S. V. NAGAEV

ON SOME LIMIT THEOREMS FOR HOMOGENEOUS MARKOV CHAINS

(Presented by Academician A. N. Kolmogorov on 13 II 1957)

Let an abstract space (X) be given, and let (\mathcal F_x) be a (\sigma)-algebra of its subsets. Let (p(\eta,A)), (\eta \in X), (A \in \mathcal F_x), be the transition-probability function. In what follows we shall assume that there exists a stationary probability distribution (p(A)) such that, for some (\rho<1) and (c),

[
|p^{(n)}(\eta,A)-p(A)|<c\rho^n
\tag{1}
]

uniformly with respect to (\eta \in X) and (A \in \mathcal F_x), where (p^{(n)}(\eta,A)) is the probability of transition in (n) steps from state (\eta) to a state belonging to the set (A). The function (p(\eta,A)), together with the initial probability distribution (\pi(A)), determines a sequence (x_1,x_2,\ldots,x_n,\ldots) of random variables connected in a homogeneous Markov chain, with

[
P(x_1\in A)=\pi(A),\qquad
P(x_n\in A)=\int_X p^{(n-1)}(\eta,A)\,\pi(d\eta).
\tag{2}
]

Let (f(\eta)) be a real function defined on (X) and measurable with respect to (\mathcal F_x).

Theorem 1. If

[
\int |f(\eta)|^2\,p(d\eta)<\infty,
]

[
\sigma^2=\lim_{n\to\infty} M\left[\frac1{\sqrt n}\sum_{m=1}^n\bigl(f(x_m)-Mf(x_m)\bigr)\right]>0
]

(the mathematical expectation is computed under the assumption that the initial distribution is stationary), then for any initial distribution (\pi(A))

[
\lim_{n\to\infty}
P\left{
\frac1{\sqrt n}\sum_{m=1}^n
\left(f(x_m)-\int_X f(\eta)\,p(d\eta)\right)<x
\right}
=
\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^{x} e^{-u^2/2\sigma^2}\,du.
\tag{3}
]

This theorem is an analogue of the well-known theorem of P. Lévy, which states that if (x_1,x_2,\ldots,x_n,\ldots) is a sequence of independent identically distributed random variables and (\sigma^2=Dx_i<\infty), then

[
\lim_{n\to\infty}
P\left(
\frac{\sum_{i=1}^n x_i-na}{\sigma\sqrt n}<x
\right)
=
\frac1{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-u^2/2}\,du,
]

where (a) is the mathematical expectation of (\chi_i). Up to the present time the central limit theorem for a homogeneous Markov chain with an arbitrary set of states has been proved under the assumption that, for some (\delta > 0) ((^1,{}^2,{}^3)),

[
\int_X |f(\eta)|^{2+\delta} p(d\eta) < \infty .
]

Theorem 2. Let (u_1, u_2, \ldots, u_n, \ldots) be a sequence of independent random variables with common distribution function (F(x)) such that

[
F(x) = p(f(\eta) < x)
]

(here (p(A)) is the stationary distribution). If, for some sequence of constants (A_n) and (B_n > 0),

[
\lim_{n \to \infty} P \left( \frac{1}{B_n} \left( \sum_{m=1}^{n} u_m - A_n \right) < x \right) = V_\alpha(x),
]

where (V_\alpha(x)) is a stable law with characteristic exponent (\alpha), and if for some (0 < \nu \le 1)

[
\lim_{n \to \infty} n^{1/2} B_n^{-\nu} \sup_{\xi} \int_{|f(\eta)| < B_n \tau} |f(\eta)|^\nu p(\xi, d\eta) = 0,
\tag{4}
]

whatever (\tau > 0) may be, then, for an arbitrary initial distribution (\pi(A)),

[
\lim_{n \to \infty} P \left( \frac{1}{B_n} \left( \sum_{m=1}^{n} f(x_m) - A_n \right) < x \right) = V_\alpha(x).
\tag{4'}
]

Condition (4) is satisfied, for example, when for some (\varepsilon < \alpha/2) the moments

[
\int_X |f(\eta)|^{\alpha-\varepsilon} p(\xi, d\eta)
]

are uniformly bounded in (\xi).

Theorem 3. If, for some (0 < \alpha < 2),

[
\int_X |f(\eta)|^\alpha p(d\eta) < \infty,
]

then, for some choice of the constants (A_n) and for an arbitrary initial distribution,

[
\lim_{n \to \infty} P \left( \frac{1}{n^{1/\alpha}} \left( \sum_{m=1}^{n} f(x_m) - A_n \right) < x \right) = E(x),
\tag{5}
]

where (E(x)) is an improper law.

Corollary. If

[
\int_X |f(\eta)| p(d\eta) < \infty,
]

then the sequence (f(x_1), f(x_2), \ldots, f(x_n), \ldots) obeys the law of large numbers.

Let now (X) be a countable set ({\omega_i}) ((i = 1, 2, \ldots)) and

[
\beta = \inf_{(i,j)} \sum_{k=1}^{\infty} \min(p_{ik}, p_{jk}),
\tag{6}
]

where (p_{ik}) is the probability of transition from (\omega_i) to (\omega_k) in one step (the meaning of condition (6) is explained in ((^3))).

Assume further that all states (\omega_i) are essential and form a positive class ((^4)). In view of (6), this class consists of a single subclass. Let (f(\omega_j) = a + k_j h), where (a) is an arbitrary real number, (k_j) is an integer, and (h > 0).

Theorem 4. If the greatest common divisor (k_j) is equal to (1),

[
\sum_{j=1}^{\infty} f^2(\omega_j)p_j<\infty
]

and (\sigma>0) ((p_j) are the final probabilities; (\sigma) is defined in the same way as in Theorem 1), then, uniformly with respect to (s),

[
\lim_{n\to\infty}\left(\frac{\sigma\sqrt n}{h}P_{\pi n}(s)-\frac{1}{\sqrt{2\pi}}e^{-z_{ns}^2/2}\right)=0,
\tag{7}
]

where (P_{\pi n}) is the probability that

[
\sum_{m=1}^{n} f(x_m)=an+sh,
]

under the condition that the initial distribution is (\pi(A)), and

[
z_{ns}=\frac{1}{\sigma\sqrt n}\left(an+sh-n\sum_{j=1}^{\infty} f(\omega_j)p_j\right).
]

Theorem 5. If the conditions of Theorem 4 are satisfied and, in addition, for some integer (k\geq 3) and some (\delta>0)

[
\sum_{j=1}^{\infty}|f(\omega_j)|^{k+\delta}p_{ij}