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The paper deals with concentration inequalities for locally small incrementsof compound empirical processes. In the asymptotic theory of $m$-estimationsuch inequalities play an essential role in deriving convergence rates forsolutions of the sample average approximation method to solve compoundstochastic programs, and in particular for $m$-estimators. We developinequalities dependent on the sample sizes with explicit terms instead ofunspecified universal constants. They are applied to study the Sample AverageApproximation method for compound stochastic programs. Nonasymptotic upperestimates for the deviation probabilities of the optimal solutions are derivedwhich are dependent on the sample sizes. They allow to conclude immediatelyconvergence rates for the optimal solutions. In the special case of classicalrisk neutral stochastic programs, we end up with upper estimates of deviationprobabilities for $m$-estimators, and their convergence rates. Moreover, we mayalso demonstrate how to apply the results to sample average approximation ofrisk averse stochastic programs. In this respect we consider stochasticprograms expressed in terms of absolute semideviations and Average Value atRisk. The investigations are based on concentration inequalities from therecent contribution Kratschmer(2024).

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