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Letbe a supercritical branching process with immigrationin a random environment  ξ . We are interested in the almost sure (a.s.) convergence rate of the submartingaleto its limitwhereis an usually used norming sequence. The result about convergence a.s. are as following. Under a moment condition of orderandwhere,a.s. for some  a  &gt; 0 that we find explicity; then assumingfor somewe havea.s.; similar conclusions hold in a varying environment, but the conditionwill be replaced bywhereis a positive sequence of real numbers.

A.s. convergence rate for a supercritical branching processes with immigration in a random environment