Fine Topology and Estimates for Potentials and Subharmonic Functions

A theorem on higher-order differences of two-state Markov chains

“…The main result of this paper, Theorem 1, is a limiting theorem for such chains: it assert the existence of the limit of kth order absolute differences taken from progressive terms of a given series (ξ n ) ∞ n=0 , when k converges to ∞ remaining on "large" subsets E ⊆ N of natural series N. The "size" of such sets E is described in terms of some discrete capacity: such sets E are thick sets, defined by means of Wiener criterion type relation from potential theory (see, e.g., [8] and [9]). The limiting process, whose existence asserts Theorem 1, is the equi-distributed random sequence.…”