Functional limit theorems for discounted exponential functional of random walk and discounted convergent perpetuity

Abstract

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ be independent and identically distributed ${\mathbb{R}}^2$-valued random vectors. Put $S_0≔0$ and $S_k≔{\xi }_1+\dots +{\xi }_k$ for $k\in \mathbb{N}$. We prove a functional central limit theorem for a discounted exponential functional of the random walk ${\sum }_{k\ge 0}{e}^{-S_k∕t}$, properly normalized and centered, as $t\to \infty$. In combination with a theorem obtained recently in Iksanov et al. (2021) this leads to an ultimate functional central limit theorem for a discounted convergent perpetuity ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\eta }_{k+1}$, again properly normalized and centered, as $t\to \infty$. The latter result complements Vervaat’s (1979) one-dimensional central limit theorem. Our argument is different from that used by Vervaat. The functional limit theorem is not informative in the case where ${\xi }_k={\eta }_k$. As a remedy, we show that ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\xi }_{k+1}$ concentrates tightly around the point $t$ in a deterministic manner.

Introduction

Let ${\xi }_1$, ${\xi }_2,\dots$ be independent copies of a real-valued random variable $\xi$. Denote by ${\left(S_k\right)}_{k\in {\mathbb{N}}_0}$, where ${\mathbb{N}}_0≔\mathbb{N}\cup \left\{0\right\}$, the zero-delayed standard random walk with jumps ${\xi }_k$, that is, $S_0≔0$ and $S_k≔{\xi }_1+\cdots +{\xi }_k$ for $k\in \mathbb{N}$. For each $t>0$, put $X\left(t\right)≔{\sum }_{k\ge 0}{e}^{-S_k∕t}$. Plainly, $X\left(t\right)<\infty$ a.s. for all/some $t>0$ if, and only if, ${lim}_{k\to \infty }{S}_k=+\infty$ a.s. Under this condition we call ${\left(X\left(t\right)\right)}_{t>0}$ discounted exponential functional of random walk. Our terminology is inherited from a continuous-time counterpart ${\int }_0^{\infty }exp(-S\left(y\right))\mathrm{d}y$, where ${\left(S\left(y\right)\right)}_{y\ge 0}$ is a Lévy process diverging to $+\infty$, known in the literature as ‘exponential functional of Lévy process’, see, for instance, (Bertoin and Yor, 2005, Carmona et al., 2001). Distributional properties of $X\left(t\right)$, with $t>0$ fixed, were investigated in Szabados and Székely (2003). The latter authors call $X\left(t\right)$ exponential functional of random walk. There are many works investigating the asymptotics of ${\sum }_{k=0}^{n}f\left(S_k\right)$ as $n\to \infty$ for various functions $f$ (for instance, exponential) in the situation that the corresponding series is divergent. This circle of problems which is very different from that related to discounted exponential functional is discussed in the books Borodin and Ibragimov (1995) and Skorokhod and Slobodenjuk (1970).

As usual, we write $\stackrel{\mathbb{P}}{\to }$ to denote convergence in probability, and $\Rightarrow$ and $\stackrel{\mathrm{d}}{⟶}$ to denote weak convergence in a function space and weak convergence of one-dimensional distributions, respectively. We prefer to use the notation $Y_t\left(u\right)\Rightarrow Y\left(u\right)$ as $t\to \infty$ in place of $Y_t\left(\cdot \right)\Rightarrow Y\left(\cdot \right)$. Also, we denote by $D(0,\infty )$ ($D[0,\infty )$) the Skorokhod space of right-continuous functions defined on $(0,\infty )$ (on $[0,\infty )$) with finite limits from the left at positive points.

Here is our first result.

Theorem 1.1

Assume that $\mu ≔\mathbb{E}\xi \in (0,\infty )$ and ${\sigma }^2≔\mathrm{Var}\xi \in (0,\infty )$. Then, as $t\to \infty$,

$$\frac{1}{t^{1∕2}}\left(\sum _{k\ge 0}{e}^{-u{S}_k∕t}-\frac{t}{\mu u}\right)\Rightarrow {\left(\frac{{\sigma }^2}{{\mu }^3}\right)}^{1∕2}{\int }_{[0,\infty )}{e}^{-uy}\mathrm{d}B\left(y\right)$$

in the $J_1$-topology on $D(0,\infty )$, where $B≔{\left(B\left(y\right)\right)}_{y\ge 0}$ is a standard Brownian motion.

Remark 1.2

Under the assumptions $\xi \ge 0$ a.s. and $\mathbb{E}{\xi }^p<\infty$ for all $p>0$ a one-dimensional central limit theorem for ${\sum }_{k\ge 0}{e}^{-S_k∕t}$ as $t\to \infty$ was proved in Theorem 2 of Dall’ Aglio (1964) with the help of the method of moments.

Remark 1.3

The limit process in Theorem 1.1 is an a.s. continuous centered Gaussian process on $(0,\infty )$ with

$$\mathbb{E}{\int }_{[0,\infty )}{e}^{-uy}\mathrm{d}B\left(y\right){\int }_{[0,\infty )}{e}^{-vy}\mathrm{d}B\left(y\right)=\frac{1}{u+v},u,v>0.$$

Integration by parts yields an alternative representation

$${\int }_{[0,\infty )}{e}^{-uy}\mathrm{d}B\left(y\right)=u{\int }_0^{\infty }{e}^{-uy}B\left(y\right)\mathrm{d}y,u>0.$$

Let $({\xi }_1,{\eta }_1)$, $({\xi }_2,{\eta }_2),\dots$ be independent copies of an ${\mathbb{R}}^2$-valued random vector $(\xi ,\eta )$ with arbitrarily dependent components. Whenever the random series ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\eta }_{k+1}$ is a.s. convergent for each $t>0$, following Iksanov et al. (2021), we call its sum discounted convergent perpetuity. We refer to the cited article for the justification of the term and standard limit theorems: a strong law of large numbers, a functional central limit theorem and a law of the iterated logarithm. In particular, Proposition 1.4 given next is Theorem 1.2 in Iksanov et al. (2021), up to the change of variable.

