首页/论文/1901.03348

arXiv 1901.033482019-01-10math.STstat.TH

On large deviations for sums of discrete m-dependent random variables

Vydas Čekanavičius, Palaniappan Vellaisamy

备注：: To appear in "Stochastics"

英文摘要：

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Large deviations for sums defined on a Galton-Watson process

In this paper we study the large deviation behavior of sums of i.i.d. random variables X_i defined on a supercritical Galton-Watson process Z. We assume the finiteness of the moments EX_1^2 and EZ_1log Z_1. The underlying interplay of the partial sums of the X_i and the lower deviation probabilities of Z is clarified. Here we heavily use lower deviation probability results on Z we recently published in [FW06].

2006-05-23math.PRmath.STstat.TH

Self-normalized Cramér type moderate deviations for the maximum of sums

Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\sum_{i=1}^nX_i$ and $V^2_n=\sum_{i=1}^nX^2_i$. A Cramér type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-Φ(x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.

2013-07-23math.STstat.TH

Cramér Type Moderate Deviations for Random Fields

We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

2019-02-07math.STstat.TH

Discrete approximations for sums of m-dependent random variables

Sums of of 1-dependent integer-valued random variables are approximated by compound Poisson, negative binomial and Binomial distributions and signed compound Poisson measures. Estimates are obtained for total variation and local metrics. The results are then applied to statistics of $m$-dependent $(k_1,k_2)$ events and 2-runs. Heinrich's method and smoothing properties of convolutions are used for the proofs.

2015-11-04math.STstat.TH

Refined Cramér Type Moderate Deviation Theorems for General Self-normalized Sums with Applications to Dependent Random Variables and Winsorized Mean

Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramér type moderate deviation theorem for the general self-normalized sum \sum_{i=1}^n X_i/(\sum_{i=1}^n Y_i^2)^{1/2}, which unifies and extends the classical Cramér (1938) theorem and the self-normalized Cramér type moderate deviation theorems by Jing, Shao and Wang (2003) as well as the further refined version by Wang (2011). The advantage of our result is evidenced through successful applications to weakly dependent random variables and self-normalized winsorized mean. Speci…

2021-07-28math.PRmath.STstat.TH

密集依赖随机变量随机和的性质与偏差

Properties and Deviations of Random Sums of Densely Dependent Random Variables

本文提出一种不依赖完整协方差结构的新方差恒等式，在密集依赖条件下建立弱大数定律和更精确的Hoeffding不等式，支持有限样本推断并推广至估计方程。

2023-10-17math.STstat.TH

更多相关论文收起更多论文

Dirichlet加权和的尖锐偏差界及其在贝叶斯算法分析中的应用

Sharp Deviations Bounds for Dirichlet Weighted Sums with Application to analysis of Bayesian algorithms

本文推导Dirichlet加权和的尖锐非渐近偏差界，基于新颖积分表示与复分析，推广Beta分布结果，并应用于贝叶斯bootstrap及多臂老虎机TS算法，使遗憾界独立于支撑大小。

2026-10-06cs.LGmath.PRmath.ST

Approximations for the boundary crossing probabilities of moving sums of normal random variables

In this paper we study approximations for boundary crossing probabilities for the moving sums of i.i.d. normal random variables. We propose approximating a discrete time problem with a continuous time problem allowing us to apply developed theory for stationary Gaussian processes and to consider a number of approximations (some well known and some not). We bring particular attention to the strong performance of a newly developed approximation that corrects the use of continuous time results in a discrete time setting. Results of extensive numerical comparisons are reported. These results show …

2019-04-27math.STstat.TH

Approximations for the boundary crossing probabilities of moving sums of random variables

In this paper we study approximations for the boundary crossing probabilities of moving sums of i.i.d. normal r.v. We approximate a discrete time problem with a continuous time problem allowing us to apply established theory for stationary Gaussian processes. By then subsequently correcting approximations for discrete time, we show that the developed approximations are very accurate even for small window length. Also, they have high accuracy when the original r.v. are not exactly normal and when the weights in the moving window are not all equal. We then provide accurate and simple approximati…

2020-01-03math.STstat.TH

Robust and efficient mean estimation: an approach based on the properties of self-normalized sums

