I need to show that $G_n \stackrel{P}{\to}_n F_0$, i.e. for any $\epsilon>0$ $$ P(|| G_n - F_0||>\epsilon) \to_n 0 $$ We know the following:
$G_n$ and $F_0$ are a bilinear functions from $\mathbb{R}^m\times \mathbb{R}^m$ to $\mathbb{R}$
$F_0(x,y)=F(Hx,Hy)$ is a deterministic function where F is a bilinear function from $\mathbb{R}^d\times \mathbb{R}^d$ to $\mathbb{R}$ and $H$ is a linear function from $ \mathbb{R}^m$ to $\mathbb{R}^d$ for $d>m$.
- $G_n(x.y)=F_n(H_nx,H_ny)$ where $(F_n)$ are random functions and $(H_n)$ are deterministic functions of which we know the following: $F_n \stackrel{P}{\to}_n F$ and $H_n \to_n H$. Where the $F_n$'s for fixed $\omega$ are bilinear as above and the $H_n$'s are linear as above.
So the statement to be shown is: if $$ F_n \stackrel{P}{\to}_n F \quad \text{and} \quad H_n \to_n H \quad \text{then} \quad G_n \stackrel{P}{\to}_n F_0 $$
I have tried to show it for some hours now, but I can't seem to get anywhere. It is a part of a proof in my notes, which doesn't go into the details of why it holds. If anyone could tell me how this is shown I would be grateful.