowledge of (continuity of) the component measures and/or of the compactness of the support of the mixing measure. Instead, we pose certain moment conditions, which may be more practical in applications. We then use the large deviation approach to study the problem of estimating the component and the mixing measures.

A large deviations heuristic made precise

2000 ◽  
Vol 128 (3) ◽  
pp. 561-569 ◽  
Author(s):  
NEIL O'CONNELL
Keyword(s):  
Large Deviations ◽  
Bin Packing ◽  
Symmetric Functions ◽  
Large Deviation ◽  
Random Variables ◽  
Packing Problem ◽  
Rate Function ◽  
Empirical Measures ◽  
Underlying Distribution ◽  
Uniformly Lipschitz

Sanov's Theorem states that the sequence of empirical measures associated with a sequence of i.d.d. random variables satisfies the large deviation principle (LDP) in the weak topology with rate function given by a relative entropy. We present a derivative which allows one to establish LDPs for symmetric functions of many i.d.d. random variables under the condition that (i) a law of large numbers holds whatever the underlying distribution and (ii) the functions are uniformly Lipschitz. The heuristic (of the title) is that the LDP follows from (i) provided the functions are ‘sufficiently smooth’. As an application, we obtain large deviations results for the stochastic bin-packing problem.

scholarly journals Cramér type large deviations for trimmed L-statistics

2018 ◽  
Vol 37 (1) ◽  
pp. 101-118 ◽  
Author(s):  
Nadezhda Gribkova
Keyword(s):  
Large Deviations ◽  
Weight Function ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Random Variables ◽  
Natural Conditions ◽  
New Approach ◽  
Probabilities Of Large Deviations ◽  
Large Deviation Problem ◽  
Deviation Problem

CRAMÉR TYPE LARGE DEVIATIONS FOR TRIMMED L-STATISTICSIn this paper, we propose a new approach to the investigationof asymptotic properties of trimmed L-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with those in Callaert et al. 1982 – the first and, as far as we know, the single article where some results on probabilities of large deviations for the trimmed L-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed L-statistic by a non-trimmed L-statistic with smooth weight function based onWinsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed L-statistics under quite mild and natural conditions.

scholarly journals Large deviations for a class of tempered subordinators and their inverse processes

2021 ◽  
pp. 1-21
Author(s):  
Nikolai Leonenko ◽  
Claudio Macci ◽  
Barbara Pacchiarotti
Keyword(s):  
Weak Convergence ◽  
Large Deviations ◽  
Large Deviation ◽  
Common Law ◽  
Random Variables ◽  
Tempered Distributions ◽  
One Dimensional ◽  
Large Deviation Principles ◽  
Time Changes ◽  
Non Gaussian

We consider a class of tempered subordinators, namely a class of subordinators with one-dimensional marginal tempered distributions which belong to a family studied in [3]. The main contribution in this paper is a non-central moderate deviations result. More precisely we mean a class of large deviation principles that fill the gap between the (trivial) weak convergence of some non-Gaussian identically distributed random variables to their common law, and the convergence of some other related random variables to a constant. Some other minor results concern large deviations for the inverse of the tempered subordinators considered in this paper; actually, in some results, these inverse processes appear as random time-changes of other independent processes.

Large-deviation asymptotics of condition numbers of random matrices

2021 ◽  
Vol 58 (4) ◽  
pp. 1114-1130
Author(s):  
Martin Singull ◽  
Denise Uwamariya ◽  
Xiangfeng Yang
Keyword(s):  
Large Deviations ◽  
Condition Number ◽  
Normal Condition ◽  
Large Deviation ◽  
Random Variables ◽  
Condition Numbers ◽  
General Distribution ◽  
Unit Variance ◽  
Standard Normal ◽  
Independent And Identically Distributed

AbstractLet $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.

scholarly journals Limit theorems for ratios of order statistics from uniform distributions

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Shoufang Xu ◽  
Changlin Mei ◽  
Yu Miao
Keyword(s):  
Order Statistics ◽  
Limit Theorems ◽  
Complete Convergence ◽  
Large Deviation ◽  
Random Variables ◽  
Independent Random Variables ◽  
Moment Conditions ◽  
Uniform Distributions ◽  
Fixed Sample ◽  
Large Numbers

AbstractLet $\{X_{ni}, 1 \leq i \leq m_{n}, n\geq 1\}${Xni,1≤i≤mn,n≥1} be an array of independent random variables with uniform distribution on $[0, \theta _{n}]$[0,θn], and $\{X_{n(k)}, k=1, 2, \ldots , m_{n}\}${Xn(k),k=1,2,…,mn} be the kth order statistics of the random variables $\{X_{ni}, 1 \leq i \leq m_{n}\}${Xni,1≤i≤mn}. We study the limit properties of ratios $\{R_{nij}=X_{n(j)}/X_{n(i)}, 1\leq i < j \leq m_{n}\}${Rnij=Xn(j)/Xn(i),1≤i<j≤mn} for fixed sample size $m_{n}=m$mn=m based on their moment conditions. For $1=i < j \leq m$1=i<j≤m, we establish the weighted law of large numbers, the complete convergence, and the large deviation principle, and for $2=i < j \leq m$2=i<j≤m, we obtain some classical limit theorems and self-normalized limit theorems.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

Sign in / Sign up

Export Citation Format

Share Document