UDC 519.21

MATHEMATICS

V. V. ANISIMOV

LIMITING DISTRIBUTIONS OF FUNCTIONALS OF A SEMI-MARKOV PROCESS DEFINED ON A FIXED SET OF STATES UP TO THE MOMENT OF FIRST EXIT

(Presented by Academician V. M. Glushkov, January 12, 1970)

Consider, for each \(t \in (0,\infty)\), a right-continuous semi-Markov process (S.M.P.) \(\varkappa_t(s) \in \{0,1,\ldots,r\}\), which is specified, following \((^{1,2})\), by the matrix of transition probabilities

\[ F_t(i,j,u)=P\{\varepsilon_{k+1}=j,\ \tau_t(\varepsilon_k)<u\mid \varepsilon_k=i\},\qquad i,j=0,\ldots,r, \]

where \(\varepsilon_k=\varkappa_t(\theta_t(k))\), \(\tau_t(\varepsilon_k)=\theta_t(k+1)-\theta_t(k)\), and \(\theta_t(k)\) is the moment of the \(k\)-th jump, i.e. \(\theta_t(0)=0\), and

\[ \theta_t(k)=\min\{s:\ s>\theta_t(k-1),\ \varkappa_t(s)\ne\varepsilon_{k-1}\},\qquad k\ge 1. \]

We shall call the set of states \(\langle \omega\rangle=\langle 1,2,\ldots,r\rangle\) a group if, for any \(i,j\in\langle\omega\rangle\), \(p_t(i,j,\langle\omega\rangle)\to 1\) as \(t\to\infty\), where \(p_t(i,j,\langle\omega\rangle)\) is the probability of reaching \(j\) from \(i\) before the S.M.P. leaves \(\langle\omega\rangle\).

Let, for each \(t\in(0,\infty)\), \(f_t^{(k)}(i,x)\), \(k=0,1,\ldots\), \(i\in\langle\omega\rangle\), \(x\in(0,\infty)\), be a family of random variables independent of the S.M.P. \(\varkappa_t(s)\), independent in the aggregate, whose distribution does not depend on the index \(k\). Denote

\[ \nu_t(j)=\min\{n:\ \varepsilon_n\in\overline{\langle\omega\rangle}\mid \varepsilon_0=j\},\qquad j\in\langle\omega\rangle. \]

Introduce an additive functional of the trajectory of the S.M.P. of the form

\[ \xi_t(j)=\sum_{k=0}^{\nu_t(j)-1} f_t^{(k)}(\varepsilon_k,\tau_t(\varepsilon_k)),\qquad \text{if } \varepsilon_0=j. \]

We shall study the limiting distributions of \(\xi_t(j)\) under the corresponding normalization, as \(t\to\infty\). Denote by \(\tau_t(l,j)\) a random variable with distribution function \(p_t(l,j)^{-1}F_t(l,j,u)\), \(l,j\in\langle\omega\rangle\), and let

\[ \gamma_t(l,j)=f_t^{(1)}(l,\tau_t(l,j)),\qquad \psi_t(l,j,\lambda)=M\exp\{i\lambda\gamma_t(l,j)\}. \]

Put

\[ g_t=\sum_{k\in\langle\omega\rangle} q_t(k,\langle\omega\rangle)\sum_{i\in\overline{\langle\omega\rangle}} p_t(k,i), \]

where \(q_t(k,\langle\omega\rangle)\), \(k\in\langle\omega\rangle\), is the stationary distribution for the chain with matrix \(\widetilde P(t,\langle\omega\rangle)=\|\widetilde p_t(k,j,\langle\omega\rangle)\|\), \(k,j\in\langle\omega\rangle\), and

\[ \widetilde p_t(k,j,\langle\omega\rangle)=p_t(k,j)\left(1-\sum_{i\in\langle\omega\rangle}p_t(k,i)\right)^{-1}. \]

Here \(p_t(k,j)=F_t(k,j,\infty)\), \(k,j=0,\ldots,r\), are the transition probabilities for the embedded Markov chain. Suppose that there exists \(\beta=\beta(g_t)\to 0\) such that, for any \(l,j\in\langle\omega\rangle\),

\[ \psi_t(l,j,\beta\lambda)=1+a_{lj}(\lambda)g_t\bigl(q_t(l,\langle\omega\rangle)p_t(l,j)\bigr)^{-1}(1+o(1)), \tag{1} \]

where \(|a_{ij}(\lambda)| \geq 0\). Note that this condition is equivalent to the condition that the quantities \(\gamma_t(l,j)\) are attracted to certain infinitely divisible laws. Denote \(\varphi_t(j,\lambda)=M\exp\{i\lambda \xi_t(j)\}\).

Theorem 1. If the set \(\langle\omega\rangle\) forms a group and (1) is satisfied, then, independently of the initial state \(j\in\langle\omega\rangle\), as \(t\to\infty\)

\[ \varphi_t(j,\beta\lambda)\to \frac{1}{1-a(\lambda)}; \]

here

\[ a(\lambda)=\sum_{i,j\in\langle\omega\rangle} a_{ij}(\lambda) \]

is the logarithm of the characteristic function of a certain infinitely divisible law.

Remark 1. In the case when \(a_{ij}(\lambda)=c_{ij}\lambda^\alpha\), \(i,j\in\langle\omega\rangle\) (\(c_{ij}\) are certain constants), the theorem has a simple probabilistic meaning:

\[ \beta \xi_t(j)\xrightarrow{\mathrm{sl}} \xi \eta^{1/\alpha} *, \]

where \(\xi\) is a stable distribution with exponent \(\alpha\), \(0<\alpha\leq 2\), \(\eta\) is an exponential distribution, and \(\xi\) and \(\eta\) are independent.

