Weak Convergence of a Sequence of Quickest Detection Problems

Abstract

Consider a production process which exists in one of two states, in-control and out-of-control. Production begins in-control and while there a chance event occurs after each item is produced so that the probability of remaining in-control is $(1 - \pi)$ and the probability of a transition out-of-control is $\pi$. Once out-of-control the process remains there until trouble is removed. Associated with each item produced is a measurable characteristic or quality which is a Gaussian random variable whose mean depends on the underlying process state. Let the mean in-control be $\mu_0$, the mean out-of-control be $\mu_1$ and the common variance be $\sigma^2$. It costs $K$ units to repair the process and the operating cost rate in-control is $c_0$ and out-of-control, $c_1$. The true process state is unknown and a rule is desired which specifies when to repair, based on the quality history, so as to minimize some function of the cost components. This model of a production process was first studied by Girshick and Rubin (1952). They show that when the criterion is long run time average total cost to repair plus operation, the optimal rule is of the form: "Stop and repair after production of the $k$th item if and only if $Z(k) \geqq \zeta$" for some critical value $\zeta$, where $Z(k)$ is a monotonic function of the posterior probability, given the observed history, that the process will be out of control for the $k +$ 1st item. Unfortunately, no easy method for computing the critical value $\zeta$ was given. Shiryaev (1963) studied a similar model but in continuous time and with a slightly different criterion. The continuous time process was a diffusion process and the operating characteristics of the control procedure could be found by solving a second order linear differential equation, thus enabling the appropriate critical value $\zeta$ to be found. Taylor (1967) applied the continuous time computations of Shiryaev to the discrete time model of Girshick and Rubin and plotted the optimal choice for $\zeta$ as a function of the costs and other system parameters. No proof of the validity of the continuous time approximations to the discrete time process was given, however. This paper fills this gap by exhibiting a sequence of Girshick and Rubin processes that converges to the continuous process of Shiryaev, and by showing that the corresponding cost functionals also converge. This procedure of obtaining a limiting continuous process from a sequence of discrete processes is very naturally carried out as an application of the theory of weak convergence of probability measures; cf. Billingsley (1968). More precisely, let $(X_i: i = 1,2,\cdots)$ be a sequence of independent Gaussian random variables with zero mean and unit variance. Let $S_0 = 0, S_k = X_1 + \cdots + X_k$ for $k = 1,2,\cdots$, and define for $n = 1,2,\cdots$ and $k = 0, 1,\cdots, n, X_k(k) = S_k/n^{\frac{1}{2}}$. Let $T_n$ be a geometric random variable with $P\lbrack T_n = j\rbrack = (\pi/n)(1 - \pi/n)^{j-1}$ for $j = 1,2,\cdots$, and $n = \lbrack\pi\rbrack + 1, \lbrack\pi\rbrack + 2,\cdots$. Define $\theta_n(k) = 0\quad\text{for} 0 \leqq k < T_n = n^{-1} (k - T_n + 1) \text{if} T_n \leqq k,$ where $k = 0, 1,\cdots, n$. Finally let $Y_n(k) = \theta_n(k) + X_n(k)$. One interprets $Y_n(k)$ as the cumulative sum up to item $k$ of observations in a truncated (at $n$) Girshick and Rubin process where $\mu_0 = 0, \mu_1 = 1/n, \sigma^2 = 1/n$ and the probability of leaving control between any two items is $\pi_n = \pi/n$. Let $U_n(k)$ be the posterior probability that the production process will be out-of-control for the production of the $(k + 1)$st item, assuming no repair is made. It is convenient to take as our control variable, $Z_n(k)$, the monotone function of $U_n(k)$ given by $Z_n(k) = U_n(k)/(1 - U_n(k))$. Then using the recursion for $U_n(k)$ (cf. Taylor (1967)) it is easy to show by induction that $Z_n(0) = 0$ and for $k = 1,2,\cdots, n$ $Z_n(k) = \pi_n(1 - \pi_n)^{-1} \sum^{k-1}_{j=0} (1 - \pi_n)^{-j} \exp \{Y_n(k) - Y_n(k - j) - j/2n\}.$ Since our analysis will treat probability measures on $C\lbrack 0, 1\rbrack$, the space of continuous, real-valued functions on [0,1] with the metric $(\rho)$ of uniform convergence, it is convenient to introduce continuous versions of the processes $\{X_n(k)\}, \{\theta_n(k)\}, \{Y_n(k)\}$, and $\{Z_n(k)\}$. We define $x_n(t) = X_n(k) + (nt - k)(X_n(k + 1) - X_n(k))$ where $kn^{-1} \leqq t \leqq (k + 1)n^{-1}, k = 0,1,\cdots, n - 1$, and $0 \leqq t \leqq 1$. Observe that $x_n(t) = X_n(k)$ for $t = k/n$ and is defined by linear interpolation for other values of $t$. In a similar manner define $\theta_n(t), y_n(t)$, and $z_n(t)$. Clearly, $y_n(t) = x_n(t) + \theta_n(t)$. Let $\zeta$ be a fixed positive number and $z \varepsilon C\rbrack 0, 1\rbrack$, then define $\tau(z) = \inf \{ t: 0 \leqq t \leqq 1, z(t) = \zeta\}$ if the set $\{t: 0 \leqq t \leqq 1, z(t) = \zeta\}$ is not empty and 1 otherwise. A cost functional whose expected value may be used to measure long run time average cost of operation plus repair (Taylor, (1967)) is given by $C_\zeta^{(n)}(z_n) = \sum^{\rbrack n\tau(z_n)\rbrack}_{k=0} \{c(n^{-1})z_n(k/n) (1 + z_n(k/n))^{-1} - \gamma/n\}$ where $\gamma$ is a constant related to $K$, the repair cost. For the continuous version of Shiryaev (1963) let $\{x(t): 0 \leqq t \leqq 1\}$ be Brownian motion and $T$, an exponential random variable, independent of $\{x(t): 0 \leqq t \leqq 1\}$, with density $\pi e^{-\pi t}, t > 0$. Define the continuous time process $(\theta(t); 0 \leqq t \leqq 1)$ by \begin{align*}\theta(t) = 0\quad\text{if} t \leqq T \\ = t - T \text{if} t > T.\end{align*} Let $y(t) = \theta(t) + x(t)$. The control variable here is $z(t) = \pi \int^t_0 \exp \{\pi s + y(t) - y(t - s) - s/2\} ds$ (Shiryaev (1963)), and the cost functional is $C_\zeta(z) = \int^{\tau(z)}_0 \lbrack cz(s)(1 + z(s))^{-1} - \gamma\rbrack ds$. This is the process whose operating characteristics concerning $C_\zeta(z)$ are computed in Taylor (1967) and assumed to hold approximately for the discrete time process. Our principal objective in this paper is to establish in a rigorous fashion the manner in which the discrete processes converge to the continuous process. We show that the measures induced on $\mathscr{C}$, the Borel sets of $C\lbrack0, 1\rbrack$, by the processes $\{y_n(t): 0 \leqq t \leqq 1\}$ and $\{z_n(t): 0 \leqq t \leqq 1\}$ converge weakly as $n \rightarrow \infty$ to the measures induced by $\{y(t): 0 \leqq t \leqq 1\}$ and $\{z(t): 0 \leqq t \leqq 1\}$ respectively. As a consequence, we also show that the distributions of $\tau(z_n)$ and $C_\zeta^{(n)}(z_n)$ converge weakly to the distributions of $\tau(z)$ and $C_\zeta(z)$, respectively.