For each $i=1,\dots,n$ let $Z_i\sim\text{Poisson}(\lambda_i)$, and suppose $\{Z_i\}$ are independent. Also for each $i=1,\dots,n$, let $\{Y_{ij}\}_{j\in\mathbb{N}}$ be an infinite sequence of iid random variables, where $Y_{ij}\sim\text{Bernoulli}(p_i)$. Define

$$ X_1=\sum_{i=1}^n\sum_{j=1}^{Z_i}Y_{ij}\quad\text{and}\quad{X_2}=\sum_{i=1}^n\nu_iZ_i $$

for some strictly positive scalars $\nu_i$. It's easy to show that

$$ X_1\sim\text{Poisson}\bigg(\sum_{i=1}^n\lambda_ip_i\bigg), $$

but that $X_2$ does not follow any well-known distribution (as a linear combination of Poisson random variables). How can I numerically compute $\mathbb{P}[\min\{X_1-X_2-3,X_1-2X_2\}\leqslant0]$ without using Monte Carlo?

What I have

For fixed scalars $\alpha,\beta\in\mathbb{R}_+$, I computed the characteristic function of $X_1-\alpha X_2-\beta$ to be

$$ \varphi(t)=\exp\bigg[-\iota\beta{t}+\sum_{i=1}^n\lambda_ip_i\big(\varphi_i(t)-1\big)\bigg] $$


$$ \varphi_i(t)=(1-p_i)e^{-\iota\alpha\nu_it}+p_ie^{\iota(1-\alpha\nu_i)t}. $$

for $\iota=\sqrt{-1}$. I can then use the Gil-Pelaez inversion formula to numerically calculate $\mathbb{P}[X_1-\alpha{X_2}-\beta\leqslant0]$ (this works quite well in practice). To calculate the minimum, I need to work out something about the joint distribution, which is where I'm stuck.

  • $\begingroup$ Hvae you tried computing $E[e^{i(t_1X_1+t_2X_2)}]$? By inverting this you'll get $P(X_1\leq a,X_2\leq b)$, and then just use inclusion-exclusion to calculate the minimum. $\endgroup$ – Alex R. Aug 8 '18 at 0:22
  • $\begingroup$ @AlexR. That would be the characteristic function of the vector-valued random variable $X=(X_1,X_2)$, right? Is there an obvious or well-known extension of the GP inversion formula to the vector-valued case? $\endgroup$ – David M. Aug 8 '18 at 0:41
  • $\begingroup$ Is this a homework question? $\endgroup$ – Sextus Empiricus Aug 8 '18 at 6:23
  • $\begingroup$ @MartijnWeterings No, it’s a stylized version of a more complex problem I encountered. If you just want to give a hint or tip, I would appreciate that too $\endgroup$ – David M. Aug 8 '18 at 12:11
  • $\begingroup$ Not that it helps much but we could write the problem more compact (and somewhat more general) as a sum of random vectors: $$R_i \sim \text{Poisson}(\lambda_i)$$ $$\vec{V} = \begin{bmatrix} V_1 \\ V_2 \end{bmatrix} = \sum_{i=1}^n \begin{bmatrix} \nu_i \\ \xi_i \end{bmatrix} R_i $$ compute $P[V_1 \leq a \text{ or } V_2 \leq b] \equiv 1-P[V_1>a \text{ and } V_2 >b]$. I guess that you should be able to express the characteristic function for $\vec{V}$ and then use some inversion formula to perform the numeric computation. $\endgroup$ – Sextus Empiricus Aug 9 '18 at 9:54

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