A compound Poisson random vector $Y$ is well defined in this site in wikipidia.

Nothing prevents me from compound strictly stationary stochastic processes instead of compound random vectors. The definition is the same: let $X= (X_t)_{t\in \mathbb Z}$ be a strictly stationary stochastic processes, so $Y= (Y_t)_{t\in \mathbb Z}$ is defined as $Y=\sum_{j=1}^N X_j$, where: $$Y_t=\sum_{j=1}^N X_{t;j}, \quad \forall t \in \mathbb Z, \quad N \sim \hbox{Poisson}(\lambda).$$ and $(X_j)_{j=1}^{\infty}$ are copies of $X=(X_{t})_{t\in \mathbb Z}$, i.e.: $$X_1=(X_{1;t})_{t\in \mathbb Z},\,\, X_2=(X_{2;t})_{t\in \mathbb Z},\,\,\cdots,\,\, X_j=(X_{j;t})_{t\in \mathbb Z},\,\,\cdots$$ are i.i.d. processes with the same distribution as $X=(X_{t})_{t\in \mathbb Z}$. The same sequence (of random processes) $(X_j)_{j=1}^{\infty}$ is also independent of $N$. Notice that $Y$ is not the classical Compound Poisson process.

So, consider two compound Poisson processes $Y^1$ and $Y^2$ compounding copies of $X^1= (X^1_{t})_{t \in \mathbb Z}$ and $X^2= (X^2_{t})_{t \in \mathbb Z}$, respectively: $$Y_t^1=\sum_{j=1}^{N_1} X_{t;j}^1 \quad Y_t^2=\sum_{j=1}^{N_2} X_{t;j}^2, \quad N_i \sim Poisson(\lambda_i), \,\, i = 1,2$$ $X^1$ and $X^2$ are strictly stationary. Define \begin{equation}\label{a}\tag{I} Z = Y^1 + Y^2, \quad Z_t = \sum_{j=1}^{N_1} X_{t;j}^1 + \sum_{j=1}^{N_2} X_{t;j}^2 \end{equation}

It straightforward to show that $Z = Y^1 + Y^2$ have the same distribution of $Z$, defined as: \begin{equation}\label{ab}\tag{II} Z= \sum_{j=1}^N \xi_j, \quad Z_t=\sum_{j=1}^N \xi_{t;j}, \quad N \sim Poisson(\lambda), \quad \lambda = \lambda_1+\lambda_2 \end{equation}

i.e. $Z$ is a compound Poisson of copies of a strictly stationary process $\xi=(\xi_t)_{t\in \mathbb Z}$ that arise from the following experiment: Let $B\sim Bernoulli(\lambda_1/\lambda)$ with values $H$ and $C$. First, draw $U$. So, if $U=H$, set $\xi=X^1$, and if $U=C$, set $\xi=X^2$.


I would like to know if there are any applications in the literature of macroeconomic, macroeconometric or finance applying this type of processes $Z=(Z_t)_{t\in \mathbb Z}$, either in its representation given by (\ref{a}) or (\ref{ab}). Do you have any literature to recommend me?



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