Weak convergence of the number of zero increments in the random walk with barrier

Abstract

We continue the line of research of random walks with barrier initiated by Iksanov and Möhle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent −α, α ∈ (0, 1), we prove that the number V n of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for V n proved in Iksanov and Negadailov (2008).

Renewal approximation for the absorption time of a decreasing Markov chain

“…, W , and Proposition 2 (1-arithmetic case) ensures that the second term converges to zero. As for the first one, it is enough to prove that e n := d 1(2) (T n , N n ) = o(c(n)) as n → ∞ (22) by property (P4) in Proposition 1.…”

“…Indeed, some extra conditions on the rate of convergence of I n to ξ in the additive case, and of n −1 I n to 1 − η in the multiplicative case are necessary. For the approach of this paper, which has already been used in [9] and [22], the rate of convergence in (Add) and (Mult) will be measured in terms of the minimal L p -distance, for the following reasons a natural choice:…”

“…The random variable T arises in different areas of applied probability and also in varying disguises, for example, the number of blocks in regenerative compositions [7], [8], [10], the number of collisions in exchangeable coalescents [11], [15], [23], [26], the number of positive jumps in a random walk with a barrier [17], [22], or the number of cuts needed to isolate the root of a random recursive tree [16].…”

“…k = n − 1} being the absorption time, provides another functional of interest and a further example involving a multiplicative renewal approximation. In [22], it was shown that the sequence V n :…”

“…where η α has a β(1 − α, α)-distribution, it was proved in [22, Theorem 1.1] that a renewal approximation can be used with a multiplicative random walk having generic step size η α (under an additional assumption in the case α ≤ 1 2 which ensures thecorrect asymptotic behavior of the discrete renewal density of a random walk with generic step size ζ , see [22,Equation (1.4…”