Abstract
Full Text
UDC 519.217
MATHEMATICS
Yu. K. BELYAEV
ON THE NUMBER OF ENTRIES OF TRAJECTORIES OF A MARKOV PROCESS INTO A GIVEN SET OF THE PHASE SPACE
(Presented by Academician A. N. Kolmogorov, 20 XII 1968)
Let \(\xi_t\), \(-\infty < t < +\infty\), be a stationary, time-homogeneous Markov process with values in the measurable space \([\mathscr E,\mathfrak B_{\mathscr E}]\).
Definition 1. We shall say that the time \(t\) is an entry of a trajectory of the process \(\xi_t\) into \(B\) from \(\bar A\), if \(\xi_t \in B \subseteq A\) and, for any \(\varepsilon > 0\), there exist \(s \in (t-\varepsilon,t)\) such that \(\xi_s \in \bar A=\mathscr E\setminus A\).
By \(\eta_A(\Delta,B)\) we denote the number of entries of \(\xi_t\) into \(B\) from \(\bar A\) that occurred in the time interval \(\Delta\), assuming that \(\eta_A(\Delta,B)\) are random variables. The random variables \(\eta_A(\Delta,B)\) form a stationary random stream \((^1)\), whose characteristics (intensity, moments, etc.) can be expressed in terms of the function \(M_{x,A}(\Delta,B)\), defined for \(\Delta \subset R^+=(0,+\infty)\) by the relation
\[ M_{x,A}(\Delta,B)=M\{\eta_A(\Delta,B)\mid \xi_0=x\}. \tag{1} \]
Below we shall call this function basic. Suppose that \(M_{x,A}(\Delta,A)<\infty\) for every interval \(\Delta \subset R^+\) having finite length \(|\Delta|<\infty\). This assumption is essential; however, it is satisfied for a wide class of Markov processes with a discrete intervention of chance \((^2)\) and sets \(B\subseteq A\subset \mathscr E\). Under the assumption made, \(M_{x,A}(\Delta,B)\) can be extended to a measure on \([A\times R^+,\mathfrak B_A\times\mathfrak B_{R^+}]\), where \(\mathfrak B_A=\mathfrak B(B:B\subseteq A,\ B\in\mathfrak B_{\mathscr E})\), \(\mathfrak B_{R^+}\) is the \(\sigma\)-algebra of Borel sets on \(R^+\). The basic functions satisfy the relation
\[ M_{x,A}(\Delta+t,B)=\int_{y\in\mathscr E} P_x(t,dy)\,M_{y,A}(\Delta,B), \tag{2} \]
where \(\Delta+t=\{s:\ s-t\in\Delta\}\), \(\Delta\subset R^+\), and \(P_x(t,A)\) are the transition probability functions.
Definition 2. A family of measures \(P_t\) on \([\mathscr E,\mathfrak B_{\mathscr E}]\) converges, as \(t\to\infty\), to \(P\) with respect to the partition \(\mathscr C=\{\mathscr E_{n,k}\}\), \(\mathscr E=\bigcup_k \mathscr E_{n,k}\), \(\mathscr E_{n,k}\cap \mathscr E_{n,l}=\varnothing\), \(k\ne l\), \(\mathscr E_{n,k}=\bigcup_{l\in I_{n,k}}\mathscr E_{n+1,l}\), if for every \(n\)
\[ \lim_{t\to\infty}\sum_k \left|P_t(\mathscr E_{n,k})-P(\mathscr E_{n,k})\right|=0. \tag{3} \]
From the convergence of \(P_t\) to \(P\) in variation \((^3)\) there follows convergence with respect to any partition \(\mathscr C\).
Theorem 1. Let \(P_x(t,A)\) converge to the stationary measure \(P_\infty(A)=P\{\xi_t\in A\}\) with respect to \(\mathscr C\), and let the basic functions be such that
\[ \lim_{n\to\infty}\max_k\left[ \sup_{x\in\mathscr E_{n,k}} M_{x,A}(\Delta_0+t_0,B) - \inf_{x\in\mathscr E_{n,k}} M_{x,A}(\Delta_0+t_0,B) \right]=0, \tag{4} \]
\[ M_{x,A}(\Delta_0+t_0,B)\leq g(x), \]
\[ \lim_{t\to\infty}\int_{y\in\mathscr E} P_t(x,dy)\,g(y) = \int_{\mathscr E} P_\infty(dy)\,g(y). \]
Then
\[ \lim_{t\to\infty} M_{x,A}(\Delta_0+t,B)=\mu_A(B)=M\eta_A(\Delta_0,B), \tag{5} \]
where \(\Delta_0=(0,1)\).
Proof follows from the uniform integrability of \(M_{y,A}(\Delta_0+t_0,B)\), the approximation of the main function by step functions on \(\mathscr C\), and relations (2), (3).
Relation (5) is an analogue of Blackwell’s theorem (4). The main difficulty in using Theorem 1 consists in verifying the convergence of \(P_t(x,A)\) to \(P_\infty(A)\) with respect to partitions \(\mathscr C=\{\mathscr C_{n,k}\}\) that are “sufficiently” fine for (4).
