Distribution of compound Poisson process

Suppose a compound Poisson process is defined as $X_{t} = \sum_{n=1}^{N_t} Y_n$, where $\{Y_n\}$ are i.i.d. with some distribution $F_Y$, and $(N_t)$ is a Poisson process with parameter $\alpha$ and also independent from $\{Y_n\}$.

  1. Is it true that as
    $t\rightarrow \infty, \, \frac{X_{t}-E(X_{t})}{\sigma(X_t)
    \sqrt(N_t)} \rightarrow \mathbf{N}(0, 1)$ in distribution, where the limit is a standard Gaussian distribution? I am considering
    using Central Limit Theorem to show it, but the
    theorem I have learned only applies when $N_t$ is
    fixed and deterministic instead of
    being a Poisson process.
  2. A side question: is it possible to derive the
    distribution of $X_{t}$, for each $t\geq 0$? Some book that has the derivation?

Thanks!

Solutions Collecting From Web of "Distribution of compound Poisson process"

Let $Y(j)$ be i.i.d. with finite mean and variance, and set
$\mu=\mathbb{E}(Y)$ and $\tau=\sqrt{\mathbb{E}(Y^2)}$.
If $(N(t))$ is an independent Poisson process with rate $\lambda$,
then the compound Poisson process is defined as
$$X(t)=\sum_{j=0}^{N(t)} Y(j).$$

The characteristic function of $X(t)$ is calculated as follows:
for real $s$ we have
\begin{eqnarray*}
\psi(s)&=&\mathbb{E}\left(e^{is X(t)}\right)\cr
&=&\sum_{j=0}^\infty \mathbb{E}\left(e^{is X(t)} \ | \ N(t)=j\right) \mathbb{P}(N(t)=j)\cr
&=&\sum_{j=0}^\infty \mathbb{E}\left(e^{is (Y(1)+\cdots +Y(j))} \ | \ N(t)=j\right) \mathbb{P}(N(t)=j)\cr
&=&\sum_{j=0}^\infty \mathbb{E}\left(e^{is (Y(1)+\cdots +Y(j))}\right) \mathbb{P}(N(t)=j)\cr
&=&\sum_{j=0}^\infty \phi_Y(s)^j {(\lambda t)^j\over j!} e^{-\lambda t}\cr
&=& \exp(\lambda t [\phi_Y(s)-1])
\end{eqnarray*}
where $\phi_Y$ is the characteristic function of $Y$.

From this we easily calculate $\mu(t):=\mathbb{E}(X(t))=\lambda t \mu$
and $\sigma(t):=\sigma(X(t))= \sqrt{\lambda t} \tau$.

Take the expansion $\phi_Y(s)=1+is\mu -s^2\tau^2 /2+o(s^2)$ and substitute it into
the characteristic function of the normalized random variable ${(X(t)-\mu(t)) /\sigma(t)}$ to obtain

\begin{eqnarray*}
\psi^*(s) &=& \exp(-is(\mu(t)/\sigma(t))) \exp(\lambda t [\phi_Y(s/\sigma(t))-1]) \
&=& \exp(-s^2/2 +o(1))
\end{eqnarray*}
where $o(1)$ goes to zero as $t\to\infty$. This gives the central limit theorem
$${X(t)-\mu(t)\over\sigma(t)}\Rightarrow N(0,1).$$

We may replace $\sigma(t)$, for example, with $\tau \sqrt{N(t)}$ to get
$${X(t)-\mu(t)\over\tau \sqrt{N(t)}}= {X(t)-\mu(t)\over\sigma(t)} \sqrt{\lambda t \over N(t)} \Rightarrow N(0,1),$$
by Slutsky’s theorem, since $\sqrt{\lambda t \over N(t)}\to 1$ in probability by the law of large numbers.


Added: Let $\sigma=\sqrt{\mathbb{E}(Y^2)-\mathbb{E}(Y)^2}$ be the standard deviation of $Y$,
and define the sequence of standardized random variables
$$T(n)={\sum_{j=1}^n Y(j) -n\mu\over\sigma\sqrt{n}},$$
so that
$${X(t)-\mu N(t)\over \sigma \sqrt{N(t)}}=T(N(t)).$$

Let $f$ be a bounded, continuous function on $\mathbb{R}$. By the usual
central limit theorem we have $\mathbb{E}(f(T(n)))\to \mathbb{E}(f(Z))$ where
$Z$ is a standard normal random variable.

