Asymptotic properties of realized power variations and related functionals of semimartingales

TL;DR

This paper investigates the asymptotic behavior of sums of functions evaluated at semimartingale increments, establishing laws of large numbers, convergence rates, and central limit theorems for these functionals.

Contribution

It provides new theoretical results on the convergence and distributional properties of realized power variations of semimartingales, including various laws of large numbers and CLTs.

Findings

Convergence in probability of sums of power functionals.

Explicit rates of convergence for these sums.

Central limit theorems describing their asymptotic distribution.

Abstract

This paper is concerned with the asymptotic behavior of sums of terms which are a test function f evaluated at successive increments of a discretely sampled semimartingale. Typically the test function is a power function (when the power is 2 we get the realized quadratic variation) . We prove a variety of ``laws of large numbers'', that is convergence in probability of these sums, sometimes after normalization. We also exhibit in many cases the rate of convergence, as well as associated central limit theorems.