541 research outputs found

### 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

### 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

### Normal approximation for nonlinear statistics using a concentration inequality approach

Let $T$ be a general sampling statistic that can be written as a linear
statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds
for $T$ are obtained. The bounds are the best possible for many known
statistics. Applications to U-statistics, multisample U-statistics,
L-statistics, random sums and functions of nonlinear statistics are discussed.Comment: Published at http://dx.doi.org/10.3150/07-BEJ5164 in the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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