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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 $R$ 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
$\hat R_n(t)=\frac{1}{n}\displaystyle\sum_{i=1}^n \mathbf 1_{\left\{\alpha^nX^i\leq t\right\}},$
which is the empirical distribution of $\alpha^nX^{1},\dots, \alpha^nX^{n}$.
Can we say that $\hat R_n(t)\to R(t)$ as $n\to\infty$?

Let $\alpha\in\mathbb R^d$, with $\alpha\neq 0$. Take a sequence of iid random variables $X^{1},\dots,X^{n}$ with values in $\mathbb R^d$ and denote $R$ 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
$\hat R_n(t)=\frac{1}{n}\displaystyle\sum_{i=1}^n \mathbf 1_{\left\{\alpha^nX^i\leq t\right\}},$
which is the empirical distribution of $\alpha^nX^{1},\dots, \alpha^nX^{n}$.
Can we say that $\hat R_n(t)\to R(t)$ as $n\to\infty$?

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 $n\to\infty$?

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 $R$ 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
$\hat R_n(t)=\frac{1}{n}\displaystyle\sum_{i=1}^n \mathbf 1_{\left\{\alpha^nX^i\leq t\right\}},$
which is the empirical distribution of $\alpha^nX^{1},\dots, \alpha^nX^{n}$.
Can we say that $\hat R_n(t)\to R(t)$ as $n\to\infty$?

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 $\hat F_N(t)\to F(t)$?

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 $n\to\infty$?