# Applications of a certain type of stochastic processes in macroeconomic, macroeconometric or finance

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 $$$$\label{a}\tag{I} Z = Y^1 + Y^2, \quad Z_t = \sum_{j=1}^{N_1} X_{t;j}^1 + \sum_{j=1}^{N_2} X_{t;j}^2$$$$

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

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 $$\xi=X^2$$.

Question

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?