On Tightness of the Skew Random Walks

Skew Brownian motion was introduced by Itô and Mckean 1 to furnish a construction of certain stochastic processes related to Feller’s classification of second-order differential operators associated with diffusion processes see also Section 4.2 in 2 . For α ∈ 0, 1 , the α-skew Brownian motion is defined as a one-dimensional Markov process with the same transition mechanism as of the usual Brownian motion, with the only exception that the excursions away from zero are assigned a positive sign with probability α and a negative sign with probability 1 − α. The signs form an i.i.d. sequence and are chosen independently of the past history of the process. If α 1/2, the process is the usual Brownian motion. Formally, the α-skew random walk on Z starting at 0 is defined as the birth-death Markov chain S α {S α k ; k ≥ 0}with S0 0 and one-step transition probabilities given by