我们考虑分叉的随机的散步在上与随机的环境及时(表示了由) 。让 Z < 潜水艇 class= “ a-plus-plus ” > n 产生 n 的粒子的数的措施,和 let\(\tilde Z_n (t)\) 是它的 Laplace 变换。我们显示出免费精力 n 的集中<啜class=“ a-plus-plus ”> 1 log\( \tilde Z_n (t)\),大偏差原则,并且为措施的顺序的中央限制定理{ Z <潜水艇class=“ a-plus-plus ”>为martingale\的限制的时刻的存在的 n },和一个必要、足够的条件( \tilde Z_n (t) /\mathbb { E }[ \tilde Z_n (t)| \xi ]\)。
We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ].