Edgeworth expansions in operator form.pdf

a r X i v : 0 7 1 2 .4 1 9 9 v 1 [ m a t h .P R ] 2 7 D e c 2 0 0 7 Edgeworth expansions in operator form Zbigniew S. Szewczak? 2 February 2008 Abstract An operator form of asymptotic expansions for Markov chains is estab- lished. Coefficients are given explicitly. Such expansions require a certain modification of the classical spectral method. They prove to be extremely useful within the context of large deviations. Key words: Asymptotic expansions, large deviations, Perron-Frobenius theorem, transition probability function. Mathematics Subject Classification (2000): 60F05, 60F10, 60J10, 47N30. 1 Introduction Let {ξk}k∈Z+ be a homogeneous Markov chain defined on a probability space (?,F ,P). Denote by S and S, respectively, the phase-space and its σ?algebra of measurable subsets. Further, denote by P (x,A), x ∈ S, A ∈ S the transition probability kernel of the chain. It means that for each A ∈ S, P (x,A) is a non- negative measurable function on S while for each x ∈ S, P (x,A) is a probability measure on S. In what follows we assume that the chain is uniformly ergodic. So, there exists a stationary distribution denoted by π. Consider the sequence of random variables X0 = f(ξ0), . . . ,Xn = f(ξn) determined by a measurable function f: S → R. In what follows we assume that σ2 = Eπ[X 2 0 ] + 2 ∞∑ n=1 Eπ[X0Xn] 0. (1.1) There exists a huge literature concerning the limit theorems for successive sums Sn = ∑n i=1Xi, n = 1, 2, . . .. For our purposes, it is enough to keep in mind only the works of S. Nagaev (1957) and (1961) and the monograph by Sirazhdinov and Formanov (1979). Despite the theory of limit theorems is well developed, some settings seem to be set aside. For example, in Szewczak (2005) it was shown that the Crame?r method of conjugate distributions assumes a special form of the local limit theorem that was not considered before. The case studied in Szewczak (2005) concerns Markov chains with a finite number of states. It worth noting that the large d