UDC 519.21

MATHEMATICS

D. S. SILVESTROV

LIMIT THEOREMS FOR FUNCTIONALS OF THE PROCESS OF STEP SUMS OF RANDOM VARIABLES DEFINED ON A SEMI-MARKOV PROCESS WITH A FINITE SET OF STATES

(Presented by Academician A. N. Kolmogorov on 11 V 1970)

Let \(T_j,\ j=1,2,\) be independent collections of random variables, defined as follows: \(T_1=\{\eta_n,\ n=0,1,2,\ldots\}\) is a homogeneous Markov chain with finite set of states \(H=\{1,2,\ldots,m\}\) and transition-probability matrix \(\|p_{ij}\|_{i,j=1}\); \(T_2=\{(\tau(n,i),\gamma(n,i)),\ n\geq 0,\ i\in H\}\) is a collection of independent random vectors, taking values in \([0,\infty)\times(-\infty,\infty)\), whose distributions do not depend on \(n\).

The random process

\[ \eta(t)=\eta_{\nu(t)},\qquad t\geq 0, \]

where

\[ \nu(t)=\max\left(n:\sum_{k=1}^{n}\tau(k-1,\eta_{k-1})\leq t\right), \]

is called a semi-Markov process \((^{1})\).

We require that \(T_j,\ j=1,2,\) satisfy the following regularity condition:

\((A_1):\) 1) \(T_1\) is ergodic (we denote its stationary distribution by \(q_j,\ j=1,2,\ldots,m\));

\[ \text{2) }\sum_{i=1}^{m} P\{\tau(0,i)>0\}>0. \]

For each \(t>0\), let us introduce the random process

\[ \xi_t(s)=\sum_{k=1}^{\nu(st)}\gamma(k-1,\eta_{k-1}),\qquad s\in[0,1], \]

which it is natural to call the process of step sums of random variables defined on the semi-Markov process \(\eta(t),\ t\geq 0\).

Let \(D_{[0,1]}\) be the space of functions on \([0,1]\) without discontinuities of the second kind, right-continuous with the uniform metric

\[ \rho(x(s),y(s))=\sup_{s\in[0,1]}|x(s)-y(s)|, \]

\[ \mu(C)=P\{w(s)\in C\},\qquad C\in\mathfrak{B}; \]

here \(\mathfrak{B}\) is the \(\sigma\)-algebra of Borel sets in \(D_{[0,1]}\), and \(w(s),\ s\in[0,1]\), is a Wiener process continuous with probability 1.

It is obvious from the construction that for all \(t>0\), with probability 1 the trajectories of the random process \(\xi_t(s),\ s\in[0,1]\), belong to \(D_{[0,1]}\).

Definition. We shall say that a measurable functional \(f(\cdot)\), defined on \(D_{[0,1]}\), is continuous in the uniform topology if there exists a set \(C\in\mathfrak{B}\) such that \(\mu(C)=1\) and for all \(x_n(s)\in D_{[0,1]}\),

\(n \geqslant 0\), if

\[ x_0(s)\in C,\ \rho(x_n(s),x_0(s))\to 0 \quad \text{as } n\to\infty, \]

then

\[ \lim_{n\to\infty} f(x_n(s))=f(x_0(s)). \]

A number of examples of \(\mu\)-continuous functionals in the uniform topology are given in \((^2)\).

Theorem 1. If condition \((A_1)\) and \((A_2)'\) are satisfied: \(M|\gamma(0,i)|^2<\infty\), \(M\tau(0,i)^2<\infty\), \(i\in H\), then all finite-dimensional distributions of the random process

\[ w_t(s)=t^{-1/2}\left(\xi_t(s)-\frac{b}{a}st\right), \qquad s\in[0,1], \]

as \(t\to\infty\), converge weakly (at continuity points) to the corresponding finite-dimensional distributions of the random process

\[ w_0(s)=\sigma w(s), \qquad s\in[0,1]; \]

here

\[ a=\sum_{i=1}^{m} q_i M\tau(0,i), \qquad b=\sum_{i=1}^{m} q_i M\gamma(0,i), \]

\[ \sigma^2=\lim_{n\to\infty}(an)^{-1}D\sum_{k=1}^{n}\left(\gamma(k-1,\eta_{k-1})-\frac{b}{a}\tau(k-1,\eta_{k-1})\right). \]

Remark 1. In \((^3)\) a simple method is given for finding the constant through \(\|p_{ij}\|_{i,j=1}^{m}\) and \(M\gamma(0,i)^{k'}\tau(0,i)^{k''}\), \(i\in H\), \(k', k''\geqslant 0\), \(k'+k''\leqslant 2\), reducing to the solution of a finite system of linear equations.

