Let $x\in(1,\infty)$ and let $Z(x)=\sum_{i\ge1}Z_i(x)$ be a sum of independent random variables $Z_1(x),Z_2(x),\dotsb$ such that each random variable is bounded as $Z_i(x)\in[-1/i,1/i]$. Moreover, it is known that $\mathbb{E} Z(x)<\infty$ and $\mathbb{V}ar Z(x)<\infty$ for each $x\in(1,\infty)$ and that $\mathbb{E}Z(x)$ and $\mathbb{C}ov(Z(x),Z(y))$ are continuous in $x$ and $y$. Furthermore, it is known that the range of $Z(x)$ is $\mathbb{R}$ for each $x$, following from $\sum_{i\ge1}1/i=+\infty$.
My question is as follows: based on the given information, is $Z$ a Gaussian random field with continuous mean $\mu(x)=\mathbb{E}Z(x)$ and covariance $C(x,y)=\mathbb{C}ov(Z(x),Z(y))$ functions? If so, why? Note that simulation of $Z$ appears to follow a Gaussian law. Intuitively it seems this should induce a central limit of sorts, but I'm not sure, given the information, how to conclude the Gaussian law.