2023/10/09 by Alexander Iksanov, Iksanov, Alexander, Valeriya Kotelnikova +1
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2310.06087
openalex publication_date 2023/10/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes 1, 2,…, with probability pk of hitting the box k. For j,n∈ℕ, denote by K^*j(n) the number of boxes containing exactly j balls provided that n balls have been thrown. We call small counts the variables K^*j(n), with j fixed. Our main result is a law of the iterated logarithm (LIL) for the small counts as the number of balls thrown becomes large. Its proof exploits a Poissonization technique and is based on a new LIL for infinite sums of independent indicators ∑k≥ 1\Bbb1Ak(t) as t→∞, where the family of events (Ak(t))t≥ 0 is not necessarily monotone in t. The latter LIL is an extension of a LIL obtained recently by Buraczewski, Iksanov and Kotelnikova (2023+) in the situation that (Ak(t))t≥ 0 forms a nondecreasing family of events.