If \(\Delta_i=(t_i,t_i+u_i)\), \(t_i+u_i<t_{i+1}\) \((\Delta_i<\Delta_{i+1})\), \(i=1,\ldots,k\), then to the \(k\)-dimensional rectangle \(\Delta_1\times\cdots\times\Delta_k\) we assign the number
\[ M_{(k),A}(\Delta_1\times\cdots\times\Delta_k,B)= \]
\[ =\int_{y_1\in B}\int_{s_1\in\Delta_1}\mu_A(dy_1)\,ds_1 \int_{y_2\in B}\int_{s_2\in\Delta_2-s_1} M_{y_1,A}(ds_2,dy_2)\cdots \]
\[ \cdots \int_{y_k\in B}\int_{s_k\in\Delta_k-(s_1+\cdots+s_{k-1})} M_{y_{k-1},A}(ds_k,dy_k). \]
We shall assume that \(M_{(k),A}\) is bounded for any bounded intervals \(\Delta_i<\Delta_{i+1}\), \(i=1,\ldots,k\). In this case it can be extended to a measure on the Borel sets of the cone
\(C_k=\{(t_1,\ldots,t_k),\,t_1<t_2<\cdots<t_k\}\). Using permutations of the values of the coordinates of the points of \(C_k\), one can extend \(M_{(k),A}\) to a measure in \(R^k\). The introduced measure has the property that, for any Borel set \(D\subset R^k\), the measure \(M_{(k),A}(D,B)\) is equal to the mathematical expectation of the number of all points \((t_1,\ldots,t_k)\in D\) whose coordinates \(t_i\), \(i=1,\ldots,k\), are entrances of \(\xi_t\) into \(B\) from \(\overline A\).
Definition 3. The measure \(M_{(k),A}\) will be called the \(k\)-leading measure of the flow of entrances into \(B\) from \(\overline A\).
Integrating the \(k\)-leading measure over appropriate sets in \(R^k\), one can obtain various moments of the random variables \(\eta_A(\Delta,B)\) (see (5)). For example, the following holds.
Theorem 2. The \(k\)-th factorial moment of the number of entrances into \(B\) from \(\overline A\) during the time \((0,t)\) is equal to
\[ M_{(k)}(t,B,A)= \]
\[ = k!\int_{y_1\in B}\int_0^t \mu_A(dy_1)\,ds_1\cdots \int_{y_k\in B}\int_0^{t-s_1-\cdots-s_{k-1}} M_{y_{k-1},A}(ds_k,dy_k). \tag{6} \]
Proof follows from the fact that the number of selections \((t_1,\ldots,t_k)\), \(t_1<t_2<\cdots<t_k\), from \(\eta_A((0,t),B)=\eta\) entrances into \(B\) from \(\overline A\) is equal to
\(\eta(\eta-1)\cdots(\eta-k+1)/k!\)
Studying the asymptotics of integrals of \(k\)-leading measures of flows of entrances into \(B\) from \(\overline A\), one can obtain limit theorems on the convergence of “thinned” flows to Poisson flows.
Definition 4. We shall say that a random flow \(\eta_\varepsilon(\Delta)\), \(\varepsilon>0\), converges in distribution to a flow \(\eta_0(\Delta)\) as \(\varepsilon\to0\) \((\eta_\varepsilon(\Delta)\Rightarrow\eta_0(\Delta))\), if for any collection of intervals \(\Delta_1,\ldots,\Delta_m\),
\[ \lim_{\varepsilon\to0} P_\varepsilon\{\eta_\varepsilon(\Delta_i)=k_i,\ i=1,\ldots,m\} = P_0\{\eta_0(\Delta_i)=k_i,\ i=1,\ldots,m\}. \]
Let \(\xi_{t,\varepsilon}\) be a stationary Markov process on \([\mathscr E_\varepsilon,\mathfrak B_\varepsilon]\), and let
\(\eta_{A_\varepsilon}(\Delta,B_\varepsilon)\) be the flow of entrances into \(B_\varepsilon\) from \(\overline A_\varepsilon\), \(\varepsilon>0\) a parameter, and \(M_{x,A_\varepsilon}(\Delta,B_\varepsilon)\) the main function of the flow \(\eta_{A_\varepsilon}(\Delta,B_\varepsilon)\). If \(\Delta=(t_1,t_2)\), then put
\[
\Delta/\mu_{A_\varepsilon}(B_\varepsilon)
=
(t_1/\mu_{A_\varepsilon}(B_\varepsilon),\,
t_2/\mu_{A_\varepsilon}(B_\varepsilon)).
