A remark on a large deviation theorem for Markov chain with a finite number of states

Abstract

Рассматривается классическая проблема больших уклонений для сумм случайных величин, заданных на состояниях однородной цепи Мар кова с конечным фазовым пространством. Устанавливается точная асим птотика для вероятностей больших уклонений порядка О [у/и). Доказа тельство основано на применении нового типа локальной теоремы. Ключевые слова и фразы: сопряженное распределение, локальная теорема, возмущение спектра, монотонная ^-аппроксимация.

Large deviations in operator form

“…The following theorem corresponds to the result obtained in [18] for a finitestate Markov chain with a strictly positive transition matrix P = (p ij ) 1≤i,j≤d . In that case the condition (Ψ) is satisfied with…”

“…By Lemma 1, Lemma 2 and the proof of Theorem 2 in [18] we have the following local limit theorem of the Shepp-Stone type in the operator form. …”

“…Let G(x) be a continuous function with a compact support. In order to prove Lemma 5 it suffices to recall Proposition 1, (3.19), (3.21) and observe that in the proof of Theorem 2 in [18] all the estimates leading to the conjugate version of (2.12) hold uniformly in s, s ∈ [c, d], [c, d] ⊂ S.…”

Limit theorems and testing hypotheses on Markov chains

“…Following [10], we consider the primitive matrix Q.s/ having the entries q rq .s/ D e s f .q/ p rq ; 1 · r; q · d; s 2 R: (9) By the Perron-Frobenius theorem (see, for example, Theorem 8.2.11 in [1]) the maximal eigenvalue of Q.s/, which is also called a spectral radius, is simple and its left and right eigenvectors are strictly positive. Denote these characteristics, respectively, by ½.s/, u.s/, and v.s/.…”

“…In [10], it is proved that ¹ .q/ n weakly converges to the product measure¸£ ¼ . Hence for any continuous compactly supported function g.u/, as n !…”