Full Text
UDC 519.21
MATHEMATICS
D. S. SILVESTROV
ASYMPTOTIC BEHAVIOR OF THE TIME OF ATTAINMENT FOR SUMS OF RANDOM VARIABLES GOVERNED BY A REGULAR SEMI-MARKOV PROCESS
(Presented by Academician V. M. Glushkov on 12 V 1969)
Let \(\xi_1(\alpha), \xi_2(\alpha)\) be, for each \(\alpha \in [0,1]\), a sequence of independent identically distributed random variables with \(M\xi_1(\alpha)=a(\alpha)\) and \(D\xi_1(\alpha)=b(\alpha)\). Introduce the random functional
\[ \tau(\alpha,s)=\min\left(n:\sum_{k=1}^{n}\xi_k(\alpha)\ge s\right),\qquad s>0. \]
From the results given in \((^1)\) it follows that in the case when:
1) \(\displaystyle \lim_{\alpha\to 1}(\mathrm{sl})\,\xi_1(\alpha)=\xi_1(1)\) *;
2) \(\displaystyle \lim_{\alpha\to 1} a(\alpha)=a(1)=0;\)
3) \(\displaystyle \lim_{\alpha\to 1} b(\alpha)=b(1)\in(0,\infty).\)
4) there exists
\[ \lim_{\alpha\to 1,\;t\to\infty}(1+a(\alpha)\sqrt{t})^{-1}=q\in[0,1], \]
\[ \lim_{\alpha\to 1,\;t\to\infty} P\left\{\frac{\tau(\alpha,w(\alpha,t))}{t}<z\right\} = \sigma(z)\sqrt{\frac{2}{\pi}} \int_{q/\sqrt{zb(1)}}^{\infty} \exp\left\{ \frac{1-q}{b(1)}-\frac{(1-q)^2}{2b^2(1)v^2}-\frac{v^2}{2} \right\}\,dv **, \]
where \(w(\alpha,t)=\sqrt{t}/(1-a(\alpha)\sqrt{t})\).
In the present paper this result is extended to a more general summation scheme, described below.
Let \(T_1(\alpha)=\{\eta_\alpha(n),\, n=0,1,\ldots\}\) be, for each \(\alpha\in[0,1]\), a homogeneous, ergodic Markov chain with a finite or countable set of states \(H=\{1,2,\ldots,m\}\), \(1\le m\le\infty\), transition probability matrix \(\|p_{ij}(\alpha)\|_{i,j=1}^{m}\) and stationary distribution \(q_j(\alpha)>0,\ j\in H\), and let \(T_2(\alpha)=\{(\tau(\alpha,n,i),\gamma(\alpha,n,i)),\, n\ge 1,\ i\in H\}\) be, independent of \(T_1(\alpha)\), a collection of random vectors independent in the aggregate such that: a) \(\tau(\alpha,n,i)\in[0,\infty)\) with probability \(1,\ n\ge 1,\ i\in H\); b) the distributions \((\tau(\alpha,n,i),\gamma(\alpha,n,i)),\ i\in H\), do not depend on \(n\).
Introduce the random functionals
\[ \xi(\eta_0,\alpha,t)=\sum_{k=0}^{\nu(\alpha,t)}\gamma(\alpha,k), \]
* The notation \(\displaystyle \lim_{\alpha\to\alpha'}(\mathrm{sl})\,\xi(\alpha)=\xi(\alpha')\) denotes weak convergence of distribution functions.
