Upper bounds on product and multiplier empirical processes

Shahar Mendelson

doi:10.1016/j.spa.2016.04.019

Upper bounds on product and multiplier empirical processes

Shahar Mendelson

Research output: Contribution to journal › Article › peer-review

36 Citations (Scopus)

Abstract

We study two empirical processes of special structure: firstly, the centred multiplier process indexed by a class F, f→|∑_i=1 ^N(ξ_if(X_i)−Eξf)|, where the i.i.d. multipliers (ξ_i)_i=1 ^N need not be independent of (X_i)_i=1 ^N, and secondly, (f,h)→|∑_i=1 ^N(f(X_i)h(X_i)−Efh)|, the centred product process indexed by the classes F and 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 γ-functionals.

Original language	English
Pages (from-to)	3652-3680
Number of pages	29
Journal	Stochastic Processes and their Applications
Volume	126
Issue number	12
DOIs	https://doi.org/10.1016/j.spa.2016.04.019
Publication status	Published - 1 Dec 2016
Externally published	Yes

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AB - We study two empirical processes of special structure: firstly, the centred multiplier process indexed by a class F, f→|∑i=1 N(ξif(Xi)−Eξf)|, where the i.i.d. multipliers (ξi)i=1 N need not be independent of (Xi)i=1 N, and secondly, (f,h)→|∑i=1 N(f(Xi)h(Xi)−Efh)|, the centred product process indexed by the classes F and 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 γ-functionals.

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JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

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Upper bounds on product and multiplier empirical processes

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