Take a sequence of iid random variables $X^{1},\dots,X^{N}$ with a given cdf $F$.   
Now define $\alpha^N=g(X^{1},\dots,X^{N})$ for some function $g$, and suppose we know that $\alpha^N\to\alpha \in \mathbb R$ almost surely as $N\to\infty$. Define    
$\hat F_N(t)=\frac{1}{N}\displaystyle\sum_{n=1}^N \mathbf 1_{\left\{\alpha^NX^n\leq t\right\}},$
which is the empirical distribution of $\alpha^NX^{1},\dots, \alpha^NX^{N}$.    
Can we say that $\hat F_N(t)\to F(t/\alpha)$ as $N\to\infty$?    
EDIT: Is the result still true in a multivariate setting? Let's say that $\alpha$, $\alpha^N$ and $X^i$ are in $\mathbb R^d$, and that $F$ is the cdf of $\alpha X^i$ (scalar product). Define $\hat F_N(t)$ in the same way, where now the product in the indicator is a scalar product. Do we have $\hat F_N(t)\to F(t)$?