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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.

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1 Answer 1

up vote 0 down vote accepted

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.

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