Proposition 1.4

Assume that $\mu =\mathbb{E}\xi \in (0,\infty )$, $\mathbb{E}\eta =0$ and ${\mathtt{s}}^2≔\mathrm{Var}\eta \in (0,\infty )$. Then, as $t\to \infty$,

$$\frac{1}{t^{1∕2}}\sum _{k\ge 0}{e}^{-u{S}_k∕t}{\eta }_{k+1}\Rightarrow {\left(\frac{{\mathtt{s}}^2}{\mu }\right)}^{1∕2}{\int }_{[0,\infty )}{e}^{-uy}\mathrm{d}B\left(y\right)$$

in the $J_1$-topology on $D(0,\infty )$, where ${\left(B\left(y\right)\right)}_{y\ge 0}$ is a standard Brownian motion.

A combination of Theorem 1.1 and Proposition 1.4 plus some additional work enables us to treat the case $\mathbb{E}\eta \ne 0$, thereby leading to an ultimate version of the functional central limit theorem for discounted convergent perpetuities.

Corollary 1.5

Assume that $\mu =\mathbb{E}\xi \in (0,\infty )$, ${\sigma }^2=\mathrm{Var}\xi \in [0,\infty )$, $\mathtt{m}≔\mathbb{E}\eta \in \mathbb{R}$, ${\mathtt{s}}^2=\mathrm{Var}\eta \in [0,\infty )$, ${\sigma }^2+{\mathtt{s}}^2>0$. Then

$$\frac{1}{t^{1∕2}}\left(\sum _{k\ge 0}{e}^{-u{S}_k∕t}{\eta }_{k+1}-\frac{\mathtt{m}t}{\mu u}\right)\Rightarrow v{\int }_{[0,\infty )}{e}^{-uy}\mathrm{d}B\left(y\right)$$

in the $J_1$-topology on $D(0,\infty )$. Here, ${\left(B\left(y\right)\right)}_{y\ge 0}$ is a standard Brownian motion,

$$v^2≔{\mu }^{-1}\mathrm{Var}({\mu }^{-1}\mathtt{m}\xi -\eta )={\sigma }^2{\mu }^{-3}{\mathtt{m}}^2-2\gamma \mathtt{m}{\mu }^{-2}+{\mathtt{s}}^2{\mu }^{-1}\in [0,\infty ),\gamma ≔\mathrm{Cov}(\xi ,\eta )=\mathbb{E}\xi \eta -\mu \mathtt{m}\in \mathbb{R}.$$

The constant $v^2$ is positive unless $\xi =c\eta$ for some $c\in \mathbb{R}$.

Remark 1.6

Putting in Corollary 1.5 $u=1$ we obtain the one-dimensional central limit theorem

$$\frac{1}{t^{1∕2}}\left(\sum _{k\ge 0}{e}^{-S_k∕t}{\eta }_{k+1}-\frac{\mathtt{m}t}{\mu }\right)\stackrel{\mathrm{d}}{⟶}{2}^{-1∕2}v\mathrm{Normal}(0,1),t\to \infty ,$$

where $\mathrm{Normal}(0,1)$ is a random variable with the standard normal distribution. This result was proved in Theorem 6.1 of Vervaat (1979). Our argument is different from that exploited in Vervaat (1979).

A perusal of the proof of Corollary 1.5 reveals that the distributional asymptotic behavior of the process on the left-hand side of (3) is driven by the competitive distributional fluctuations of the functional versions of the random walks $\left(S_k\right)$ and $({\eta }_1+\cdots +{\eta }_k)$. The latter are represented in the limit by the same integral functional of (generally) dependent Brownian motions $-B_1$ and $B_2$. Because of the minus sign and the dependence the contributions of the two random walks compensate each other. When $\xi$ and $\eta$ are linearly dependent, the Brownian motion $B_1$ is a multiple of $B_2$. As a result, the contributions of the random walks get mutually neutralized, so that the limit variance $v^2$ is equal to $0$. This means that $t^{-1∕2}$ is not a proper normalization in this case. According to Theorem 1.7, when $\eta =c\xi$ for nonzero real $c$, no normalization is needed at all, and ${\sum }_{k\ge 0}{e}^{-S_k∕t}{\eta }_{k+1}$ concentrates tightly around the point $ct$ in a deterministic manner for $t$ large enough.

Theorem 1.7

Assume that $\mu =\mathbb{E}\xi \in (0,\infty )$ and ${\sigma }^2=\mathrm{Var}\xi \in [0,\infty )$. Then

$$\underset{t\to \infty }{lim}\left(\sum _{k\ge 0}{e}^{-S_k∕t}{\xi }_{k+1}-t\right)=\frac{\mathbb{E}{\xi }^2}{2\mu }\text{a.s.}$$

The rest of the paper is structured as follows. We prove Theorem 1.1, Corollary 1.5 and Theorem 1.7 in Sections 2, 3 and 4 respectively. The appendix collects some auxiliary facts.