Central Limit Theorem and Moderate deviation for nonhomogenenous Markov chains

Moderate deviations for nonhomogeneous Markov chains

“…Central limit theorem and moderate deviation for empirical means for countable nonhomogeneous Markov chains were established by Xu et al [14] (cf. also Xu et al [15]). Large deviations for empirical measures for independent random variables under sublinear expectation were investigated by Gao and Xu [16].…”

“…By (2.17) and (2.15) in Xu et al [14] (cf. also Xu et al [15]) with m l=1 z l f l in place of f , we have…”

“…Then lim n→∞ sup m≥0 In Section 2, we prove Theorem 1.4. The ideas of proofs of Theorem 1.4 come from Gao [8], Gao, Xu [16] and Xu et al [14,15,17].…”

“…Remark 2.1. The idea of proofs of Theorem 1.4 comes from Xu et al ( [14,15]), Gao [8], and Gao and Xu [16]. The key difference between the proofs of Theorem 1.4 and that of Gao and Xu [16] is that we could not directly use Gärtner-Ellis theorem.…”

“…The key difference between the proofs of Theorem 1.4 and that of Gao and Xu [16] is that we could not directly use Gärtner-Ellis theorem. We need to combine Theorem 1.1 in Gao [8], the results of Xu et al ( [14,15]), and the method of Gao and Xu [16] to conjecture and obtain Theorem 1.4.…”