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I am looking for similar results for point-wise convergence of quantile function, but with a different setting.

Suppose $\{B_n\}$ is a series of random variables, $B_n>0$ for each $n$. Let $Z$ be a standard normal variable, and $Z$ is independent from the whole $\{B_n\}$. We know that $B_n\to B$ a.s., where $B$ is a positive definite constant.

Will it be true that $$ P\{ B_n Z\leq x \vert B_n\} \to P\{ B Z\leq x\vert B\} $$ in some sense (a.s., in probability or other ways), and the limit $P\{ B Z\leq x\vert B\}=P\{ B Z\leq x\}$ since $B$ is a constant matrix?

I appreciate any guidance!

Percy W
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First, using that $\mathbb E\left[h(U,V)\mid U\right]=g(U)$ with $g(u)=\mathbb E\left[h(u,V)\right]$, we find that $$ \mathbb P\left(f(B_nZ)\leqslant x\mid B_n\right)=g(B_n), $$ where $g(u)=\mathbb P(f(uZ)\leqslant x)$. Notice also that $g$ is continuous, except maybe at $0$ (write $g$ as an integral involving the density of $uZ$). Therefore, $g(B_n)\to g(B)$.

Davide Giraudo
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