Abstract Generated abstract
This paper studies the asymptotic distribution of the first time at which a cumulative process exceeds a high level, extending a known limit theorem for sums of independent identically distributed random variables to sums governed by a regular semi-Markov process. The construction uses an embedded ergodic Markov chain, state-dependent holding times and increments, and regeneration at returns to a fixed state to reduce the process to independent identically distributed cycle variables whose characteristic functions and moments are described. Under convergence, tightness, and moment conditions on transition probabilities, holding times, and increments, the paper proves a limit theorem for the normalized attainment time in the critical case where the mean increment tends to zero. Two remarks indicate extensions to increments generated by processes with independent increments and to a version including the current unfinished sojourn.
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.