Limit theorems for mixed max-sum processes with renewal stopping (Q1769414)

The authors consider the sequence of i.i.d. random variables \(\{(\xi _{\varepsilon ,n},\gamma _{\varepsilon ,n},k_{\varepsilon ,n}); n=1,2,\dots\}\) with positive \(k_{\varepsilon ,n}.\) They introduce the processes \[ \xi _\varepsilon (t)=\max _{k\leq 1\vee tn_\varepsilon }\xi _{\varepsilon ,k},\quad \gamma _\varepsilon (t)=\sum _{k\leq tn_\varepsilon }\gamma _{\varepsilon ,k},\quad k_\varepsilon (t)=\sum _{k\leq tn_\varepsilon }k_{\varepsilon ,k}, \] where \(n_\varepsilon\) is the nonrandom function such that \(n_\varepsilon \rightarrow +\infty ,\varepsilon \rightarrow 0.\) For the renewal stopping process \(\tau _\varepsilon (t)=\sup (s:k_\varepsilon (s)\leq t)\), \(t\geq 0\), the following process is studied \((\xi _\varepsilon (\tau _\varepsilon (t)), \gamma _\varepsilon (\tau _\varepsilon (t))\), \(k_\varepsilon (\tau _\varepsilon (t)))\), \(t>0.\) The article contains results about the weak convergence of this process under \(\varepsilon \rightarrow 0.\)