Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience

Abstract

The aim of this note is to correct the errors in the formulation and proof of Lemma 4.1 in [1] and some claims that are based on that lemma. The correct formulation of the aforementioned lemma should be as follows.

Conditions for recurrence and transience for one family of random walks

“…Therefore, we provide another classification of random walks that is closely related to one given in [1]. For any vector n = (n (1) , n (2) , . .…”

“…+ |n (d) |. Since in our case, the vectors are assumed to be two-dimensional with positive components, the notation reduces to n = n (1) + n (2) . For simplicity, a family of random walks having index ψ will be called ψ-random walks.…”

“…In the present paper, we study the following new family of twodimensional random walks. The set A is the set of infinite sequences, the elements a of which are specified as indexed sequences {α n }, where n = {n (1) , n (2) } is the set of ordered pairs of nonnegative integers, n (1) ≤ n (2) . So, α n = α (n (1) ,n (2) ) , and in the rest of the paper each of the notation α n or α (n (1) ,n (2) ) can be used interchangeably.…”

Extension of the Bertrand–De Morgan Test and Its Application

“…Birth-and-death processes. In this section, we demonstrate an application of Theorem 2 in the theory of birth-and-death processes improving Lemma 4.1 in [11]. Namely, we prove the following theorem.…”

“…In particular, if lim n→∞ r n exists, the theorem does not provide information about the situation when lim n→∞ r n = 1. It follows from the proof of Lemma 4.1 in [11] that if this convergence is from the below, then (1) diverges. If it is not, the series may either converge or diverge.…”