Central Limit Theorem for Nonstationary Markov Chains. I

1956/01/01 by R. L. Dobrushin · 4 citations
Mathematics · #Advanced Banach Space Theory #Advanced Topics in Algebra #advanced mathematical theories

paper · doi:10.1137/1101006

crossref issued 1956/01/01 · crossref published 1956/01/01 · crossref published-print 1956/01/01 · openalex publication_date 1956/01/01 · crossref created 2005/03/07 · crossref deposited 2017/01/29 · openalex created_date 2025/10/10 · crossref indexed 2026/07/31 · openalex updated_date 2026/07/31

Abstract

This paper is a continuation of investigations on the central limit theorem for nonstationary Markov chains, carried out by Markov (1910), Bernstein (1922–1936), Sapogov (1947–1949) and Linnik (1948–1949). Let Ω _i ,i = 1,2, be sets of states of the chain, and \mathfrakA_i be σ -algebras of measurable subsets of these sets. A function \bf P(x,B) (x ∈ Ω ₁ ,B ∈ \mathfrakA₂ ) is called a stochastic transition function if it has the following properties: 1) if \bf P(x,B) for fixed B is a measurable function of x; 2) if \bf P(x,B) for fixed x is a probability measure in B. Let Λ _i ,i = 1,2, be Banach spaces of completely additive functions of sets λ (B_i )(B_i ∈ \mathfrakA_i ) with λ (Ω _i ) = 0 = 0 and with norm ‖ λ ‖ = \mathop sup _B ∈ \mathfrakA_i |λ (B)|.By means of the equation \bf Pλ (B) = ∫_{Ω i} P(x,B)λ (dx) ,any stochastic transition function defines the operator \bf P taking Λ ₁ into Λ ₂ . Let N(\bf P) be the norm of this operator \bf P The number α (\bf P) = 1 - N(\bf P) is called the ergodic coefficient of this stochastic transition function. If we have a denumerable chain, then \bf P is a stochastic matrix (p_ij ) and α (\bf P) = \mathop inf _k,l ∑_{m = 1} min (p_lm ,p_km ) .A nonstationary chain with n moments of time is defined by a sequence of pairs (Ω _i⁽ⁿ⁾ ⋅ \mathfrakB_i⁽ⁿ⁾ ),i = i, … ,n, transition functions \bf P_i⁽ⁿ⁾ (x,B), (x ∈ Ω _i⁽ⁿ⁾ ,B ∈ Ω _{i + 1}⁽ⁿ⁾ ,i = 1, … ,n - 1) and the initial probability distribution \bf P⁽ⁿ⁾ (B)P⁽ⁿ⁾ (B)(B ∈ \mathfrakA₁ ). In a natural manner the probability measure is given in the space Ω ⁽ⁿ⁾ = Ω ₁⁽ⁿ⁾ × ⋯ × Ω _n⁽ⁿ⁾ . Let X_i⁽ⁿ⁾ be a measurable function defined on Ω _i⁽ⁿ⁾ and continued onto Ω ⁽ⁿ⁾ , and let S⁽ⁿ⁾ = X₁⁽ⁿ⁾ + X₂⁽ⁿ⁾ + ⋯ X_n⁽ⁿ⁾ . We shall consider the limit distribution for a sequence of random variables S⁽ⁿ⁾ = \fracS⁽ⁿ⁾ - \bf ES⁽ⁿ⁾ √ \bf DS⁽ⁿ⁾ . By definition α ⁽ⁿ⁾ = \mathop min _i α (\bf P_i⁽ⁿ⁾ ).Theorem 1: If 1) \bf DX_i⁽ⁿ⁾ > c > 0, 2) |X_i⁽ⁿ⁾ | < C < ∞ , 3) n^{1 / 3} \mathop min _i α (\bf P_i⁽ⁿ⁾ ) → ∞ (n → ∞ )then S⁽ⁿ⁾ is an asymptotically normal sequence of random variables. If we replace condition (2) by (2') DX_i⁽ⁿ⁾ < C < ∞ ,then the limit distribution for S(n)will become an infinitely divisible distribution if it exists. This theorem is not valid if we replace condition (3) by n^1 / 3 \mathop min _i α (\bf P_i⁽ⁿ⁾ ) → k < 0 (n → ∞ ) Theorem 8: If conditions (1), (2') and (3) are satisfied, F_i⁽ⁿ⁾ (t) = \bf P\ X_i⁽ⁿ⁾ - \bf EX_i⁽ⁿ⁾ < t \and (4) \mathop lim _{n → ∞} \frac1n(α ⁽ⁿ⁾ )² \mathop ∑ _{i = 1}ⁿ ∫_|t| > rn^1 / 2 (α ⁽ⁿ⁾ )^3 / 2 t² dF_i⁽ⁿ⁾ (t) = 0,then , S⁽ⁿ⁾ is an asymptotically normal sequence of random variables. If we replace condition (4) by a less stringent condition, Theorem 8 is not valid. A criterion for the ergodic properties of nonstationary Markov chains is also given.

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