Some limit theorems for rescaled Wick powers

TL;DR

This paper proves the strong L2 convergence of rescaled Wick powers and characterizes their limits as normal or log-normal distributions, using estimates of Wick products and second quantization operators.

Contribution

It introduces new convergence results for rescaled Wick powers and provides explicit representations of their limits, advancing the understanding of their asymptotic behavior.

Findings

Rescaled Wick powers converge strongly in L2(P) as the power index increases.

The limits of these Wick powers are normally or log-normally distributed.

The paper offers explicit formulas for the limiting distributions.

Abstract

We establish the strong L2(P)-convergence of properly rescaled Wick powers as the power index tends to infinity. The explicit representation of such limit will also provide the convergence in distribution to normal and log-normal random variables. The proofs rely on some estimates for the L2(P)-norm of Wick products and on the properties of second quantization operators.