Background
Let $X_t$ be the continuous time Markov process on the state space {Working, Broken} with failure rate $\alpha$ and repair rate $\beta$. By elementary calculations [1]
$$
\begin{align*}
P\left\{ X_t = \text{Working}\ | \ X_s = \text{Working} \right\} &= p + (1-p) \cdot e^{-(\alpha+\beta)(t-s)} \\
P\left\{ X_t = \text{Broken}\ | \ X_s = \text{Broken} \right\} &= 1-p + p \cdot e^{-(\alpha+\beta)(t-s)}
\end{align*}
$$
where $p = \frac{\beta}{\alpha+\beta}$.
Without loss of generality, we assert that the process has converged to its stationary distribution at time $t = 0$. (Equivalently, from any start time, let the process run until it is arbitrarily close to stationary, then reset the time origin.)
Let $U$ be the cumulative sojourn Working during an interval $[0,\tau]$ (the process's uptime). Then $U$ is asymptotically normal as $ \tau \rightarrow \infty $, with a mean $\mu_U$ and variance $\sigma^2_U$ that can be calculated from $\alpha$ and $\beta$ [2].
My conjecture
Suppose we reward the process at rate $g(t)$ if it is Working at time $t \in [0,\tau]$,
where $g$ is bounded and $F(x) = \frac{1}{\tau} g^{-1}(x)$ is a well-defined cumulative distribution function. Let $\mu_R$ and $\sigma^2_R$ be the mean and variance of the distribution defined by $F$.
Let $Q$ be the reward accumulated during $[0,\tau]$. My conjecture is that $Q$ is asymptotically normal as $ \tau \rightarrow \infty $, with mean $\mu_Q$ and variance $\sigma^2_Q$ calculated as
$$
\begin{align*}
\mu_Q &= \mu_R \mu_U \\
\sigma^2_Q &\approx \left( \sigma^2_R + \mu_R^2 \right) \sigma^2_U
\end{align*}
$$
Edit: Originally conjectured `=' for the calculation of $\sigma^2_Q$ but I'm not so sure now.
The conjecture has empirical support from experiments with the following
-
$g(t) = \frac{k}{\tau}\left(\frac{t}{\tau}\right)^{k-1}$ for $k > 1$
-
$g(t) = \frac{k}{\tau}\left(1 – \frac{t}{\tau}\right)^{k-1}$ for $k > 1$
-
$F$ specifying a uniform distribution
-
$F$ specifying a triangular distribution
-
$F$ specifying an arcsine distribution
-
$F$ specifying a U-quadratic distribution
What I'm trying
Setting $\delta t = \frac{\tau}{n}$, $t_k = (k-1)\delta t$ and using $1_{t}$ to indicate that $X_t$ is Working at time $t$, we have
$$
Q = \lim_{\delta t \rightarrow 0} \sum_{k = 1}^{n} g(t_k) \cdot 1_{t_k} \ \delta t
$$
Now suppose that $R$ is distributed according to $F$. Then choosing $Y$ uniformly from $[0,1]$ and calculating $g(Y\tau)$ is equivalent to sampling $R$ (by the inverse probability integral transform). So
$$
Q = \lim_{\delta t \rightarrow 0} \sum_{\ell = 1}^{M} R_{\ell} \ \delta t
$$
where $M = \frac{U}{\delta t}$ and the $R_{\ell}$ are distributed according to $F$.
Thus $Q$ is the sum of a random number of identically distributed but dependent random variables.
My question
What theorems exist on asymptotic distributions for the sum of a random number of identically distributed but dependent random variables?
(I am aware of [3] for example on the asymptotic distribution for the sum of a random number of independent identically distributed random variables.)
Many thanks in advance.
References
[1] Nicolas Privault, 2013, Understanding Markov chains: Examples and applications, Springer Undergraduate Mathematics Series, vol. IX, Springer, Singapore.
[2] Lajos Takács, 1959, On a sojourn time problem in the theory of stochastic
processes, Transactions of the American Mathematical Society, 93(3), 531–540.
[3] Herbert Robbins, 1948, The asymptotic distribution of the sum of a random number of random variables, Bulletin of the American Mathematical Society), 54, 1151-1161.
Best Answer
I [think I have] proved the calculation for $\mu_Q$ and $\sigma^2_Q$ by applying the methods I used in a related problem.
I then [also think I have] proved that $Q$ is asymptotically normal (under certain conditions) using the Berry-Esseen theorem derived at [1].
References
[1] Gutti Jogesh Babu, Malay Ghosh, Kesar Singh, 1978, On rates of convergence to normality for $\phi$-mixing processes, The Indian Journal of Statistics, 40(A)(3), 278-293.