Some limit theorems of runs to the continuousvalued sequence

Purpose The purpose of this paper is to give some limit theorems on neighborhood and increasing runs of continuousvalued dependent random sequence. In the main result no assumptions are made concerning the random variables. As corollary a result on independent case is obtained. Designmethodologyapproach The crucial part of the proof is to construct a nonnegative suppermartingale depending on a parameter by using the moment generating function, and then applying the Doob's martingale convergence theorem. Findings The upper and lower bounds of the deviations from the sums of arbitrary continuousvalued random variables from the reference distributions are obtained. Research limitationsimplications The computation of asymptotic loglikelihood ratio hPQ is the main limitations, and it is difficult to obtain the rigorous bounds of the deviations. Practical implications A useful method to study the property for runs of dependent random sequence. Originalityvalue The new approach of study strong limit behavior for dependent random sequence.