A random variable $X$ dominates another random variable $Y$ in first order stochastic dominance, denoted as $X\geq_1 Y$, if ti holds that $\mathbb{E}\left(\phi(X)\right)\geq \mathbb{E}\left(\phi(Y)\right)$ for all increasing functions $\phi$ for which the expectation is well defined. Given a bounded random variable $X$we denote by $\max[X]=\min\{a:\mathbb{P}[X\leq a]=1\}$, the essential maximum of $X$ and by $\min[X]=\max\{a:\mathbb{P}[X\geq a]=1\}$, the essential minimum of $X$ whica are the maximum and mimimun of the support of $X$. We have the following definition of stochastic dominance for sums of i.i.d. random variables.
$\textbf{Definition:}$ Let $X$ and $Y$ be random variables and let $(X_i)$ and $(Y_i)$ be i.i.d. copies of $X$ and $Y$, respectively. The random variable $X$ first-order dominates $Y$ in the aggregate if for all $n$ large enough
$$\tag{1} X_1+X_2\dots +X_n \geq Y_1+Y_2\dots +Y_n$$
If the observations of each random variable where generating a multivaraite random vector with mean $(\mu_i)_{1\times n}$ and symmetric and positive semi-definite variance-covariance matrix $(\Sigma_i)_{n\times n}$ where i={X,Y} whould the above definition still hold and under which conditions?