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Ivan Matsak

Publications and source records attributed to Ivan Matsak.

6 recordsLinked to original sources
arXiv2026

Statistical analysis of the remaining human life expectancy. Mathematical models of the type Gompertz-Makeham and others

Many researchers believe that human lifetime on the interval (30,80) years is described by the Gompertz-Makeham distribution. In this work, the task is set to test this hypothesis. We consider a number of distributions: Gompertz-Makeham, Gompertz-Weibull, Logistic Kannisto-Makeham, Logistic Beard-Makeham and Logistic Beard-Weibull, some of which are partially known, for which comparative calculations on real mortality data are performed using the $χ^2$ criterion. The data were selected for three countries: England \& Wales, Norway and Sweden, cohorts born in 1900. As it turned out, on the interval (30,80) years the Gompertz-Makeham distribution is far from being the best among those listed above. In addition, for the same data, a similar problem was studied for the interval (80,120) years.

stat.AP↗
arXiv2025

Estimates of the probability of a regenerative process reaching a high level

The problem of estimating the probability of a random process reaching a certain level is well known. In this article, two-sided estimates are established for the probability that a regenerative process reaches a high level. Two auxiliary results for geometric sums with delay will play an important role. Examples of application to random processes describing queue lengths in queueing theory are also given.

math.PR↗
arXiv2025

On asymptotic behavior of almost surely extreme values of independent random variables

The article studies the almost surely asymptotics of extreme values $\barξ_n = \max_{1\leq i \leq n} ξ_i$, where $ ξ, ξ_1 , ξ_2 , \ldots$ are discrete identically distributed random variables. One of the main results on this topic is related to the law of the iterated logarithm for the lim sup (LIL) and a law of the triple logarithm for the lim inf (LTL). But, taking into account the specifics of the discrete case, necessary and sufficient conditions are established for $\mathbf{P}(\barξ_n = a_n +l \quad\mbox{infinitely often} ) =1$, where $a_n$ is some given increasing sequence of integers and $l$ is a fixed integer. Note that in the case of discrete random variables whose distribution tails are close to the tails of the Poisson distribution or fall off even faster, Theorems 1-3 of this article are significantly more informative than LIL and LTL. For the geometric distribution (or discrete random variables whose tails fall off even more slowly), the results are given in Theorem 3 and Theorem B, which are an important complement to LIL and LTL.

math.PR↗
arXiv2020

The laws of iterated and triple logarithms for extreme values of regenerative processes

We analyze almost sure asymptotic behavior of extreme values of a regenerative process. We show that under certain conditions a properly centered and normalized running maximum of a regenerative process satisfies a law of the iterated logarithm for the $\limsup$ and a law of the triple logarithm for the $\liminf$. This complements a previously known result of Glasserman and Kou [Ann. Appl. Probab. 5(2) (1995), 424--445]. We apply our results to several queuing systems and a birth and death process.

math.PR↗
arXiv2020

On Distributions of One Class of Random Sums and their Applications

We propose results of the investigation of properties of the random sums of random variables. We consider the case, where the number of summands is the first moment of an event occurrence. An integral equation is presented that determines distributions of random sums. With the help of the obtained results we analyse the distribution function of the time during which the Geiger-Muller counter will not lose any particles, the distribution function of the busy period of a redundant system with renewal, and the distribution function of the sojourn times of a single-server queueing system.

math.PR↗