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](http://en.wikipedia.org/wiki/Inverse_transform_sampling)). 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.



Sum of a random number of identically distributed but dependent random variables?
I [*think I have*] proved the calculation for $\mu\_Q$ and $\sigma^2\_Q$ by applying the [methods I used in a related problem](https://mathoverflow.net/questions/184280/publishing-an-elementary-proof-of-a-less-general-and-less-useful-version-of-a-cl).


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.


