The Discounted Central Limit Theorem and its Berry-Esseen Analogue

Abstract

Let $X_0, X_1, X_2, \cdots$ be a sequence of independent random variables with common distribution function $F(x)$. Let $\nu$ be a discount factor $(0 < \nu < 1)$. Then we define \begin{equation*}\tag{1.1} X_\nu = \sum^\infty_{k=0} \nu^k X_k,\end{equation*} which may be interpreted as the present value of a sum of certain periodic and identically distributed payments $X_k$. We assume that the first three moments of $X_k$ are finite: \begin{align*}\tag{1.2}\mu &= \int^{+\infty}_{-\infty} x dF(x) < \infty,\quad \sigma^2 = \int^{+\infty}_{-\infty} (x - \mu)^2 dF(x) < \infty, \\ \rho &= \int^{+\infty}_{-\infty}|x - \mu|^3 dF(x) < \infty.\\ \end{align*} It will be shown that the normalized random variable \begin{equation*}\tag{1.3} Z_\nu = \frac{(1 - \nu)^{\frac{1}{2}}}{\sigma} \big(X_\nu - \frac{\mu}{1 - \nu}\big)\end{equation*} is asymptotically normal for $\nu \rightarrow 1$. The analogue of the Berry-Esseen theorem (see [1], [2], [3], [4]) will be established for the difference $F_\nu(x) - \mathcal{N}_\nu(x), F_\nu(x)$ being the distribution of $Z_\nu$, whereas $\mathcal{N}_\nu(x)$ is the normal distribution with zero mean and variance $(1 + \nu)^{-1}$: \begin{equation*}\tag{1.4} \mathcal{N}_\nu(x) = \big(\frac{1 + \nu}{2\pi} \big)^{\frac{1}{2}}\int^x_{-\infty} \exp \big{-\frac{1 + \nu}{2}t^2\big)dt.\end{equation*} The proof uses those Fourier techniques which are masterfully presented in Feller's book [3] for the proof of the "ordinary" Berry-Esseen theorem. In the last section the corresponding estiamte is established for the compound Poisson process. Other aspects of the Discounted Central Limit Theorem are treated in a paper by Whitt [5].