Time vs. Ensemble Averages for Nonstationary Time Series

TL;DR

This paper investigates the convergence of sliding window time averages in stationary increment processes and argues that Tchebyshev's Theorem can be used to construct ensemble averages from single time series with periodicity.

Contribution

It demonstrates the limitations of Tchebyshev's Theorem for time averages and proposes its application for ensemble averages in nonstationary series with periodicity.

Findings

Tchebyshev's Theorem conditions are violated for time averages of stationary increments.

Ensemble averages can be constructed from single series with periodicity.

Time averages may not converge to a limit in probability for certain processes.

Abstract

We analyze the question whether sliding window time averages applied to stationary increment processes converge to a limit in probability. The question centers on averages, correlations, and densities constructed via time averages of the increment x(t,T)=x(t+T)-x(t)and the assumption is that the increment is distributed independently of t. We show that the condition for applying Tchebyshev's Theorem to time averages of functions of stationary increments is strongly violated. We argue that, for both stationary and nonstationary increments, Tchebyshev's Theorem provides the basis for constructing emsemble averages and densities from a single, historic time series if, as in FX markets, the series shows a definite statistical periodicity on the average.