Normal approximations of the number of records in geometrically distributed random variables

On Asymptotics of Exchangeable Coalescents with Multiple Collisions

“…It can be shown that P(I k = 1) ∼ b/k as k → ∞. From this, as n → ∞, the expectation and the variance of X n are both asymptotically equal to b log n. By the central limit theorem, the distribution of (X n − b log n)/ √ b log n converges to the standard normal law, which again agrees with Theorem 3.1, case (1). The latter result is classics which dates back to [17].…”

Asymptotic normality for the counting process of weak records and δ-records in discrete models

“…The above result was obtained by Bai et al [3], using generating function methods. With some extra effort, our results could be extended to functional central limit theorems such as…”

On the number of jumps of random walks with a barrier

“…In this latter situation some results on the asymptotic behavior of recursion (1.1) with (1.3) are available, for example, in the context of random composition structures [4], [18], [20], [21], of coalescent theory [19], [24], [26] (see also Section 7 of the present work), and in the context of random trees [11], [14], [24], [29], [30]. We also refer the 208 A. IKSANOV AND M. MÖHLE reader to [3] for a number of interpretations of the random recursion (1.1), where…”