Let $E$ be a $\mathbb R$-Banach space, $k\in\mathbb N$ and $\mu_1,\ldots,\mu_k$ be probability measures on $\mathcal B(E)$ with finite second moments. Moreover, let $\mu:=\mu_1\ast\cdots\ast\mu_k$ denote the convolution$^1$ of $\mu_1,\ldots,\mu_k$.
Assuming that $X$ is a $E$-valued random variable with $X\sim\mu$. Can we derive formulas for $\operatorname E[X]$ and $\operatorname{Var}[X]$?
The expectation is easily computed: $$\operatorname E[X]=\sum_{i=1}^k\int\nu_i({\rm d}y_i)y_i\tag1;$$ at least if I'm not missing something.
But the variance seems to be more complicated. For simplicity, assume $k=2$. If $E$ is a $\mathbb R$-Hilbert space $H$, then it should hold $$\left\|\operatorname E[X]\right\|_H^2=\left\|\int\mu_1({\rm d}x_1)x_1\right\|_H^2+\left\|\operatorname E[X]\right\|_H^2=\left\|\int\mu_2({\rm d}x_2)x_2\right\|_H^2+2\left\langle\int\mu_1({\rm d}x_1)x_1,\int\mu_2({\rm d}x_2)x_2\right\rangle_H\tag2$$ and $$\operatorname E\left[\left\|X\right\|_H^2\right]=\int\mu_1({\rm d}x_1)\left\|x_1\right\|_H^2+\int\mu_2({\rm d}x_2)\left\|x_2\right\|_H^2+2\left\langle\int\mu_1({\rm d}x_1)x_1,\int\mu_2({\rm d}x_2)x_2\right\rangle_H\tag3.$$
So, it seems like $\operatorname{Var}[X]$ should be equal to the sum of the "variances" of the $\mu_i$. But can we generalize this to the Banach space case?
$^1$ i.e. if $$\tau:E^k\to E\;,\;\;\;x\mapsto x_1+\cdots+x_k,$$ then $$\mu=\tau\left(\mu_1\otimes\cdots\otimes\mu_k\right)$$ is the pushforward of the product measure $\mu_1\otimes\cdots\otimes\mu_k$ under $\tau$.