In the work necessary and sufficient conditions of weak convergence of one-dimensional distributions of the generalized Koks processes centralized by nonrandom values are resulted. Estimations of the speed of convergence and asymptotic decomposition for one-dimensional functions of distributions of these processes are under construction. Such processes describe the behavior of a reserve of the insurance company. A translation theorem specification for the generalized processes of risk at observance of necessary and sufficient conditions of weak convergence of one-dimensional distributions of the generalized processes of risk is resulted.