Limit Theorems for Mixed Max–sum Processes with Renewal Stopping

This article is devoted to the investigation of limit theorems for mixed max–sum processes with renewal type stopping indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems. 1. Introduction. The main object of this article is the derivation of a number of limit theorems for mixed max–sum processes with renewal stopping. Such processes are constructed from the three-component sequences of i.i.d. random vectors taking values in R 1 × R 1 × [0, ∞) in the following way. The first component of the sequence is used to construct an extremal max process of i.i.d. random variables. The second is used as a traditional real-valued sum process of i.i.d. random variables. Finally, the third component is introduced by a nonnegative sum process of i.i.d. random variables. It induces the stopping renewal process that is a process of the first exceeding times over a specific level t > 0. The first two components are then stopped using this renewal process. The overall process so obtained will be called a max–sum process with renewal stopping. Note that at this point we do not restrict possible dependencies between the three components. Max–sum processes with renewal stopping of the above type naturally appear in various applications. To help visualize such processes, we give a few concrete examples.