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Jonas Šiaulys

Publications and source records attributed to Jonas Šiaulys.

11 recordsLinked to original sources
arXiv2026

Right-tail asymptotics for products of independent normal random variables

Let $X_1,\dots,X_n$ be independent normal random variables with $X_i\sim N(μ_i,σ_i^2)$, and set $Z=\prod_{i=1}^n X_i$. We derive asymptotic approximations for the right tail probability $\mathbb{P}(Z>x)$ as $x\to\infty$. When at least one mean is nonzero, the asymptotic formula remains explicit and involves a finite multiplicative factor arising from admissible sign patterns (reflecting the different ways the product can be positive); it includes an explicit first relative correction term of order $x^{-1/n}$, with remaining relative error $O(x^{-2/n})$. The proof uses a boundary saddle-point/Laplace method: first a multidimensional Laplace approximation near the boundary saddle, then a one-dimensional endpoint Laplace approximation.

math.PR↗
arXiv2026

Asymptotic formulas for products of Poisson distributions

In this paper, we study the asymptotic behaviour of the product tail probability $ \mathbb{P}(ξ_1\cdotsξ_N \geqslant n), $ where $\{ξ_1,\ldots,ξ_N\}$ is a finite collection of independent Poisson random variables with positive parameters $λ_1,\ldots,λ_N$. We derive a refined Laplace-type asymptotic formula for the tail probability, based on Stirling's logarithmic approximation, a constrained saddle-point method, the Lambert function, and a careful evaluation of the constrained Gaussian prefactor. This yields an explicit approximation with an $O(\log n)$ remainder term in the exponent.

math.PR↗
arXiv2023

Ruin probability for renewal risk models with neutral net profit condition

In ruin theory, the net profit condition intuitively means that the incurred random claims on average do not occur more often than premiums are gained. The breach of the net profit condition causes guaranteed ruin in few but simple cases when both the claims' inter-occurrence time and random claims are degenerate. In this work, we give a simplified argumentation for the unavoidable ruin when the incurred claims on average occur equally as the premiums are gained. We study the discrete-time risk model with $N\in\mathbb{N}$ periodically occurring independent distributions, the classical risk model, also known as the Cramér-Lundberg risk process, and the more general E. Sparre Andersen model.

math.PR↗
arXiv2022

Multi seasonal discrete time risk model revisited

In this work we set up the distribution function of $\mathcal{M}:=\sup_{n\geqslant1}\sum_{i=1}^{n}{(Z_i-1)}$, where the random walk $\sum_{i=1}^{n}Z_i, n\in\mathbb{N},$ is generated by $N$ periodically occurring distributions and the integer-valued and non-negative random variables $Z_1,\,Z_2,\,\ldots$ are independent. The considered random walk generates so-called multi seasonal discrete time risk model, and a known distribution of random variable $\mathcal{M}$ enables to calculate ultimate time ruin or survival probability. Verifying obtained theoretical statements we demonstrate several computational examples for survival probability $\mathbb{P}(\mathcal{M}< u)$ when $N=2,\,3$ or $10$.

math.PR↗
arXiv2018

Exponential bounds for the tail probability of the supremum of an inhomogeneous random walk

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent but not necessarily identically distributed random variables. In this paper, the sufficient conditions are found under which the tail probability $\mathbb{P}(\sup_{n\geqslant0}\sum_{i=1}^nξ_i>x)$ can be bounded above by $\varrho_1\exp\{-\varrho_2x\}$ with some positive constants $\varrho_1$ and $\varrho_2$. A way to calculate these two constants is presented. The application of the derived bound is discussed and a Lundberg-type inequality is obtained for the ultimate ruin probability in the inhomogeneous renewal risk model satisfying the net profit condition on average.

math.PR↗
arXiv2017

Randomly stopped maximum and maximum of sums with consistently varying distributions

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables, and $η$ be a counting random variable independent of this sequence. In addition, let $S_0:=0$ and $S_n:=ξ_1+ξ_2+\cdots+ξ_n$ for $n\geqslant1$. We consider conditions for random variables $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution functions of the random maximum $ξ_{(η)}:=\max\{0,ξ_1,ξ_2,\ldots,ξ_η\}$ and of the random maximum of sums $S_{(η)}:=\max\{S_0,S_1,S_2,\ldots,S_η\}$ belong to the class of consistently varying distributions. In our consideration the random variables $\{ξ_1,ξ_2,\ldots\}$ are not necessarily identically distributed.

math.PR↗
arXiv2016

Randomly stopped sums with consistently varying distributions

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables, and $η$ be a counting random variable independent of this sequence. We consider conditions for $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution function of the random sum $S_η=ξ_1+ξ_2+\cdots+ξ_η$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{ξ_1,ξ_2,\ldots\}$ are not necessarily identically distributed.

math.PR↗
arXiv2016

Random convolution of inhomogeneous distributions with $\mathcal{O}$-exponential tail

Let $\{ξ_1,ξ_2,\ldots\}$ be a sequence of independent random variables (not necessarily identically distributed), and $η$ be a counting random variable independent of this sequence. We obtain sufficient conditions on $\{ξ_1,ξ_2,\ldots\}$ and $η$ under which the distribution function of the random sum $S_η=ξ_1+ξ_2+\cdots+ξ_η$ belongs to the class of $\mathcal{O}$-exponential distributions.

math.PR↗
arXiv2016

Ruin probability in the three-seasonal discrete-time risk model

This paper deals with the discrete-time risk model with nonidentically distributed claims. We suppose that the claims repeat with time periods of three units, that is, claim distributions coincide at times $\{1,4,7,\ldots\}$, at times $\{2,5,8,\ldots\}$, and at times $\{3,6,9,\ldots\}$. We present the recursive formulas to calculate the finite-time and ultimate ruin probabilities. We illustrate the theoretical results by several numerical examples.

math.PR↗
arXiv2015

A Lundberg-type inequality for an inhomogeneous renewal risk model

We obtain a Lundberg-type inequality in the case of an inhomogeneous renewal risk model. We consider the model with independent, but not necessarily identically distributed, claim sizes and the interoccurrence times. In order to prove the main theorem, we first formulate and prove an auxiliary lemma on large values of a sum of random variables asymptotically drifted in the negative direction.

math.PR↗