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<p>Consider a branching process {<i>Z</i><sub><i>n</i></sub>}<sub><i>n</i>≥0</sub> with immigration in varying environments. For <i>a</i> ∈ {0, 1, 2, …}, let <i>C</i>(<i>a</i>) = {<i>n</i> ≥ 0: <i>Z</i><sub><i>n</i></sub> = <i>a</i>} be the collection of times at which the population size of the process attains level <i>a</i>. We give a criterion to determine whether the set <i>C</i>(<i>a</i>) is finite or not. For the critical Galton–Watson process, based on a moment method, we show that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10114_2025_4035_Article_IEq1.gif" Format="GIF" Height="27" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">\({{| {C(a) \cap [1,n]} |} \over {\log \;n \to S}}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mfrac> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <mrow> <mi>C</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mrow> <mo stretchy="false">|</mo> </mrow> </mrow> <mrow> <mi>log</mi> <mspace width="thickmathspace" /> <mi>n</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mrow> </mfrac> </mrow> </math></EquationSource> </InlineEquation> in distribution, where <i>S</i> is an exponentially distributed random variable with <i>P</i>(<i>S</i> &gt; <i>t</i>) = e<sup>−<i>t</i></sup>, <i>t</i> &gt; 0.</p>

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Times of a Branching Process with Immigration in Varying Environments Attaining a Fixed Level

  • 期刊论文

  • 发表日期：

    2025年7月15日

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  • Huaming Wang

摘要

Consider a branching process {Zn}n≥0 with immigration in varying environments. For a ∈ {0, 1, 2, …}, let C(a) = {n ≥ 0: Zn = a} be the collection of times at which the population size of the process attains level a. We give a criterion to determine whether the set C(a) is finite or not. For the critical Galton–Watson process, based on a moment method, we show that \({{| {C(a) \cap [1,n]} |} \over {\log \;n \to S}}\) | C ( a ) ∩ [ 1 , n ] | log n → S in distribution, where S is an exponentially distributed random variable with P(S > t) = e−t, t > 0.