Theorem 2. If the conditions \((A_j)\), \(j=1,2\), are satisfied, then for all functionals \(f(\cdot)\) on \(D_{[0,1]}\) that are \(\mu\)-continuous in the uniform topology,

\[ P\{f(w_t(s))<u\}\to P\{f(w_0(s))<u\} \quad \text{as } t\to\infty \]

for all continuity points of the distribution function standing on the right.

Remark 2. For the case when condition (B) is satisfied:
1) \(\tau(0,i)=1,\ i\in H\) with probability 1;
2) \(m=1\) (control by the Markov chain is absent).
The corresponding results are contained, for example, in \((^2)\).

The proof of Theorem 1 is carried out analogously to how this is done for one-dimensional distributions in \((^4)\).

The proof of Theorem 2 contains two stages.

It is not difficult to prove that, when the conditions \((A_j)\), \(j=1,2\), are satisfied,

\[ \sup_{s\in[0,1]}\left|\frac{\nu(st)}{t}-a^{-1}s\right|\xrightarrow{P}0 \quad \text{as } t\to\infty. \]

Next, using a simple assertion,

Lemma 1. If, for a sequence of random processes \(\xi_n(s)\), \(s\geqslant 0\), \(n=0,1,\ldots\), whose trajectories with probability 1 belong to the space \(D_{[0,\infty)}\) of functions on \([0,\infty)\) without discontinuities of the second kind and continuous from the right, and a sequence of random processes \(\nu_n(s)\), \(s\geqslant 0\), \(n=0,1,\ldots\), taking nonnegative values with probability 1 and whose trajectories with probability 1 belong to \(D_{[0,\infty)}\), the relations

\[ \text{a)}\quad \rho_n(t)=\sup_{s\in[0,t]}|\xi_n(s)-\xi_0(s)|\xrightarrow{P}0 \quad \text{as } n\to\infty, \]

where \(\xi_0(s),\ s \geq 0\), is a random process continuous with probability 1, \(t \geq 0\);

b)
\[ \hat{\rho}_n(T)=\sup_{s\in[0,T]}|v_n(s)-v_0(s)| \xrightarrow{P} 0 \quad \text{as } n\to\infty,\ T\geq 0; \]

c)
\[ P\left\{\sup_{s\in[0,T]} v_0(s)\geq t\right\}\to 0 \quad \text{as } t\to\infty, \]

then
\[ \sup_{s\in[0,T]}|\xi_n(v_n(s))-\xi_0(v_0(s))| \xrightarrow{P}0 \quad \text{as } n\to\infty; \]

and by the identity
\[ \{v(t)\geq x\}=\left\{\sum_{k=1}^{[x]}\tau(k-1,\eta_{k-1})\leq t\right\}, \]

the proof can be reduced to the case where all \(\tau(0,i)=1,\ i\in H\), with probability 1.

The remainder of the proof is carried out analogously to that given in \({}^{2}\) for the case in which condition (B) holds.

Kyiv State University
named after T. G. Shevchenko

Received
27 IV 1970

REFERENCES

\({}^{1}\) I. I. Ezhov, V. S. Korolyuk, Kibernetika, No. 5 (1967). \({}^{2}\) A. V. Skorokhod, N. P. Slobodenyuk, Limit Theorems for Random Walks, Kyiv, 1970. \({}^{3}\) Chung Kai-lai, Homogeneous Markov Chains, Moscow, 1964. \({}^{4}\) R. Pyke, R. Schanfele, Ann. Math. Statist., 35, 1746 (1964).