  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.
