A limit theorem for sums of random number of i.i.d. random variables and its application to occupation times of Markov chains (Q1060490)

Let \(\{(\xi_ i,\tau_ i)\}^{\infty}_{i=1}\) be a sequence of independent, identically distributed random variables with values in \({\mathbb{R}}\times [0,\infty)\). (We do not assume that \(\{\tau_ i\}_ i\) is independent of \(\{\xi_ i\}_ i.)\) A functional limit theorem for the following process is proved. \(S(t)=\xi_ 1+\xi_ 2+...+\xi_{T(t)},\) \(t\geq 0\), where T(t) is the right-continuous inverse of \(U(t)=\tau_ 1+\tau_ 2+...\tau_{[t]},\) \(t\geq 0\). As an application the occupation-time problem for Markov chains is considered.