541 research outputs found

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

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    Let (W,W′)(W,W') be an exchangeable pair. Assume that E(W−W′∣W)=g(W)+r(W),E(W-W'|W)=g(W)+r(W), where g(W)g(W) is a dominated term and r(W)r(W) is negligible. Let G(t)=∫0tg(s) dsG(t)=\int_0^tg(s)\,ds and define p(t)=c1e−c0G(t)p(t)=c_1e^{-c_0G(t)}, where c0c_0 is a properly chosen constant and c1=1/∫−∞∞e−c0G(t) dtc_1=1/\int_{-\infty}^{\infty}e^{-c_0G(t)}\,dt. Let YY be a random variable with the probability density function pp. It is proved that WW converges to YY in distribution when the conditional second moment of (W−W′)(W-W') given WW 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/n1/\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

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    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

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    Let X1,X2,...X_1,X_2,... be independent random variables with zero means and finite variances, and let Sn=∑i=1nXiS_n=\sum_{i=1}^nX_i and Vn2=∑i=1nXi2V^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≤k≤nSk/Vn\max_{1\leq k\leq n}S_k/V_n is obtained. In particular, for identically distributed X1,X2,...,X_1,X_2,..., it is proved that P(max⁡1≤k≤nSk≥xVn)/(1−Φ(x))→2P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-\Phi (x))\rightarrow2 uniformly for 0<x≤o(n1/6)0<x\leq\mathrm{o}(n^{1/6}) under the optimal finite third moment of X1X_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

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    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

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    Let TT 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 TT 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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