Abstract Generated abstract
This paper establishes limit theorems for sums of functions of homogeneous Markov chains on general state spaces under a uniform geometric convergence condition to a stationary distribution. It proves a central limit theorem assuming only square integrability, extends convergence to stable laws under corresponding domain of attraction and moment conditions, and derives a law of large numbers as a corollary. For countable state spaces, it also obtains local limit theorems and asymptotic expansions for lattice-valued functions of the chain. The results are obtained using spectral theory of linear operators in Banach spaces, generalizing earlier matrix and integral-equation methods.
Full Text
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 \(\delta>0\)
\[ \sum_{j=1}^{\infty}|f(\omega_j)|^{k+\delta}p_{ij}<M<\infty \]
uniformly for all \(i\), then
\[ P_{\pi n}(s)=\frac{h}{\sigma\sqrt n}\left\{\varphi(z_{ns})+\sum_{m=1}^{k-2}\frac{1}{n^{m/2}}T_{\pi m}\bigl(\varphi(z_{ns})\bigr)+O\left(\frac{1}{n^{(k-2)/2}}\right)\right\}. \tag{8} \]
Here
\[ \varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2},\qquad T_{\pi m}(\varphi(x))=\sum_{i=m+2}^{3m} a_{\pi mi}\frac{d^i}{dx^i}\varphi(x), \]
where the coefficients \(a_{\pi mi}\) depend on the initial distribution \(\pi(A)\) and on \(p(\eta,A)\).
Theorems 4 and 5 are a generalization of results of S. Kh. Sirazhdinov \((^5)\). All the theorems formulated above are obtained by means of a method based on the application of the spectral theory of linear operators in a Banach space \((^7)\). This method is a natural generalization of the matrix method and the method of integral equations, presented, for example, in \((^6)\); it can also be used for proving multidimensional limit theorems.
In conclusion, the author thanks R. L. Dobrushin and V. M. Zolotarev for valuable comments.
Moscow State University
named after M. V. Lomonosov
Received
23 XII 1957
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