Discrete Bessel process and its properties

This paper considers a discrete analogue of a three-dimensional Bessel process — a certain discrete random process, which converges to a continuous Bessel process in the sense of the Donsker–Prokhorov invariance principle, and which has an elementary path structure such as in the case of a simple random walk.
The paper introduces four equivalent definitions of a discrete Bessel process, which describe this process from different points of view. The study of this process shows that its relationship to the simple random walk repeats the well-known properties which connect the continuous three-dimensional Bessel process with the standard Brownian motion. Thus, hereby we state and prove discrete versions of Pitman's theorem, Williams theorem on Brownian path decomposition, and some other statements related to these two processes.