On convergence of tail probability series in law of large numbers (Q910089)

Let \(\{X_ m\}\) be a sequence of r.v. and \(S_ n = \sum^{n}_{j=1}X_ j\). Assume that the positive functions \(H(t)\) and \(\varphi(t)\) are defined on \((0,+\infty)\) and \(H(t) \uparrow +\infty\) (\(t\to+\infty\)). Denote \[ \begin{gathered} \hat\varphi(t) = \int^{t}_{0} \varphi(u) du, \quad t>0, \qquad \nu(\varepsilon) = \sum^\infty_{n=1} I(| S_ n| \geq \varepsilon H(n)), \\ \eta(\varepsilon) = \sup \{(| S_ n| - \varepsilon H(n))^+ /\varepsilon : \;n\geq 1\} \text{\quad and\quad} \chi(\varepsilon) = \sup \{n\geq1: \;| S_ n| \geq \varepsilon H(n)\}. \end{gathered} \] The following three problems concerned with the relations between the convergence of tail probability series in the law of large number and the moments of some functions of \(S_ n\) are investigated: \[ \begin{aligned} \sum^\infty_{n=1} \varphi(n) P\{| S_ n| \geq \varepsilon H(n)\} < \infty \quad &\Leftrightarrow \quad E \hat\varphi (\nu(\varepsilon)) < \infty? \tag{1} \\ \sum^\infty_{n=1} \varphi(n) P\{\max_{1\leq k\leq n} | S_ k| \geq \varepsilon H(n)\} <\infty \quad &\Leftrightarrow \quad E \hat\varphi(H^{-1}(\eta(\varepsilon))) < \infty? \tag{2} \\ \sum^\infty_{n=1} \varphi(n) P \{\sup_{k\geq n} \{| S_ k| / H(k)\} \geq \varepsilon \} < \infty \quad &\Leftrightarrow \quad E \hat\varphi(\chi(\varepsilon)) < \infty? \tag{3} \end{aligned} \]