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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.
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 res
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
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 $mathcal{R}^p$ uniform