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If I have $N$ independent random variables, all identically distributed according to a Laplace distribution with mean $0$ and variance $\sigma^2$ (or a scale parameter of $\sigma/\sqrt{2}$), will their sum have a mean of $0$ and a variance of $N \sigma^2$?

What distribution will this sum have?

If there are two Laplace-distributed random variables and they are correlated with a correlation coefficient of $\rho$, will the variance of their sum be $(\sigma_A^2+\sigma_B^2+2\rho \sigma_A \sigma_B)$, as if they were normal?

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The variance of a sum of independent random variables is the sum of the variances: they don't need to have identical distribution or any special kind of distribution for this to hold, just that each random variable has finite variance. Similarly for the identity $$\sigma_{A+B}^2=\sigma_A^2+\sigma_B^2+2\rho \sigma_A \sigma_B;$$ there is no requirement that the random variables be normally distributed for this to hold.

Finally, for the distribution of the sum of independent Laplacian random variables, the sum is not Laplacian. It will likely be easiest to use characteristic functions or moment-generating functions to deduce the distribution of the sum than to do the $n$-fold convolution.

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