Upper bounds on product and multiplier empirical processes

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• 作者：Shahar Mendelson ^{shahar@tx.technion.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Empirical processes ; Generic chaining
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：126
• 期：12
• 页码：3652-3680
• 全文大小：343 K

文摘

We study two empirical processes of special structure: firstly, the centred multiplier process indexed by a class F$F$, $f\to \left|{\sum }_{i=1}^N({\xi }_{i}f\left(X_i\right)-\mathbb{E}\xi f)\right|$, where the i.i.d. multipliers ${\left({\xi }_i\right)}_{i=1}^N$ need not be independent of 5a16ef9d6d8d7b8a8ac21">${\left(X_i\right)}_{i=1}^N$, and secondly, $(f,h)\to \left|{\sum }_{i=1}^N(f\left(X_i\right)h\left(X_i\right)-\mathbb{E}fh)\right|$, the centred product process indexed by the classes F$F$ and H$H$.

We use chaining methods to obtain high probability upper bounds on the suprema of the two processes using a natural variation of Talagrand’s 35a98" title="Click to view the MathML source">γ$\gamma$-functionals.