A limit theorem for the time of \(\varepsilon\)-convergence of two independent renewal processes. (English. Russian original) Zbl 0838.60073

Kushnir, A. O.

Theory Probab. Math. Stat. 45, 67-70 (1992); translation from Teor. Veroyatn. Mat. Stat., Kiev 45, 69-74 (1991).

Let \((S_k : k \geq 0)\) and \((T_k : k \geq 0)\) be two independent renewal processes with \(S_0 = - S\) and \(T_0 = 0\) and let \(\theta_\varepsilon (S)\) be the \(\varepsilon\)-coupling time \[ \theta_\varepsilon (S) = \inf_{k \geq 1} \bigl \{T_k : S_l \in (T_k - \varepsilon, T_k] \text{ for some } l\geq 0 \bigr\}. \] The author provides conditions such that \(\varepsilon \cdot \theta_\varepsilon (S)\) converges weakly to an exponential distribution with expectation equal to the product of mean recurrence times of the two renewal processes.

Reviewer: H.Wegmann (Darmstadt)

MSC:

60K05	Renewal theory
60F05	Central limit and other weak theorems

Keywords:

coupling time; renewal processes; mean recurrence times