Rates of convergence for tail series (Q1345548)

Let \(X_ 1,X_ 2,\dots\) be independent random variables with partial sums \((S_ n)\). Assume that \(\sum^ \infty_{i = 1} X_ i\) is a.s. convergent, and define \(T_ n = \sum^ \infty_{i = n + 1} X_ i\) for \(n \geq 0\). Let \((a_ n)\) be a positive, nondecreasing, divergent real sequence. Define the numbers \[ \alpha = \inf\left\{r : \sum^ \infty_{i = 1} P(a_{n_ i} T_{n_ i} > r) < \infty\quad \text{for every admissible sequence } (n_ i)\right\}, \] \[ \beta = \inf\left\{r : \sum^ \infty_{i = 1} P(a_{n_ i}(S_{n_{i + 1}} - S_{n_ i}) > r) < \infty \quad \text{for every admissible sequence } (n_ i)\right\}. \] Suppose that \[ \liminf_{n \geq m \to \infty} P(a_ m(S_ n - S_ m) > -t)>0 \quad\text{for every } t>0. \] Then \[ \limsup_{n\to \infty} a_ n T_ n \leq \alpha \quad \text{a.s.},\tag{1} \] \[ \limsup_{n \to \infty} a_ n T_ n = \alpha = \beta \quad \text{a.s. if } \alpha < \infty.\tag{2.} \]