We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the proce...We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.展开更多
In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the rene...In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the renewal risk model.More precisely,we demonstrate the expressibility of the distribution function n P(sup n≥1^(n)∑_(i=1)(X_(i)-cθ_(i))<u),u∈N_(0)using the roots of the probability-generating function,expectation E(X-cθ)X-cθ,and probability mass function of.We assume that the random X_(1),X_(2),...cθ_(1),cθ_(2),...variables of the mutually independent sequences and are cθc>0 X cθindependent copies of X and respectively,wherein,and are independent,θnonnegative,and integer.We also assume that the support of is finite.To illustrate the applicability of the proven theoretical statements we present a few numerical outputs when the mentioned random variables adopt some particular distributions.展开更多
基金the National Natural Sciente Foundation of China (Grant Nos. 10771021, 10471012)Scientific Research Foundation for Returned Scholars, Ministry of Education of China (Grant No. [2005]564)
文摘We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.
文摘In this work,we propose an alternative to the Pollaczek-Khinchine formula for the ultimate time survival(or ruin)probability calculation in exchange for a few assumptions on the random variables that generate the renewal risk model.More precisely,we demonstrate the expressibility of the distribution function n P(sup n≥1^(n)∑_(i=1)(X_(i)-cθ_(i))<u),u∈N_(0)using the roots of the probability-generating function,expectation E(X-cθ)X-cθ,and probability mass function of.We assume that the random X_(1),X_(2),...cθ_(1),cθ_(2),...variables of the mutually independent sequences and are cθc>0 X cθindependent copies of X and respectively,wherein,and are independent,θnonnegative,and integer.We also assume that the support of is finite.To illustrate the applicability of the proven theoretical statements we present a few numerical outputs when the mentioned random variables adopt some particular distributions.
