= Skellam distribution
{c}
The distribution of $N_+-N_-$ for independent <Poisson random variables> with means $\lambda_+,\lambda_-$ is the Skellam distribution. Its <characteristic function> is $\exp(\lambda_+(e^{iu}-1)+\lambda_-(e^{-iu}-1))$, its mean is $\lambda_+-\lambda_-$ and its variance is $\lambda_++\lambda_-$. For equal means $\lambda>0$,
$$
\mathbb P(N_+-N_-=k)=e^{-2\lambda}\sum_{j=0}^\infty\frac{\lambda^{2j+|k|}}{j!(j+|k|)!},\qquad k\in\mathbb Z.
$$
This is the fixed-time law of a <symmetric Poisson difference process>.
// Target: markov-process.bigb
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