Abstract <p> Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\{Z_n,\,n=0,1,2,\dots\}\)</EquationSource> </InlineEquation> be a critical branching process in an i.i.d. random environment, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(Z_{r,n}\)</EquationSource> </InlineEquation> be the number of particles in the process at moment <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(0\leq r\leq n-1\)</EquationSource> </InlineEquation> that have a positive number of descendants in generation <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n\)</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\{S_n,\,n=0,1,2,\dots\}\)</EquationSource> </InlineEquation> be the associated random walk of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\{Z_n,\,n=0,1,2,\dots\}\)</EquationSource> </InlineEquation>. It is known that if the increments of the associated random walk have zero mean and finite variance <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\sigma^2\)</EquationSource> </InlineEquation>, then, for any <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(t\in [0,1]\)</EquationSource> </InlineEquation>, <Equation ID="Equi"> <EquationSource Format="TEX">\(\lim_{n\to\infty}\mathbf P\biggl(\frac{\ln Z_{[nt],n}}{\sigma \sqrt{n}}\leq x\biggm|Z_n>0\biggr) =\mathbf P\Bigl(\,\min_{t\leq s\leq 1}B_s^+\leq x\Bigr),\qquad x\in [0,\infty),\)</EquationSource> </Equation> where <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\{B_t^+,\,0\leq t\leq 1\}\)</EquationSource> </InlineEquation> is a Brownian meander. In the present paper we supplement this result by describing the distribution of the properly scaled random variable <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(\ln Z_{r,n}\)</EquationSource> </InlineEquation> under the condition <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(\{S_n\leq t\sqrt{k},\,Z_n>0\}\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(t>0\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(r,k\to\infty\)</EquationSource> </InlineEquation> in such a way that <InlineEquation ID="IEq14"> <EquationSource Format="TEX">\(k=o(n)\)</EquationSource> </InlineEquation> as <InlineEquation ID="IEq15"> <EquationSource Format="TEX">\(n\to\infty\)</EquationSource> </InlineEquation>. We also consider the case when the distribution of the increments of the associated random walk belongs (without centering) to the domain of attraction of a stable law. </p>

Reduced Critical Branching Processes in Non-favorable Random Environment

期刊论文
发表日期：

2025年9月1日

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V. A. Vatutin,
E. E. Dyakonova

摘要

Abstract

Let \(\{Z_n,\,n=0,1,2,\dots\}\) be a critical branching process in an i.i.d. random environment, \(Z_{r,n}\) be the number of particles in the process at moment \(0\leq r\leq n-1\) that have a positive number of descendants in generation \(n\) , and \(\{S_n,\,n=0,1,2,\dots\}\) be the associated random walk of \(\{Z_n,\,n=0,1,2,\dots\}\) . It is known that if the increments of the associated random walk have zero mean and finite variance \(\sigma^2\) , then, for any \(t\in [0,1]\) , \(\lim_{n\to\infty}\mathbf P\biggl(\frac{\ln Z_{[nt],n}}{\sigma \sqrt{n}}\leq x\biggm|Z_n>0\biggr) =\mathbf P\Bigl(\,\min_{t\leq s\leq 1}B_s^+\leq x\Bigr),\qquad x\in [0,\infty),\) where \(\{B_t^+,\,0\leq t\leq 1\}\) is a Brownian meander. In the present paper we supplement this result by describing the distribution of the properly scaled random variable \(\ln Z_{r,n}\) under the condition \(\{S_n\leq t\sqrt{k},\,Z_n>0\}\) , where \(t>0\) and \(r,k\to\infty\) in such a way that \(k=o(n)\) as \(n\to\infty\) . We also consider the case when the distribution of the increments of the associated random walk belongs (without centering) to the domain of attraction of a stable law.