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arXiv · 2606.07194

Survival Probability of Markov Linear Reccurence Sequence in a Random Environment: Subcritical Case

Alexander Shklyaev
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Abstract

The Markov linear recurrence sequence in a random environment (MRSRE) is a particular case of Markov chain on non-negative integers. It is a natural generalization of several models with branching. We consider MRSRE with absorption at zero and study the survival time. We introduce a classification of subcritical MRSRE corresponding to that of branching process in a random environment. We show that general R-positivity theory can be applied to this model in a strongly subcritical case and prove the analogues of the Kolmogorov theorem and the Yaglom theorem. In other words, we describe the asymptotic behavior of the survival probability over a long time and the conditional distribution of the sequence, conditioned on the survival event. The results are applied to particular branching models.

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Alexander Shklyaev. 2026-06-05. Survival Probability of Markov Linear Reccurence Sequence in a Random Environment: Subcritical Case. https://arxiv.org/abs/2606.07194

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