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 0}|v_n(s)-v_0(s)| \xrightarrow{P
\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).