#StackBounty: #normal-distribution #poisson-distribution #gaussian-mixture-distribution Gaussian distribution with poisson variance

Bounty: 50

Let’s $r_tsim mathcal{N}(0,V_t)$ where $V_t = (1+m)^{J_t}$, $mgeq0$ and $J_tsim mathcal{P}oi(lambda)$. How can we compute the distribution of $r_t$ when the parameters are $m$ and $lambda$. In other words I would like to compute $$sum_{jgeq0}frac{e^lambda}{j!}lambda^jfrac{1}{sqrt{2pi(1+m)^{j}}}expleft{-frac{r_t^2}{2(1+m)^{j}}right},.$$

I know this sum is convergent as for all $j$, $expleft{-frac{r_t^2}{2(1+m)^{j}}right}leq 1$ then $$sum_{jgeq0}frac{e^lambda}{j!}lambda^jfrac{1}{sqrt{2pi(1+m)^{j}}}expleft{-frac{r_t^2}{2(1+m)^{j}}right} leq sum_{jgeq0}frac{e^lambda}{j!}lambda^jfrac{1}{sqrt{2pi(1+m)^{j}}} = frac{expleft{lambda+lambda(1+m)^{-1/2}right}}{sqrt{2pi}},.$$

My objective at the end will be to estimate this model by MCMC. So, I would like to sample parameters $m$ and $lambda$ after computing their posterior. An accepted answer should then help me to compute those posteriors.

Thanks


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