Mathematics

V. A. Statulevičius

ASYMPTOTIC EXPANSION FOR NONHOMOGENEOUS MARKOV CHAINS

(Presented by Academician A. N. Kolmogorov, 20 VIII 1956)

A nonhomogeneous Markov chain is studied with a finite number (s>1) of possible states (e_1,\ldots,e_s) and transition probabilities (p_{\alpha\beta}^{(k)}) from state (e_\alpha) at the ((k-1))-st step to state (e_\beta) at the (k)-th step.

Let the random variable (\zeta_n^{(\alpha)}) denote the number of visits to state (e_\alpha) during the first (n) steps. Then, for the probability (P_\gamma(m)) that the random vector (\zeta_n=(\zeta_n^{(1)},\ldots,\zeta_n^{(s)})) assumes the value (m=(m_1,\ldots,m_s)), under the condition that (e_\gamma) is the initial state, the following theorem is valid.

Theorem. Suppose that the following conditions are satisfied:

A. (p_{\alpha\beta}^{(l)} \geq \lambda p_{\alpha\beta}^{(k)}) for all (\alpha,\beta,k,l), where the constant (\lambda>0).

B. The set of states of the chain forms one essential class.

C. The rank (r) of the chain is equal to (s) ((^1)).

Then, for any (\gamma) and integer (k>0), uniformly in all ((m_1,\ldots,m_s)), we have

[
\sqrt{D_n^{(1)}\cdots D_n^{(s-1)}}\,\mathcal P_\gamma(m)
=
g_{s-1}(x)
+
\sum_{j=1}^{k} n^{-j/2} P_{\gamma j}!\left(-\frac{\partial}{\partial x}g(x)\right)
+
O!\left(\frac{1}{n^{(k+1)/2}}\right).
\tag{1}
]

Here

[
g_{s-1}(x)
=
\frac{1}{\sqrt{(2\pi)^{s-1}\Delta_n}}
\exp\left[-\frac12 Q_n^{-1}(x)\right]
]

is the density of the ((s-1))-dimensional normal distribution,

[
x=
\left(
\frac{m_1-E_n^{(1)}}{\sqrt{D_n^{(1)}}},
\ldots,
\frac{m_{s-1}-E_n^{(s-1)}}{\sqrt{D_n^{(s-1)}}}
\right);
]

(E_n^{(\alpha)} \asymp n,\ D_n^{(\alpha)} \asymp n); the quadratic form (Q_n(x)=Q_n(x_1,\ldots,x_{s-1})) is positive definite; (\Delta_n) is the determinant of this form. (P_{\gamma j}(it)) is a polynomial of degree not exceeding (3j) in the components of the vector (it=(it_1,\ldots,it_{s-1})). The coefficients of the polynomial are real, depend on (\gamma) and (n), but are uniformly bounded for all (n). (P_{\gamma j}!\left(-\frac{\partial}{\partial x}g_{s-1}(x)\right)) means that, in place of the powers (-it_\alpha), derivatives of (g_{s-1}(x)) with respect to the corresponding component (x_\alpha) are taken ((^2)).

If condition C is violated, but (r>1), then in equality (1) (s) should be replaced by (r).

Leningrad State University
named after A. A. Zhdanov

Received
20 VIII 1956

References

(^1) V. A. Statulevičius, DAN, 107, No. 4 (1956).
(^2) S. Kh. Sirazhdinov, Limit Theorems for Homogeneous Markov Chains, Tashkent, 1955.