Small counts in nested Karlin's occupancy scheme generated by discrete\n Weibull-like distributions

2022/03/16 by Alexander Iksanov, Iksanov, Alexander, Valeriya Kotelnikova +1
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2203.08918

openalex publication_date 2022/03/16 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

A nested Karlin's occupancy scheme is a symbiosis of classical Karlin's\nballs-in-boxes scheme and a weighted branching process. To define it, imagine a\ndeterministic weighted branching process in which weights of the first\ngeneration individuals are given by the elements of a discrete probability\ndistribution. For each positive integer j, identify the jth generation\nindividuals with the jth generation boxes. The collection of balls is one and\nthe same for all generations, and each ball starts at the root of the weighted\nbranching process tree and moves along the tree according to the following\nrule: transition from a mother box to a daughter box occurs with probability\ngiven by the ratio of the daughter and mother weights.\n Assume that there are n balls and that the discrete probability\ndistribution responsible for the first generation is Weibull-like. Denote by\n\K_n^(j)(l) and \K_n^*(j)(l) the number of the jth\ngeneration boxes which contain at least l balls and exactly l balls,\nrespectively. We prove functional limit theorems (FLTs) for the matrix-valued\nprocesses \(\K_[ rm≠^T+\⋅]^(j)(l)\)_j,l\∈\ℕ and \( K_[ rm≠^T+\⋅]^*(j)(l)\)_j,l\∈\ℕ, properly normalized and\ncentered, as T\→ \∞. The present FLTs are an extension of a FLT proved\nby Iksanov, Kabluchko and Kotelnikova (2022) for the vector-valued process\n\(\K_[ rm e^T+\⋅]^(j)(1)\)_j\∈\ℕ. While\nthe rows of each of the limit matrix-valued processes are independent and\nidentically distributed, the entries within each row are stationary Gaussian\nprocesses with explicitly given covariances and cross-covariances. We provide\nan integral representation for each row. The results obtained are new even for\nKarlin's occupancy scheme.\n

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