Nonnormal approximation by Stein's method of exchangeable pairs with application to the Curie--Weiss model

Let $(W,W')$ be an exchangeable pair. Assume that $$E(W-W'|W)=g(W)+r(W),$$ where $g(W)$ is a dominated term and $r(W)$ is negligible. Let $G(t)=\int_0^tg(s)\,ds$ and define $p(t)=c_1e^{-c_0G(t)}$, where $c_0$ is a properly chosen constant and $c_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt$. Let $Y$ be a random variable with the probability density function $p$. It is proved that $W$ converges to $Y$ in distribution when the conditional second moment of $(W-W')$ given $W$ satisfies a law of large numbers. A Berry-Esseen type bound is also given. We use this technique to obtain a Berry-Esseen error bound of order $1/\sqrt{n}$ in the noncentral limit theorem for the magnetization in the Curie-Weiss ferromagnet at the critical temperature. Exponential approximation with application to the spectrum of the Bernoulli-Laplace Markov chain is also discussed.Comment: Published in at http://dx.doi.org/10.1214/10-AAP712 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cram\'er type moderate deviation theorems for self-normalized processes

Cram\'er type moderate deviation theorems quantify the accuracy of the relative error of the normal approximation and provide theoretical justifications for many commonly used methods in statistics. In this paper, we develop a new randomized concentration inequality and establish a Cram\'er type moderate deviation theorem for general self-normalized processes which include many well-known Studentized nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized $U$-statistics.Comment: Published at http://dx.doi.org/10.3150/15-BEJ719 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Chung type law of the iterated logarithm for subsequences of a Wiener process

AbstractLet {W(t), t ⩾ 0} be a standard Wiener process and {tn, n ⩾ 1} be an increasing sequence of positive numbers with tn → ∞. We consider the limit inf for the maximum of a subsequence |W(ti)|. It is proved in this paper that the Chung law of the iterated logarithm holds, i.e., lim infn→∞(tnlog log tn)−12 maxi〈n |W(ti)| = π√8 a.s. if tn − tn−1 = o(tnlog log tn) and that the assumption tn − tn−1 = o(tnlog log tn) cannot be weakened to tn − tn−1 = O(tnlog log tn)

Self-normalized Cram\'{e}r 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 Cram\'{e}r 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 $P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-\Phi (x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ415 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

On non-stationary threshold autoregressive models

In this paper we study the limiting distributions of the least-squares estimators for the non-stationary first-order threshold autoregressive (TAR(1)) model. It is proved that the limiting behaviors of the TAR(1) process are very different from those of the classical unit root model and the explosive AR(1).Comment: Published in at http://dx.doi.org/10.3150/10-BEJ306 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm