Some limit theorems of runs to the continuous‐valued sequence

Abstract

PurposeThe purpose of this paper is to give some limit theorems on ε‐neighborhood and ε‐increasing runs of continuous‐valued dependent random sequence. In the main result no assumptions are made concerning the random variables. As corollary a result on independent case is obtained.Design/methodology/approachThe crucial part of the proof is to construct a non‐negative supper‐martingale depending on a parameter by using the moment generating function, and then applying the Doob's martingale convergence theorem.FindingsThe upper and lower bounds of the deviations from the sums of arbitrary continuous‐valued random variables from the reference distributions are obtained.Research limitations/implicationsThe computation of asymptotic log‐likelihood ratio h(P|Q) is the main limitations, and it is difficult to obtain the rigorous bounds of the deviations.Practical implicationsA useful method to study the property for runs of dependent random sequence.Originality/valueThe new approach of study strong limit behavior for dependent random sequence.