Limit theorems for maxima of sums and renewal processes (Q1781275)

Let \(X_1,X_2,\dots\) be a sequence of i.i.d. random variables and let \(S_n= X_1+\cdots+ X_n\). It is assumed throughout that \[ S_n\to\infty \text{ a.s. as }t\to\infty.\tag{\(*\)} \] The corresponding renewal counting process can be defined (a.s.) in the following two different ways: \[ N(t)= \sup\{n: S_n\leq t\},\;t\geq 0,\quad\text{or}\quad \nu(t)= \text{inf}\{n: S_n> t\},\;t\geq 0. \] Furthermore put \(M_n= \max_{1\leq k\leq n}\,S_k\), \(T_n= \inf_{k\geq n}S_k.\) Typical results are the following: Theorem 2. Assume \((*)\). The following statements are equivalent: (2a) \(M_n/b_n\to 1\) a.s. for some positive sequence \((b_n)\); (2b) \(T_n/b_n\to 1\) a.s. for some positive sequence \((b_n)\); (2c) \(S_n/b_n\to 1\) a.s. for some positive sequence \((b_n)\); (2d) \(0< E[X_1]< \infty\). Theorem 3. Assume \((*)\). The following statements are equivalent: (3a) \(N(t)/t\to 1/a\) a.s. for some positive \(a\); (3b) \(\nu(t)/t\to 1/a\) a.s. for some positive \(a\); (3c) \(M_n/n\to a\) a.s. for some positive \(a\); (3d) \(T_n/n\to a\) a.s. for some positive \(a\); (3e) \(0< E[X_1]<\infty\). Theorem 5. Assume \((*)\). Let \(1< r< 2\). The following statements are equivalent: (5a) \(t^{-1/r}(N(t)- t/a)\to 0\) a.s. for some positive \(a\); (5b) \(t^{-1/r}(\nu(t)- t/a)\to 0\) a.s. for some positive \(a\); (5c) \(n^{-1/r}(T_n- na)\to 0\) a.s. for some positive \(a\); (5d) \(n^{-1/r}(M_n- na)\to 0\) a.s. for some positive \(a\); (5e) \(0< E[X_1]< \infty\) and \(E[|X_1|^r]< \infty\). Partial results were obtained e.g. by \textit{B. A. Rogozin} [Theory Probab. Appl. 21, 375--379 (1976); translation from Teor. Veroyatn. Primen. 21, 383--387 (1976; Zbl 0381.60043)], \textit{H. Kesten} and \textit{R. A. Maller} [Ann. Inst. Henri Poincaré, Probab. Stat. 35, No. 6, 685--734 (1999; Zbl 0940.60064)], \textit{R. A. Maller} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 43, 141--148 (1978; Zbl 0366.60035)] and \textit{A. Gut}, \textit{O. Klesov} and \textit{J. Steinebach} [Stat. Probab. Lett. 35, No. 4, 381--394 (1997; Zbl 0885.60026)].