Sharp tail distribution estimates for the supremum of a class of sums of i.i.d. random variables

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• 作者：Pé ; ter Major ^{major.peter@renyi.mta.hu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60F10 ; 60G50 ; 60G60
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：126
• 期：1
• 页码：118-137
• 全文大小：279 K

文摘

We take a class of functions F$\mathcal{F}$ with polynomially increasing covering numbers on a measurable space (X,X)$(X,\mathcal{X})$ together with a sequence of i.i.d. X$X$-valued random variables ξ₁,…,ξ_n${\xi }_1,\dots ,{\xi }_n$, and give a good estimate on the tail behaviour of ${sup}_{f\in \mathcal{F}}{\sum }_{j=1}^{n}f\left({\xi }_j\right)$ if the relations sup_x∈X|f(x)|≤1${sup}_{x\in X}\left|f\left(x\right)\right|\le 1$, Ef(ξ₁)=0$Ef\left({\xi }_1\right)=0$ and Ef(ξ₁)²<σ²$Ef{\left({\xi }_1\right)}^2<{\sigma }^2$ hold with some 0≤σ≤1$0\le \sigma \le 1$ for all f∈F$f\in \mathcal{F}$. Roughly speaking this estimate states that under some natural conditions the above supremum is not much larger than the largest element taking part in it. The proof heavily depends on the main result of paper Major (2015). We also present an example that shows that our results are sharp, and compare them with results of earlier papers.