** \(\sigma(z)=0\) for \(z<0\); \(\sigma(z)=1\) for \(z\ge 0\).
where
\[ \nu(\alpha,t)=\max\left(n:\sum_{k=0}^{n}\tau(\alpha,k)\leq t\right),\qquad \eta_0=\eta_\alpha(0)=\mathrm{const}\in H; \]
\[ \tau(\alpha,n)=0 \text{ for } n=0;\qquad \tau(\alpha,n)=\tau(\alpha,n,\eta_\alpha(n-1)) \text{ for } n\geq 1 \]
\[ \gamma(\alpha,n)=0 \text{ for } n=0;\qquad \gamma(\alpha,n)=\gamma(\alpha,n,\eta_\alpha(n-1)) \text{ for } n\geq 1; \]
\[ \tau(\eta_0,\alpha,s)=\inf(t:\xi(\eta_0,\alpha,t)\geq s),\qquad s>0. \]
Denote
\[ \theta_0(\alpha)=\inf(t:\eta_\alpha(\nu(\alpha,t))=i), \]
\[ \theta_n(\alpha)=\inf(t:\nu(\alpha,t)>\nu(\alpha,\theta_{n-1}(\alpha)),\ \eta_\alpha(\nu(\alpha,t))=i),\qquad n\geq 1, \]
the times of successive entries of the chain \(T_1(\alpha)\) into the state \(i\in H\), and
\[ (\widetilde{\gamma}(\alpha,n,i),\widetilde{\tau}(\alpha,n,i))= \begin{cases} (0,0), & \text{for } n=0,\\[6pt] \left(\theta_n(\alpha)-\theta_{n-1}(\alpha),\ \displaystyle\sum_{k=\nu(\alpha,\theta_{n-1}(\alpha))+1}^{\nu(\alpha,\theta_n(\alpha))} \gamma(\alpha,k)\right), & \text{for } n\geq 1. \end{cases} \]
Obviously, the random vectors \((\widetilde{\gamma}(\alpha,n,i),\widetilde{\tau}(\alpha,n,i))\), \(n\geq 1\), are independent and identically distributed, and
\[ M\exp\{\sqrt{-1}\,(s\widetilde{\gamma}(\alpha,1,i)+t\widetilde{\tau}(\alpha,1,i))\}= \]
\[ = g_\alpha(s,t,i)\left(\sum_{\substack{k\in H\\ k\ne i}} f_\alpha(s,t,k,i)p_{ik}(\alpha)+p_{ii}(\alpha)\right), \]
where
\[ g_\alpha(s,t,j)=M\exp\{\sqrt{-1}\,(s\gamma(\alpha,1,j)+t\tau(\alpha,1,j))\},\quad j\in H; \]
\[ f_\alpha(s,t,j,i)= M\exp\left\{\sqrt{-1}\left(s\sum_{k=0}^{\Delta_i(\alpha)}\gamma(\alpha,k)+t\sum_{k=0}^{\Delta_i(\alpha)}\tau(\alpha,k)\right)\mid \eta_\alpha(0)=j\right\}, \]
\[ j\in H,\ j\ne i, \]
\[ \Delta_i(\alpha)=\min(n:\eta_\alpha(n)=i), \]
and the functions \(f_\alpha(s,t,j,i)\), \(j\in H\), \(j\ne i\), satisfy the system of linear equations
\[ f_\alpha(s,t,j,i)=g_\alpha(s,t,j)\left(\sum_{\substack{k\in H\\ k\ne i}} f_\alpha(s,t,k,i)p_{jk}(\alpha)+p_{ji}(\alpha)\right),\qquad j\in H,\ j\ne i, \]
and if the corresponding moments for the random variables \(\tau(\alpha,1,i)\), \(\gamma(\alpha,1,i)\), \(i\in H\), exist, then it is not difficult to find the quantities
\[ a_1(\alpha)=q_i(\alpha)M\widetilde{\gamma}(\alpha,1,i),\qquad a_2(\alpha)=q_i(\alpha)D\widetilde{\gamma}(\alpha,1,i), \]
\[ b(\alpha)=q_i(\alpha)M\widetilde{\tau}(\alpha,1,i). \]
Theorem. Suppose that for \(T_j(\alpha)\), \(j=1,2\), the following conditions are satisfied:
(A): 1. \(\displaystyle \lim_{\alpha\to 1}p_{ij}(\alpha)=p_{ij}(1),\ i,j\in H.\)
- \(\displaystyle \lim_{u\to\infty}\lim_{\alpha\to 1}\sup_{j\in H}P\{\Delta_i(\alpha)>u\mid \eta_\alpha(0)=j\}=0.\)
(B): 1. \(\displaystyle \lim_{\alpha\to 1}(\mathrm{d})\bigl(\tau(\alpha,1,i),\gamma(\alpha,1,i)\bigr)=\bigl(\tau(1,1,i),\gamma(1,1,i)\bigr),\ i\in H.\)
-
\(\displaystyle \lim_{\alpha\to 1}\sup_{j\in H}M\tau(\alpha,1,j)<\infty.\)
-
\(\displaystyle \lim_{\alpha\to 1}M\tau(\alpha,1,i)=M\tau(1,1,i),\ i\in H.\)
-
\(\displaystyle \lim_{\alpha\to 1}\sup_{j\in H}M(\gamma,1,j)^2<\infty.\)
-
\(\displaystyle \lim_{\alpha\to 1}M(\gamma(\alpha,1,i))^j=M(\gamma(1,1,i))^j,\ i\in H,\ j=1,2.\)
(C): 1. For \(\alpha>\alpha_0\), \(a_1(\alpha)\geq 0\).
