What is a formula for covariance of poisson process
2026-03-30 04:00:20.1774843220
Probability theory of Vapnik
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Let $Y_t:=(-1)^{N_t}$ and $s<t$. Then \begin{align} \mathsf{E}Y_t&=\mathsf{P}(N_t \text{ is even})-\mathsf{P}(N_t \text{ is odd}) \\ &=e^{-\lambda t}(\sinh(\lambda t)-\cosh(\lambda t))=e^{-2\lambda t} \end{align} and \begin{align} \mathsf{E}Y_tY_s&=\mathsf{P}(Y_t=1\mid Y_s=1)\mathsf{P}(Y_s =1) \\ &-\ldots \\ &+\mathsf{P}(Y_t=-1\mid Y_s=-1)\mathsf{P}(Y_s =-1)\\ &=e^{-2\lambda(t-s)}. \end{align} Thus, $$ \operatorname{Cov}(Y_s,Y_t)=e^{-2\lambda(t-s)}-e^{-2\lambda t}e^{-2\lambda s}. $$