# Convolutions of joint random variables

I have two discrete dependent random variables $$X,Y$$, where both $$X$$ and $$Y$$ can take values either $$0$$ or $$1$$. Furthermore, I know their joint distribution $$f_{X,Y}(X,Y)$$.

Now let's say I have an iid sequence $$(X_1, Y_1) \dots, (X_K,Y_K)$$, with $$K >0$$ known. I'm interested in computationally obtaining the probability distribution of $$f_{\sum_{k =1}^K X_k,\sum_{k =1}^K Y_k} \left(\sum_{k =1}^K X_k, \sum_{k =1}^K Y_k \right).$$

Does anyone know if I can do this via convolutions/Fast Fourier transforms? And if so how this can be done?

I'd ideally like to generate a $$K+1$$ by $$K+1$$ matrix whose $$i,j$$th entry is $$\mathbb{P}(\sum_{k =1}^K X_k = i-1, \sum_{k =1}^K Y_k = j-1)$$.

Thanks!

• Convolution generalizes naturally to multivariate arrays and the FFT method similarly generalizes. – whuber Nov 14 '19 at 23:50