Upper bounds for series involving moderate and small deviations

Abstract

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