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Register a new userCramer-type Moderate Deviation Theorems for Nonnormal Approximation
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A Cramer-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramer-type moderate result using Steins method when the limiting was a normal distribution. In this paper, Cramer-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Steins coupling. In particular, a Cramer-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.
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In this paper we study the moderate deviations for the magnetization of critical Curie-Weiss model. Chen, Fang and Shao considered a similar problem for non-critical model by using Stein method. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cramer-type result.
Let {(X_i,Y_i)}_{i=1}^n be a sequence of independent bivariate random vectors. In this paper, we establish a refined Cramer 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 Cramer (1938) theorem and the self-normalized Cramer 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. Specifically, by applying our new framework on general self-normalized sum, we significantly improve Cramer type moderate deviation theorems for one-dependent random variables, geometrically beta-mixing random variables and causal processes under geometrical moment contraction. As an additional application, we also derive the Cramer type moderate deviation theorems for self-normalized winsorized mean.
Let $(xi_i,mathcal{F}_i)_{igeq1}$ be a sequence of martingale differences. Set $S_n=sum_{i=1}^nxi_i $ and $[ S]_n=sum_{i=1}^n xi_i^2.$ We prove a Cramer type moderate deviation expansion for $mathbf{P}(S_n/sqrt{[ S]_n} geq x)$ as $nto+infty.$ Our results partly extend the earlier work of [Jing, Shao and Wang, 2003] for independent random variables.
Let $(X _i)_{igeq1}$ be a stationary sequence. Denote $m=lfloor n^alpha rfloor, 0< alpha < 1,$ and $ k=lfloor n/m rfloor,$ where $lfloor a rfloor$ stands for the integer part of $a.$ Set $S_{j}^circ = sum_{i=1}^m X_{m(j-1)+i}, 1leq j leq k,$ and $ (V_k^circ)^2 = sum_{j=1}^k (S_{j}^circ)^2.$ We prove a Cramer type moderate deviation expansion for $mathbb{P}( sum_{j=1}^k S_{j}^circ /V_k^circ geq x)$ as $nto infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed.
Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$ S^kappa_t=frac{1}{t^kappa}int_0^tH(X_s)ds, ttoinfty $$ for an ergodic diffusion process $X_t$ under $0.5<kappa<1$ and appropriate $H$. We mean a decomposition with ``corrector: $$ frac{1}{t^kappa}int_0^tH(X_s)ds={rm corrector}+frac{1}{t^kappa}underbrace{M_t}_{rm martingale}. $$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$ frac{1}{t^kappa}M_t, ttoinfty. $ Starting from Bayer and Freidlin, cite{BF}, and finishing by Wus papers cite{Wu1}-cite{WuH}, in the MD study Laplaces transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ....
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