Small counts in nested Karlin’s occupancy scheme generated by discrete Weibull-like distributions

Iksanov, Alexander; Kotelnikova, Valeriya

doi:10.1016/j.spa.2022.08.006

Cited by 3 publications

(5 citation statements)

References 9 publications

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“…In the case α ∈ (0, 1], according to Theorems 3, 5 and 5' in [12], both K * j (t) and K * j (n), centered by their means and normalized by their standard deviations, converge in distribution to a random variable with the standard normal distribution. In the case ρ ∈ , ∞ , Corollary 1.6 in [11] provides functional central limit theorems for K * j (t) and K * j (n), properly scaled. Our purpose is to prove laws of the iterated logarithm (LILs) for K * j (t) as t → ∞ and K * j (n) as n → ∞.…”

Section: Resultsmentioning

confidence: 99%

“…Following the referee's suggestion, for the sake of comparison, we now provide a verbal description of the LILs obtained in [3]. Under the assumptions of Theorems 1, 2 and 3, the limit relations (4), ( 5), (7), (8), (11), (12), (20) and (21) hold true with K j (t), EK j (t) and Var K j (t) replacing K * j (t), EK * j (t) and Var K * j (t). Also, these limit relations hold true with K j (n), EK j (n) and Var K j (n) replacing K * j (t), EK * j (t) and Var K * j (t), and n → ∞ replacing t → ∞.…”

Section: Resultsmentioning

confidence: 99%

“…According to the penultimate inequality in the proof of Lemma 2.13 in [11], for large enough n and any j ≤ n,…”

Section: Proofs Of Theoremmentioning

confidence: 99%

“…where the case-dependent constant C is equal to the right-hand side of ( 4), ( 7), (11) or (20) (( 5), ( 8), (12) or ( 21)), respectively. This taken together with the conclusion of Step 1, formula (65) and Proposition 3 enables us to obtain lim sup…”

Section: Proofs Of Theoremmentioning

confidence: 99%

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A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

Iksanov,

Kotelnikova

2024

Modern Stochastics: Theory and Applications

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In the Karlin infinite occupancy scheme, balls are thrown independently into an infinite array of boxes $1,2,\dots $ , with probability ${p_{k}}$ of hitting the box k. For $j,n\in \mathbb{N}$, denote by ${\mathcal{K}_{j}^{\ast }}(n)$ the number of boxes containing exactly j balls provided that n balls have been thrown. Small counts are the variables ${\mathcal{K}_{j}^{\ast }}(n)$, with j fixed. The 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 ${\textstyle\sum _{k\ge 1}}{1_{{A_{k}}(t)}}$ as $t\to \infty $, where the family of events ${({A_{k}}(t))_{t\ge 0}}$ 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 when ${({A_{k}}(t))_{t\ge 0}}$ forms a nondecreasing family of events.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

“…According to the penultimate inequality in the proof of Lemma 2.13 in [11], for large enough n and any j ≤ n,…”

Section: Proofs Of Theoremmentioning

confidence: 99%

Section: Proofs Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

A law of the iterated logarithm for small counts in Karlin’s occupancy scheme

Iksanov,

Kotelnikova

2024

Modern Stochastics: Theory and Applications

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“…. ., properly scaled, normalized and centered, are proved in Iksanov, Kabluchko, and Kotelnikova (2022a) and Iksanov and Kotelnikova (2022), respectively for a nested occupancy scheme with deterministic probabilities.…”

Section: Related Literaturementioning

confidence: 99%

A Limit Theorem for a Nested Infinite Occupancy Scheme in Random Environment

Braganets,

Iksanov

2023

AJS

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We investigate an infinite balls-in-boxes scheme, in which boxes are arranged in nested hierarchy and random probabilities of boxes are defined in terms of iterated fragmentation of a unit mass. Gnedin and Iksanov (2020) obtained a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. We prove a counterpart of their result, in which centering is not needed and the limit processes are not Gaussian. An application is given to the scheme generated by a residual allocation model.

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