An Ornstein-Uhlenbeck process associated to self-normalized sums

TL;DR

This paper demonstrates that self-normalized sums of symmetric i.i.d. variables converge to an Ornstein-Uhlenbeck process, highlighting its stationarity, and provides methods for simulating functionals of this process.

Contribution

It establishes the convergence of self-normalized sums to an Ornstein-Uhlenbeck process and introduces simulation techniques for its functionals.

Findings

Self-normalized sums converge to OU process in $C[0, \infty)$.

OU process is stationary, unlike Brownian motion.

Method for simulating functionals of the OU process.

Abstract

We consider an Ornstein-Uhleneck (OU) process associated to self-normalised sums in i.i.d. symmetric random variables from the domain of attraction of $N(0, 1)$ distribution. We proved the self-normalised sums converge to the OU process (in $C[0, \infty)$). Importance of this is that the OU process is a stationary process as opposed to the Brownian motion, which is a non-stationary distribution (see for example, the invariance principle proved by Csorgo et al (2003, Ann Probab) for self-normalised sums that converges to Brownian motion). The proof uses recursive equations similar to those that arise in the area of stochastic approximation and it shows (through examples) that one can simulate any functionals of any segment of the OU process. The similar things can be done for any diffusion process as well.