\]
Introduce the majorant
\[ \overline M_{A_\varepsilon}(B_\varepsilon) = \sup_{s>0}\sup_{y\in B_\varepsilon} M_{y,A_\varepsilon}(\Delta_0+s,B_\varepsilon). \tag{7} \]
Theorem 3. If the fundamental functions \(M_{y,A_\varepsilon}(\Delta,B_\varepsilon)\) satisfy the conditions
\[ \lim_{\varepsilon\to 0}\overline{M}_{A_\varepsilon}(B_\varepsilon)=0 \]
and, uniformly in \(y\in B_\varepsilon\) and \(\varepsilon\leq \varepsilon_0\),
\[ \lim_{s\to\infty} M_{y,A_\varepsilon}(\Delta_0+s,B_\varepsilon)/\mu_{A_\varepsilon}(B_\varepsilon)=1, \tag{9} \]
then
\[ \eta_{A_\varepsilon}(\Delta/\mu_{A_\varepsilon}(B_\varepsilon),B_\varepsilon)\Rightarrow \eta_0(\Delta), \tag{10} \]
where \(\eta_0(\Delta)\) is a Poisson flow with unit intensity.
The proof is based on verifying the convergence of the moments of the random variables
\(\eta_{A_\varepsilon}(\Delta_i/\mu_{A_\varepsilon}(B_\varepsilon),B_\varepsilon)\),
\(\Delta_i\cap\Delta_j=\varnothing\), \(i\ne j\), \(i,j=1,\ldots,k\), \(k=1,2,\ldots\), to the moments of mutually independent Poisson random variables with means equal to \(|\Delta_i|\). For example, for \(k=1\) it is first proved that asymptotically the integral (6), with \(A,B,t\) replaced by \(A_\varepsilon,B_\varepsilon,t_\varepsilon=t/\mu_{A_\varepsilon}(B_\varepsilon)\), is equivalent to the integral
\[ \begin{aligned} M_{(k)}(\tau_\varepsilon,t_\varepsilon,B_\varepsilon,A_\varepsilon) &=k!\int_{y_1\in B_\varepsilon}\int_{0}^{t_\varepsilon-(k-1)\tau_\varepsilon} \mu_{A_\varepsilon}(dy_1)\,ds_1 \times \\ &\quad \times \int_{y_2\in B_\varepsilon}\int_{\tau_\varepsilon}^{t_\varepsilon-s_1-(k-2)\tau_\varepsilon} M_{y_1,A_\varepsilon}(ds_2,dy_2)\cdots \int_{y_k\in B_\varepsilon}\int_{\tau_\varepsilon}^{t-s_1-\cdots-s_{k-1}} M_{y_{k-1},A_\varepsilon}(ds_k,dy_k) \end{aligned} \]
for a corresponding growth of \(\tau_\varepsilon\). From (8) we find that, uniformly in \(y\in B_\varepsilon\),
\[ \lim_{\varepsilon\to 0,\ \tau_\varepsilon\to\infty,\ u_\varepsilon\to\infty} \int_{\tau_\varepsilon}^{\tau_\varepsilon+u_\varepsilon} (\tau_\varepsilon+u_\varepsilon-s)^k M_{y,A_\varepsilon}(ds,B_\varepsilon)/ \mu_{A_\varepsilon}(B_\varepsilon)(u_\varepsilon^{k+1}/k+1)=1. \]
Hence, in turn, we obtain that
\[ \lim_{\varepsilon\to 0} M(\tau_\varepsilon,t_\varepsilon,B_\varepsilon,A_\varepsilon)=t^k. \]
The convergence of the finite-dimensional distributions of the quantities
\(\eta_{A_\varepsilon}(\Delta_i/\mu_{A_\varepsilon}(B_\varepsilon),B_\varepsilon)\)
to Poisson distributions is obtained as a consequence of the theorem on convergence of moments [6].
Definition 5. An entrance \(t\) into \(B\) from \(\overline{A}\) will be called a \(\tau\)-entrance (\(\bar{\tau}\)-entrance) if in the interval \((t-\tau,t)\) there are no entrances (there are entrances) into \(B\) from \(\overline{A}\). A \(\tau\)-entrance into \(B\) from \(\overline{A}\) will be called the beginning of a \(\tau\)-packet if it is followed by a \(\bar{\tau}\)-entrance into \(B\) from \(\overline{A}\).
Suppose that the original flow is a superposition of \(m\) renewal flows having renewal intensities \((1)\), \(h(t)\to h_0\), \(t\to\infty\). Then the subflow of initial moments of \(\varepsilon\)-packets, as \(\varepsilon\downarrow 0\), converges in distribution to a Poisson flow in a time scale normalized by the intensity of occurrence of \(\varepsilon\)-packets. This result can be obtained by verifying conditions (8), (9) for the flow of entrances of the Markov process
\(\xi_t=(u_{1t},\ldots,u_{mt})\), where \(u_{it}\) are the backward recurrence times (1) into the set
\[ A=\{(x_1,\ldots,x_m),\ x_i=0,\ x_j<\tau,\ x_k>0,\ i\ne j\ne k,\ i,j,k=1,\ldots,m\} \]
from \(\overline{A}\).
Moscow State University
named after M. V. Lomonosov
Received
20 XII 1968
CITED LITERATURE
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