Remark 2. If, for the quantities \(\gamma_t(k,j)M_t^{-1}\), \(k,j\in\langle\omega\rangle\), condition (1) is satisfied, where \(a_{kj}(\lambda)=i\lambda c_{kj}\), and, in addition, the quantities \(\gamma_t(k,j)\) have an \(n\)-th moment \((n\geq 1)\), with

\[ M_t^{-l}g_t^{\,l-1}q_t(k,\langle\omega\rangle)p_t(k,j)M\gamma_t(k,j)^l=o(1), \]

\[ 1<l\leq n,\qquad k,j\in\langle\omega\rangle, \]

then under the conditions of Theorem 1

\[ M_t^{-1}g_t\xi_t(j)\xrightarrow{\mathrm{sl}}\eta \]

and all moments up to order \(n\), inclusive, converge. Here

\[ M_t=\sum_{i\in\langle\omega\rangle} q_t(i,\langle\omega\rangle)M f_t^{(1)}(i,\tau_t(i)), \]

\(\eta\) is exponential and \(M\eta=1\). In particular, if we set \(f_t^{(k)}(i,x)=1\) or \(f_t^{(k)}(i,x)=x\), \(i\in\langle\omega\rangle\), we obtain that

\[ g_t\nu_t(j)\xrightarrow{\mathrm{sl}}\eta,\qquad m_t^{-1}g_t\Omega_t(i)\xrightarrow{\mathrm{sl}}\eta, \]

where \(\Omega_t(j)\) is the time spent by the s.m.p. in the group \(\langle\omega\rangle\) before exit, under the condition that \(\varepsilon_0=j\in\langle\omega\rangle\), and

\[ m_t=\sum_{i\in\langle\omega\rangle} q_t(i,\langle\omega\rangle)M\tau_t(i). \]

This agrees with the results of \((^3,^4)\).

Let us now assume that the quantities \(\gamma_t(k,j)\) have a first moment and that there exists \(\beta=\beta(g_t)\) such that, for all \(k,j\in\langle\omega\rangle\),

\[ \psi_t(k,j,\beta\lambda)=1+i\lambda m_t(k,j)\beta+a_{kj}(\lambda)g_t\frac{1+o(1)}{q_t(k,\langle\omega\rangle)p_t(k,j)}. \tag{2} \]

Theorem 2. If the set \(\langle\omega\rangle\) forms a group and (2) is satisfied, then, independently of the initial state \(l\in\langle\omega\rangle\), the distribution of the random variable

\[ \alpha_t\bigl(\xi_t(l)-M_t\nu_t(l)\bigr) \]

converges weakly to the distribution with characteristic function

\[ \frac{1}{1-c_1a(\lambda)+c_2\lambda^2}, \]

* In the sense of weak convergence of distribution functions.

where \(\alpha_t=\min\{\sqrt{g_t}M_t^{-1},\,\beta\}\); \(a(\lambda)\) is defined as in Theorem 1, and \(c_1\geqslant0\) and \(c_2\geqslant0\) are certain constants, with, if

\[ \beta^2 M_t^2 g_t^{-1}\to\mu, \]

then for \(\mu=\infty\), \(c_1=0\), and for \(\mu=0\), \(c_2=0\).

In conclusion we indicate an effective algorithm which, in a finite number of steps, not exceeding \(r-1\), makes it possible to check whether the given set \(\langle\omega\rangle\) will form a group.

Call a transition from \(i\) to \(j\) “proper” if \(P_t(i,j)\nrightarrow0\). We perform the following operation with our embedded chain: at the first step we carry out all “proper” transitions. We obtain a certain directed graph. It is obvious that connected subsets from which there are no transitions form groups. Replace each such group \(\langle d\rangle\) by one state with transition probabilities

\[ p_t(\langle d\rangle,j)= \left(\sum_{k\in\langle d\rangle} q_t(k,\langle d\rangle)\sum_{l\in\langle d\rangle}p_t(k,l)\right)^{-1} \sum_{i\in\langle d\rangle}q_t(i,\langle d\rangle)p_t(i,j); \]

\[ p_t(j,\langle d\rangle)=\sum_{k\in\langle d\rangle}p_t(j,k);\qquad j\in\overline{\langle d\rangle}. \]

Thus, after the first step we obtain a new chain with a smaller number of states. With it we perform the same operation. Then it is clear that \(\langle\omega\rangle\) forms a group if and only if all of \(\langle\omega\rangle\) can be reduced to a single state.

It is of interest to note that the types of distributions found were indicated earlier in \((^5,^6)\) and, through the construction of \(\xi_t(j)\), are naturally connected with the problem considered there.

Kyiv State University
named after T. G. Shevchenko

Received
1 XII 1970

REFERENCES

\(^1\) P. Levy, Proc. Ind. Congress Math., 3, 416 (1954).
\(^2\) R. Pyke, Ann. Math. Stat., 32, No. 4 (1961).
\(^3\) V. S. Korolyuk, Ukr. matem. zhurn., No. 6 (1969).
\(^4\) O. P. Vinogradov, Matem. zametki, 3, issue 5 (1968).
\(^5\) B. V. Gnedenko, Rev. Roumaine Math. Pures et appl., 12, No. 9 (1967).
\(^6\) B. V. Gnedenko, Hussein Fahim, DAN, 187, No. 1 (1969).