Let $X$ be a random variable with unknown mean and finite variance. We present a new estimator of the mean of $X$ that is robust with respect to the possible presence of outliers in the sample, provides tight sub-Gaussian deviation guarantees without any additional assumptions on the shape or tails of the distribution, and moreover is asymptotically efficient. This is the first estimator that provably combines all these qualities in one package. Our construction is inspired by robustness properties possessed by the self-normalized sums. Theoretical findings are supplemented by numerical simula…

2021-12-30math.STstat.TH

A Large Deviation Inequality for $β$-mixing Time Series and its Applications to the Functional Kernel Regression Model

We give a new large deviation inequality for sums of random variables of the form $Z_k = f(X_k,X_t)$ for $k,t\in \mathbb{N}$, $t$ fixed, where the underlying process $X$ is $β$-mixing. The inequality can be used to derive concentration inequalities. We demonstrate its usefulness in the functional kernel regression model of Ferraty et al. (2007) where we study the consistency of dynamic forecasts.

2017-07-04math.STstat.TH

Stability of Random Sums

When the distribution of a random (N) sum of independent copies of a r.v X is of the same type as that of X we say that X is N-sum stable. In this paper we consider a generalization of stability of geometric sums by studying distributions that are stable under summation w.r.t Harris law. We show that the notion of stability of random sums can be extended to include the case when X is discrete. Finally we propose a method to identify the probability law of N for which X is N-sum stable. See also Satheesh and Nair (2002), (Some classes of distributions on ther non-negative lattice, J. Ind. Stati…

2003-11-20math.PRmath.STstat.TH

Dimension-free Bounds for Sums of Independent Matrices and Simple Tensors via the Variational Principle

We consider the deviation inequalities for the sums of independent $d$ by $d$ random matrices, as well as rank one random tensors. Our focus is on the non-isotropic case and the bounds that do not depend explicitly on the dimension $d$, but rather on the effective rank. In an elementary and unified manner, we show the following results: 1) A deviation bound for the sums of independent positive-semi-definite matrices. This result complements the dimension-free bound of Koltchinskii and Lounici [Bernoulli, 2017] on the sample covariance matrix in the sub-Gaussian case. 2) A new bound for truncat…

2022-05-26math.PRmath.STstat.TH

部分和自正则化最大值的精确Cramér型中偏差界

Sharp Cramér-type moderate deviation bounds for the self-normalized maximum of partial sums

Dalian University of Technology（大连理工大学）；Southern University of Science and Technology（南方科技大学）

本文针对$2<p\le3$的有限$p$阶绝对矩条件，建立部分和自正则化最大值的精确Cramér型中偏差界，证明Shao2010的猜想，通过反射游走表示等方法完成证明。

2026-10-16math.PRmath.STstat.TH

可计算的强逼近界及其应用

Computable Bounds for Strong Approximations with Applications

Harvard University（哈佛大学）

本文提出了一种可计算的KMT不等式，用于有界独立同分布随机变量的部分和，同时给出了在标准差未知时的经验版本，并展示了其在在线突变点检测和首次击中时间概率中的应用。

2026-05-18math.PRmath.STstat.TH

Shannon-McMillan theorems for discrete random fields along curves and lower bounds for surface-order large deviations

The notion of a surface-order specific entropy h_c(P) of a two-dimensional discrete random field P along a curve c is introduced as the limit of rescaled entropies along lattice approximations of the blowups of c. Existence is shown by proving a corresponding Shannon-McMillan theorem. We obtain a representation of h_c(P) as a mixture of specific entropies along the tangent lines of c. As an application, the specific entropy along curves is used to refine Foellmer and Ort's lower bound for the large deviations of the empirical field of an attractive Gibbs measure from its ergodic behavior in th…

2007-10-29math.PRmath.STstat.TH

Sharp Concentration Results for Heavy-Tailed Distributions

We obtain concentration and large deviation for the sums of independent and identically distributed random variables with heavy-tailed distributions. Our concentration results are concerned with random variables whose distributions satisfy $\mathbb{P}(X>t) \leq {\rm e}^{- I(t)}$, where $I: \mathbb{R} \rightarrow \mathbb{R}$ is an increasing function and $I(t)/t \rightarrow α\in [0, \infty)$ as $t \rightarrow \infty$. Our main theorem can not only recover some of the existing results, such as the concentration of the sum of subWeibull random variables, but it can also produce new results for th…

2022-07-25cs.LGmath.PRmath.ST