On the rates of convergence of first passage times and their associated processes (Q1390183)

For i.i.d.\ random variables \(X_k\), \(k\in\mathbb N\), set \(S_n =\sum_{k=1}^n, \bar {S}_n = \max_{1 \leq k \leq n} S_k\). The authors study the convergence rate in the LLN for the first passage time \( N(c) \) over the barrier \(c\geq 0\) (i.e., \(N(c)=\min\{n\geq 1, S_n > c\}) \) using the relation \( \{ N(c) > n \} = \{ \bar S_n<c \} \). When \( EX_1 = \theta > 0 \) and \( r > 1, s > 1, 1/2 < s/r \leq 1 \), each of the following three properties \((j): \sum_{n=1}^\infty n^{s-2} P\{|\Delta(j)|> n^{s/r} \varepsilon \} < \infty \) for all \( \varepsilon > 0\) \((j=1,2,3)\), where \( \Delta (1) = S_n - n\theta, \Delta(2) = \overline {S}_n - n\theta, \Delta (3) = N(n) - n/\theta\), is equivalent to (0) \( E|X_1|^r < \infty \) and to (4) \( \int_1^\infty c^{s-2} P\{|N(c) - c/\theta|> c^{s/r}\varepsilon \}dc < \infty \) for all \( \varepsilon > 0 \). \textit{Y. S. Chow} proved that \((0)\Leftrightarrow(1)\Leftrightarrow(2)\) [Bull. Inst. Math., Acad. Sinica 1, 207-220 (1973; Zbl 0296.60014)]. A similar extension of the results referring to the case \( r > 0, s > 1, s/r > 1 \) is also obtained.