Branching processes in random environment with freezing

It is well known that a branching process in random environment (BPRE) can be analyzed via the associated random walk <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>ξ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>,</m:mo> </m:math> equation*Sₙ = ξ₁ + + ξₙ,equation* where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>ξ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>=</m:mo> <m:mo form="prefix">ln</m:mo> <m:msup> <m:msub> <m:mi>φ</m:mi> <m:mrow> <m:msub> <m:mi>η</m:mi> <m:mi>k</m:mi> </m:msub> </m:mrow> </m:msub> <m:mo>′</m:mo> </m:msup> </m:math> ξₖ = ln φηₖ' (1). Here <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo fence="false" stretchy="false">{</m:mo> <m:msub> <m:mi>η</m:mi> <m:mi>k</m:mi> </m:msub> <m:mrow> <m:msubsup> <m:mo fence="false" stretchy="false">}</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> \ ηₖ = 1∞ is the random environment and φ x ( t ) is the generating function of the number of descendants of a particle for given environment x . We study the probability of extinction of a branching process in random environment with freezing: in constrast to classic BPRE, in this process every state η k of the environment lasts for given number τ k of generations. It turns out that this variant of BPRE is also closely related to a random walk <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:msub> <m:mi>S</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mi>τ</m:mi> <m:mn>1</m:mn> </m:msub> <m:msub> <m:mi>ξ</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>+</m:mo> <m:mo>⋯</m:mo> <m:mo>+</m:mo> <m:msub> <m:mi>τ</m:mi> <m:mi>n</m:mi> </m:msub> <m:msub> <m:mi>ξ</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>.</m:mo> </m:math> equation*Sₙ = τ₁ ξ₁ + + τₙ ξₙ.equation* We find several sufficient conditions for extinction probability of such process to be one or less than one correspondingly.