arXiv subjects

Anatolii Nikitin

Publications and source records attributed to Anatolii Nikitin.

3 recordsLinked to original sources

arXiv2025

Some central limit theorems for critical beta-splitting tree

We further explore a connection initially unveiled in Iksanov (2025) between critical beta-splitting trees and infinite `balls-in-boxes' schemes. Using the connection, we derive a new joint central limit theorem for components of the height of a leaf chosen uniformly at random in the discrete version of a critical beta-splitting tree. Also, we obtain a joint central limit theorem for the heights in the discrete and continuous versions of a critical beta-splitting tree.

math.PR↗

arXiv2022

Limit theorems for discounted convergent perpetuities II

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. Assuming that $ξ_1$ has zero mean and finite variance and imposing three distinct groups of assumptions on the distribution of $η_1$ we prove three functional limit theorems for the logarithm of convergent discounted perpetuities $\sum_{k\geq 0}e^{ξ_1+\ldots+ξ_k-ak}η_{k+1}$ as $a\to 0+$. Also, we prove a law of the iterated logarithm which corresponds to one of the aforementioned functional limit theorems. The present paper continues a line of research initiated in the paper Iksanov, Nikitin and Samoillenko (2022), which focused on limit theorems for a different type of convergent discounted perpetuities.

math.PR↗

arXiv2021

Limit theorems for discounted convergent perpetuities

Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{R}^2$-valued random vectors. We prove a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm for convergent perpetuities $\sum_{k\geq 0}b^{ξ_1+\ldots+ξ_k}η_{k+1}$ as $b\to 1-$. Under the standard actuarial interpretation, these results correspond to the situation when the actuarial market is close to the customer-friendly scenario of no risk.

math.PR↗