Convergence of Baum–Katz Series with OSV-Functions

Abstract

In this paper conditions for the convergence of series \[\sum_{n=1}^{\infty}n^{t-1}L(n)P(\left | S_{n} \right |\geq n^{1/ r}\varepsilon )\]  for arbitrary  \[\varepsilon > 0.\]  different values of parameters \[t\geq 0, 0< r< 2\]  and functions L are considered. Such series appear by investigating complete convergence, as well as by studying different questions on large deviations in limit theorems of probability theory. Sufficient conditions for the convergence of such a series for not necessarily monotone and continuous slowly varying functions L are obtained. For \[r=1\] and non-monotone function L condition \[E\left [ \left | X \right |^{t+1}L(\left | X \right |) \right ]< \infty\]  does not imply the existence of the first moment. This mean that in general case it is necessary to consider series \[\sum_{n=1}^{\infty }n^{t-1}L(n)P(\left | S_{n}-\mathrm{med}(S_{n})\right |> \varepsilon n^{1/r}),\] which includes medians of sums \[{\mathrm {med}}(S_{n}).\] instead of generalized Baum–Katz series \[\sum_{n=1}^{\infty }n^{t-1}L(n)P(\left | S_{n}\right |\geq \varepsilon n^{1/r})\] In order to get rid of medians it is necessary to add an assumption on the finiteness of the first moment. This mean that obtained results extend one result of Heyde and Rohatgi to the case of non-monotone slowly varying functions L for \[t\geq 0.\] Moreover, we enlarge the class of functions for which sufficient conditions for the convergence of introduced series, for \[t\geq 0.\] are found. It turns out that appropriate results hold true not only for monotone and for continuous slowly varying functions, but also for a more wide class of functions, namely, OSV-functions. Generalization of main result for the case of normalizing sequences, that are Marcinkiewicz–Zygmund sequences, is also presented. In this case, depending on r two additional moment assumptions are imposed in order to avoid medians.