- \(a_1(1)=0,\ a_2(1),\ b(1)\in(0,\infty)\).
It is not difficult to show that, under conditions (A) and (B),
\[ \lim_{\alpha\to1} a_j(\alpha)=a_j(1),\quad j=1,2,\qquad \lim_{\alpha\to1} b(\alpha)=b(1). \]
- There exists
\[ \lim_{\alpha\to1,\ t\to\infty} (1+a_1(\alpha)\sqrt{t})^{-1}=g\in[0,1]. \]
Then
\[ \lim_{\alpha\to1,\ t\to\infty} P\left\{ \frac{\tau(\eta_0,\alpha,w(\alpha,t))}{b(1)t}<z \right\} = \]
\[ = \sigma(z)\sqrt{\frac{2}{\pi}} \int_{q/\sqrt{za_2(1)}}^{\infty} \exp\left\{ \frac{1-q}{a_2(1)} -\frac{(1-q)^2}{2a_2^2(1)v^2} -\frac{v^2}{2} \right\}\,dv, \]
where
\[ w(\alpha,t)=\sqrt{t}/(1+a_1(\alpha)\sqrt{t}). \]
Remark 1. Let \(\gamma(\alpha,n,i)=\xi(\alpha,n,i,\tau(\alpha,n,i))\), \(n\ge1,\ i\in\bar H\), where \(\{\{\xi(\alpha,n,i,t),\ t\ge0\},\ n\ge1,\ i\in\bar H\}\), independently of \(T_1(\alpha)\) and \(\{\tau(\alpha,n,i),\ n\ge1,\ i\in\bar H\}\), is a collection of mutually independent, stochastically continuous homogeneous processes with independent increments and finite-dimensional distributions not depending on \(n\). Then condition (B) will take the form:
-
\(\displaystyle \lim_{\alpha\to1}(\mathrm{d})\,\tau(\alpha,1,i)=\tau(1,1,i),\ i\in\bar H.\)
-
\(\displaystyle \lim_{\alpha\to1}\sup_{j\in\bar H} M\bigl(\tau(\alpha,1,j)\bigr)^2<\infty.\)
-
\(\displaystyle \lim_{\alpha\to1} M\bigl(\tau(\alpha,1,i)\bigr)^j = M\bigl(\tau(1,1,i)\bigr)^j,\ i\in\bar H,\ j=1,2.\)
-
\(\displaystyle \lim_{\alpha\to1}(\mathrm{d})\,\xi(\alpha,1,i,1)=\xi(1,1,i,1),\ i\in\bar H.\)
-
\(\displaystyle \lim_{\alpha\to1}\sup_{j\in\bar H} M\bigl(\xi(\alpha,1,i,1)\bigr)^2<\infty.\)
-
\(\displaystyle \lim_{\alpha\to1} M\bigl(\xi(\alpha,1,i,1)\bigr)^j = M\bigl(\xi(1,1,i,1)\bigr)^j,\ i\in\bar H,\ j=1,2.\)
Remark 2. If one introduces the functionals
\[ \hat{\xi}(\eta_0,\alpha,t)= \sum_{k=0}^{\nu(\alpha,t)} \gamma(\alpha,k) +\xi\bigl(\alpha,\nu(\alpha,t)+1,\eta_\alpha(\nu(\alpha,t)),t-\theta_{\nu(\alpha,t)}(\alpha)\bigr), \]
\[ \hat{\tau}(\eta_0,\alpha,s)= \inf\{t:\hat{\xi}(\eta_0,\alpha,t)\ge s\},\qquad s>0, \]
then for \(\hat{\tau}(\eta_0,\alpha,s)\) the result of the theorem holds if, in addition to conditions (A), (B), (C), the condition
\[ \lim_{u\to\infty}\lim_{\alpha\to1}\sup_{x\in[0,\infty)}\sup_{j\in\bar H} P\{\tau(\alpha,1,j)>u+x\mid \tau(\alpha,1,j)>x\}=0. \]
is fulfilled.
Kyiv State University
named after T. G. Shevchenko
Received
21 IV 1969
References
- A. V. Skorokhod, N. P. Slobodyanyuk, Limit Theorems for Random Walks, Kyiv, 1969.