We have for any $N>1$,
$$\begin{eqnarray*}
|\mathbb{E}(f(T(N(t)))) – \mathbb{E}(f(Z))|
&=& \sum_{n=0}^\infty |\mathbb{E}(f(T(n)) – \mathbb{E}(f(Z))|\ \mathbb{P}(N(t)=n) \cr
&\leq& 2\|f\|_\infty \mathbb{P}(N(t)\leq N) +\sup_{n>N} |\mathbb{E}(f(T(n)))- \mathbb{E}(f(Z)) |.
\end{eqnarray*}
$$
First choosing $N$ large to make the right hand side small, then letting $t\to\infty$ so
that $\mathbb{P}(N(t)\leq N)\to 0$, shows that
$$ \mathbb{E}(f(T(N(t)))) \to \mathbb{E}(f(Z)). $$
This shows that $T(N(t))$ converges in distribution to a standard normal as $t\to\infty$.

Derivation of the formula for the distribution of $X_t$. The formula
$$
P_{X_t } = e^{ – t\nu (\mathbb{R})} \sum\nolimits_{k = 0}^\infty {(k!)^{ – 1} t^k \nu ^k } ,
$$
where $\nu$ is the L\’evy measure of $X$ and $\nu^k$ is the $k$-fold convolution of $\nu$, is very easy to derive. Indeed, using the notation in the question, the law of total probability gives
$$
{\rm P}(X_t \in B) = \sum\limits_{k = 0}^\infty {{\rm P}(X_t \in B|N_t = k){\rm P}(N_t = k)} .
$$
Thus,
$$
{\rm P}(X_t \in B) = \sum\limits_{k = 0}^\infty {{\rm P}(Y_1 + \cdots + Y_k \in B)\frac{{e^{ – \alpha t} (\alpha t)^k }}{{k!}}} = e^{ – \alpha t} \sum\limits_{k = 0}^\infty {(k!)^{ – 1} t^k \alpha ^k F_Y^k (B) } ,
$$
where $F_Y^k$ is the $k$-fold convolution of $F_Y$. Now, since the corresponding L\’evy measure $\nu$ is given by $\nu (dx) = \alpha F_Y (dx)$, it holds $\nu(\mathbb{R}) = \alpha$ and $\nu^k = \alpha^k F_Y ^k$, and so the original formula is established.

Remark. As should be clear, the distribution of $X_t$ is very complicated in general.

The distribution of $X_t$ is given by
$$
P_{X_t } = e^{ – t\nu (\mathbb{R})} \sum\nolimits_{k = 0}^\infty {(k!)^{ – 1} t^k \nu ^k } ,
$$
where $\nu$ is the L\’evy measure of $X$, and $\nu^k$ is the $k$-fold convolution of $\nu$. This is contained in Remark 27.3 in the book [L\’evy Processes and Infinitely Divisible Distributions], by Sato.

For a compound Poisson process with rate $\alpha$ and jump distribution $F_Y$, the L\’evy measure $\nu$ is finite and given by $\nu(dx)=\alpha F_Y (dx)$.

EDIT:

For the first question, note that if $t =n \in \mathbb{N}$, then
$$
X_t = X_n = (X_1 – X_0) + (X_2 – X_1) + \cdots + (X_n – X_{n-1})
$$
(note that $X_0 = 0$). Thus, $X_t$ is a sum of $n$ i.i.d. variables, each with expectation ${\rm E}(X_1)$ and variance ${\rm Var}(X_1)$. Now, as is well known and easy to show,
$$
{\rm E}(X_1) = \alpha {\rm E}(Y_1) = \alpha \int {xF_Y (dx)}
$$
and
$$
{\rm Var}(X_1) = \alpha {\rm E}(Y_1^2) = \alpha \int {x^2 F_Y (dx)},
$$
provided that $Y_1$ has finite second moment. Thus, by the central limit theorem,
$$
\frac{{X_t – n\alpha {\rm E}(Y_1 )}}{{\sqrt {\alpha {\rm E}(Y_1^2 )} \sqrt n }} \to {\rm N}(0,1),
$$
as $n \to \infty$.

EDIT: Put it another way,
$$
\frac{{X_t – {\rm E}(X_t )}}{{\sqrt {{\rm Var}(X_t )} }} \to {\rm N}(0,1)
$$
(shown here for the case $t \to \infty$ integer).

EDIT: Some more details in response to the OP’s request.

A compound Poisson process is a special case of a L\’evy process, that is, a process $X=\{X_t: t \geq 0\}$ with stationary independent increments, continuous in probability and having sample paths which are right-continuous with left limits, and starting at $0$. In particular, for any $t \geq 0$ and any $n \in \mathbb{N}$, $X_t$ can be decomposed as a sum of $n$ i.i.d. random variables, which means that $X_t$ is infinitely divisible. There is a vast literature available online on this important topic.