Some limit theorems of runs to the continuous‐valued sequence

The 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.

The 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.

The upper and lower bounds of the deviations from the sums of arbitrary continuous‐valued random variables from the reference distributions are obtained.

The 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.

A useful method to study the property for runs of dependent random sequence.

The new approach of study strong limit behavior for dependent random sequence.