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 5001969&_mathId=si1.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=db203c61493405195e6adbf851f0ebdc" title="Click to view the MathML source">F$\mathcal{F}$ with polynomially increasing covering numbers on a measurable space 5001969&_mathId=si2.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=0403d946b7d0f4fa553fe391e305c9a7" title="Click to view the MathML source">(X,X)$(X,\mathcal{X})$ together with a sequence of i.i.d. 5001969&_mathId=si3.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=4d0102339f9fc371d6d94d56e98dac26" title="Click to view the MathML source">X$X$-valued random variables 5001969&_mathId=si4.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=cbbb908d6d0fb4fee8f93770aef7ee01" title="Click to view the MathML source">ξ₁,…,ξ_n${\xi }_1,\dots ,{\xi }_n$, and give a good estimate on the tail behaviour of 5001969&_mathId=si5.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=f924b26a597f6974900452f779e327f0">5001969-si5.gif">${sup}_{f\in \mathcal{F}}{\sum }_{j=1}^{n}f\left({\xi }_j\right)$ if the relations 5001969&_mathId=si6.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=5fdbad4aedb695231ce8feac2deb449b" title="Click to view the MathML source">sup_x∈X|f(x)|≤1${sup}_{x\in X}\left|f\left(x\right)\right|\le 1$, 5001969&_mathId=si7.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=b6e949af33a3a102c010f64c35cbdd99" title="Click to view the MathML source">Ef(ξ₁)=0$Ef\left({\xi }_1\right)=0$ and 5001969&_mathId=si8.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=b356b2f2c622042535c6fef0820199d0" title="Click to view the MathML source">Ef(ξ₁)²<σ²$Ef{\left({\xi }_1\right)}^2<{\sigma }^2$ hold with some 5001969&_mathId=si9.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=9ec1fba11c0a2b5f5b2a09df75739f9d" title="Click to view the MathML source">0≤σ≤1$0\le \sigma \le 1$ for all 5001969&_mathId=si10.gif&_user=111111111&_pii=S0304414915001969&_rdoc=1&_issn=03044149&md5=9e423e46d71f1343a5720a1f65320948" title="Click to view the MathML source">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.