Some Limit Theorems for Stationary Markov Chains

Let X be a space of points, FX a σ-algebra of its subsets, and p(ξ ,A), ξ ∈ X, A ∈ FX, a stochastic transition function satisfying the following condition: an integer k 1 exists such that \[ {(1)} { }η ,ξ ∈ X,A ∈ F_X | {p⁽ᵏ⁾ (ξ ,A) - p⁽ᵏ⁾ (η ,A)} | < 1. \] Let us define the sequence of random variables x₁ ,x₂ , ⋯ ,xₙ , ⋯ as follows: \[ ( {x_1 ∈ A_1 ,x_2 ∈ A_2 , ⋯ ,x_n ∈ A_n } ) = ∫A_1 {π (dξ _1 )} ∫A_2 {p(ξ _1 ,dξ _2 ) ⋯ } ∫A_n {p(ξ n - 1 ,dξ _n )} , \] where π ( · ) is the initial distribution. Let f(ξ ) be a real function of ξ ∈ X measurable with respect to FX. In Chapter I the asymptotic behaviour of the characteristic function of ∑ ₁ⁿ f(xᵢ ) is studied. Chapter II is devoted to limit theorems. The central limit theorem is proved under the assumption that \[ {(2)} ∫_X {f^2 (ξ )p(dξ ) < ∞ } , \] where p( · ) is a stationary absolute probability distribution corresponding to p( · , · ). The sufficient conditions for convergence to stable laws are given. In chapter III the local limit theorem is proved, and asumptotic expansions are given. The characteristic function method is the basic one used.