Large Deviations of Bisexual Branching Process in Random Environment

We study large deviation probabilities of bisexual branching process in a random (i.i.d.) envrionment. Under several conditions on the mating function we introduce the associated random walk of the process. We also assume Cramer conditon for the step of the walk and moment conditions on the number of descendants of one pair. We find asymptotics of $\mathbf{P}(\ln N_n \in [x,x+\Delta_n))$ as $n\to\infty$ for $x/n$ from some domain and all $\Delta_n$, tending to zero sufficiently slowly. Similar results for bisexual branching process with immigration in a random envrionment are proved too.