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Register a new userNonuniform Berry-Esseen bound for self-normalized martingales
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We give a nonuniform Berry-Esseen bound for self-normalized martingales, which bridges the gap between the result of Haeusler (1988) and Fan and Shao (2018). The bound coincides with the nonuniform Berry-Esseen bound of Haeusler and Joos (1988) for standardized martingales. As a consequence, a Berry-Esseen bound is obtained.
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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.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
We prove an Erdos-Feller-Kolmogorov-Petrowsky law of the iterated logarithm for self-normalized martingales. Our proof is given in the framework of the game-theoretic probability of Shafer and Vovk. As many other game-theoretic proofs, our proof is self-contained and explicit.
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.
In this article, we are interested in the normal approximation of the self-normalized random vector $Big(frac{sum_{i=1}^{n}X_{i1}}{sqrt{sum_{i=1}^{n}X_{i1}^2}},dots,frac{sum_{i=1}^{n}X_{ip}}{sqrt{sum_{i=1}^{n}X_{ip}^2}}Big)$ in $mathcal{R}^p$ uniformly over the class of hyper-rectangles $mathcal{A}^{re}={prod_{j=1}^{p}[a_j,b_j]capmathcal{R}:-inftyleq a_jleq b_jleq infty, j=1,ldots,p}$, where $X_1,dots,X_n$ are non-degenerate independent $p-$dimensional random vectors with each having independent and identically distributed (iid) components. We investigate the optimal cut-off rate of $log p$ in the uniform central limit theorem (UCLT) under variety of moment conditions. When $X_{ij}$s have $(2+delta)$th absolute moment for some $0< deltaleq 1$, the optimal rate of $log p$ is $obig(n^{delta/(2+delta)}big)$. When $X_{ij}$s are independent and identically distributed (iid) across $(i,j)$, even $(2+delta)$th absolute moment of $X_{11}$ is not needed. Only under the condition that $X_{11}$ is in the domain of attraction of the normal distribution, the growth rate of $log p$ can be made to be $o(eta_n)$ for some $eta_nrightarrow 0$ as $nrightarrow infty$. We also establish that the rate of $log p$ can be pushed to $log p =o(n^{1/2})$ if we assume the existence of fourth moment of $X_{ij}$s. By an example, it is shown however that the rate of growth of $log p$ can not further be improved from $n^{1/2}$ as a power of $n$. As an application, we found respecti
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