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Given an ARMA(1,1) process

$$X_t = \phi X_{t-1} + \varepsilon_t + \theta \varepsilon_{t-1},\quad \varepsilon_t \sim WN(0,\sigma^2)$$

Let $N \sim Po(\lambda)$ a poisson random variable. Consider the compound sums: $$Y_t = \sum_{j=1}^N X_{t;j}$$ where $\{(X_t)_{t \in \mathbb{Z}}\}_{j\in \mathbb{N}}$ are copies (i.i.d. along $j$) of $(X_t)_{t \in \mathbb{Z}}$. Note that $N$ is the same for all $t$.

I am trying to show that the process $(Y_t)_{t \in \mathbb{Z}}$ is also an ARMA(1,1) process. My intuition say to me that it is true.

Note that for any $j$:

$$X_{t,j} = \phi X_{t-1;j} + \varepsilon_{t,j} + \theta \varepsilon_{t-1;j}$$

Taking the sum, we have:

$$\sum_{j=1}^N X_{t,j}= \phi \sum_{j=1}^N X_{t-1;j}+ \underbrace{\sum_{j=1}^N \varepsilon_{t,j}}_{:=\, \epsilon_t} + \theta \,\,\, \underbrace{\sum_{j=1}^N \varepsilon_{t-1;j}}_{ := \, \epsilon_{t-1} }$$

that is:

$$Y_t = \phi Y_{t-1} + \epsilon_t + \theta \epsilon_{t-1}$$

I think that the sum of independent white noise is a white noise and according to this, we have that $E(\epsilon_t) = \lambda E(\varepsilon_t)$ and $Var(\epsilon_t) = \lambda Var (\varepsilon_t)$. Thus, I can say that $\varepsilon_t \sim WN(0,\lambda \sigma^2)$

I don't know if my argument is right. I would like to receive a feedback from you, in case I'm missing a more rigorous argument.

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    Conditional on $N$ you can find the orders of $Y_t$ via https://www.jstor.org/stable/2345178? – Jarle Tufto Jan 13 '22 at 08:26
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    Interesting question, but I have to admit I am skeptical about the claim: when your Poisson has a realization of zero, $Y_t=0$. If $\lambda$ is small, this can happen quite often, and the result will not look very ARMA(1,1). Looking forward to "real" answers. @RichardHardy, any thoughts? – Stephan Kolassa Jan 13 '22 at 08:27
  • This is very similar to the *thinning operator* used often in integer time series analysis, where instead of ARMA series $X$ one would add random numbers of (typically) IID Bernoulli variables, yielding "binomial thinning". So this could be called "ARMA thinning", but searching for that term yields pretty much nothing at all. – Stephan Kolassa Jan 13 '22 at 08:40
  • (Also, simulating this for a small value of $\lambda$, e.g. $\lambda=0,2$, indeed gives something that does not look very ARMA to me...) – Stephan Kolassa Jan 13 '22 at 08:42
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    Wait, I think I'm misunderstanding. You have a *single* realization of $N$, so you add a fixed number of ARMA(1,1) up, right? In that case, Jarle's comment should stand, and you can disregard my comments, I was thinking about drawing a new $N$ every $t$... – Stephan Kolassa Jan 13 '22 at 08:47
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    I think what you suggest is correct. When all the processes you add have the same autoregressive and moving average parameter, the resulting sum is indeed also ARMA(1,1) (and this is not in disagreement with the paper I linked to). The resulting noise terms $\epsilon_t$ marginally have Poisson compound distributions with a point mass at zero. They are not independent but still uncorrelated and hence qualifies as white noise. – Jarle Tufto Jan 13 '22 at 09:08
  • @StephanKolassa, I suppose the OP and Jarle have figured it out correctly. – Richard Hardy Jan 13 '22 at 18:35
  • @user346624 Please register &/or merge your accounts (you can find information on how to do this in the **My Account** section of our [help]), then you will be able to edit & comment on your own question. – Sycorax Jan 13 '22 at 22:27
  • Yes, I have a single realization of $N$, so I add a fixed number of ARMA(1,1). – user346624 Jan 13 '22 at 21:51
  • Related to the coments above: Yes, I have a single realization of $N$, so I add a fixed number of ARMA(1,1). – user346624 Jan 13 '22 at 21:51
  • Hi: It seems that Eduardo Engle proved it ( I've decided that anything interesting has already been proven if you look hard enough !!!! LOL ) but the p and q of the result changes in some non-intuitive way. https://www.researchgate.net/publication/229706727_A_unified_approach_to_the_study_of_sums_products_time-aggregation_and_other_functions_of_ARMA_processes – mlofton Jan 14 '22 at 01:19

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