Some small deviation theorems for arbitrary continuous random sequence

Abstract

Let (Xn)n∈N be a sequence of arbitrary continuous random variables, by the notion of relative entropy \(h_\mu ^{\tilde \mu } (\omega )\) as a measure of dissimilarity between probability measure μ and reference measure \(\tilde \mu \), the explicit, general bounds for the partial sums of arbitrary continuous random variables under suitable conditions are developed. The argument uses the known and elementary lemma of convergence for likelihood ratio.