Let $\alpha\in\mathbb R^d$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $F$ the cdf of $\alpha X^1$ (where the product is a scalar product). Now consider $\alpha^n=g(X^{1},\dots,X^{n})$ for some function $g$, and suppose we know that $\alpha^n\to\alpha$ almost surely as $n\to\infty$. Define
$\hat F_n(t)=\frac{1}{n}\displaystyle\sum_{i=1}^n \mathbf 1_{\left\{\alpha^nX^i\leq t\right\}},$
which is the empirical distribution of $\alpha^nX^{1},\dots, \alpha^nX^{n}$.
Can we say that $\hat F_n(t)\to F(t)$ as $n\to\infty$?