Concentration inequalities for locally small increments of compound empirical processes with applications to solutions of compound and risk averse stochastical programming

The paper deals with concentration inequalities for locally small increments of compound empirical processes. In the asymptotic theory of \(m\)-estimation such inequalities play an essential role in deriving convergence rates for solutions of the sample average approximation method to solve compound...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2024-08
Main Author: Kratschmer, Volker
Format: Article
Language:English
Subjects:
Approximation
Asymptotic properties
Continuity (mathematics)
Divergence
Empirical analysis
Mathematical analysis
Mixed integer
Risk
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The paper deals with concentration inequalities for locally small increments of compound empirical processes. In the asymptotic theory of \(m\)-estimation such inequalities play an essential role in deriving convergence rates for solutions of the sample average approximation method to solve compound stochastic programs, and in particular for \(m\)-estimators. We develop inequalities dependent on the sample sizes with explicit terms instead of unspecified universal constants. They are applied to study the Sample Average Approximation method for compound stochastic programs. Nonasymptotic upper estimates for the deviation probabilities of the optimal solutions are derived which are dependent on the sample sizes. They allow to conclude immediately convergence rates for the optimal solutions. In the special case of classical risk neutral stochastic programs, we end up with upper estimates of deviation probabilities for \(m\)-estimators, and their convergence rates. Moreover, we may also demonstrate how to apply the results to sample average approximation of risk averse stochastic programs. In this respect we consider stochastic programs expressed in terms of absolute semideviations and Average Value at Risk. The investigations are based on concentration inequalities from the recent contribution Kratschmer(2024).
ISSN:2331-8422