Abstract
We use the Malliavin calculus to prove a new abstract concentration inequality result for zero mean, Malliavin differentiable random variables which admit densities. We demonstrate the applicability of the result by deriving two new concrete concentration inequalities, one relating to an integral functional of a fractional Brownian motion process, and the other relating to the centered maximum of a finite sum of Normal random variables. These concentration inequalities are, to the best of our knowledge, largely unattainable via existing methods other than those which are the subject of this paper.
Highlights
IntroductionConcentration inequalities characterize the rate of decay of the tail distribution of a random variable
We demonstrate the applicability of the result by deriving two new concrete concentration inequalities, one relating to an integral functional of a fractional Brownian motion process, and the other relating to the centered maximum of a finite sum of Normal random variables
Concentration inequalities characterize the rate of decay of the tail distribution of a random variable
Summary
Concentration inequalities characterize the rate of decay of the tail distribution of a random variable. If Z is a random variable, concentration inequalities are typically some variant of an upper or lower bound on the quantity P (Z ≥ z). Z where Z is any random variable and z > 0, and Bernstein’s inequality: 1n nz. I=1 where Z1, ..., Zn are independent Bernoulli random variables uniformly distributed on the set {−1, +1}. We derive new abstract concentration inequalities for Malliavin differentiable random variables. This paper is organized as follows: in Section 2 we derive and present concentration inequalities which are the main result of the paper; and in Section 3 we apply the main result to compute new bounds on the tail distributions of concrete random variables
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