Upper bounds for series involving moderate and small deviations

Abstract:

Let $X,~X_{1},~X_{2},...$ be i.i.d. random variables with $0<EX^{2}=\sigma ^{2}<\infty$ and $EX=0,$ and set $S_{n}=X_{1}+\cdots +X_{n}.$ We prove Paley-type inequalities for series involving probabilities of moderate deviations $P(\left \vert S_{n}\right \vert \geq \lambda \sqrt {n\log n}),$ $\lambda >0,$ and probabilities of small deviations $P(\left \vert S_{n}\right \vert \geq$ $\lambda \sqrt {n\log \log n})$, $\lambda >\sigma \sqrt {2}.$