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Consider two continuous function $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$ and another real random variable $X$. Then $T_n :=f(X_n,\cdot)$ and $V :=g(X,\cdot)$ can be interpreted as $\mathcal{L}^2$-valued random variables. Denote the distribution of $T_n$ by $\mathbb{P}_n$ and the distribution of $V$ by $\mathbb{P}$ (then $\mathbb{P}_n$ and $\mathbb{P}$ are defined on the Borel-$\sigma$-field on $\mathcal{L}^2$).

Now if we know that $(\mathbb{P}_n)_{n \in \mathbb{N}}$ is tight and that $(f(X_n,t_1), \ldots ,f(X_n,t_k)) \longrightarrow (g(X,t_1), \ldots ,g(X,t_k))$ in distribution (as $\mathbb{R}^k$-valued random variables) for all $t_1,\ldots,t_k \in \mathbb{R},k \in \mathbb{N}$, can we conclude that $\mathbb{P}_n$ is converging weakly to $\mathbb{P}$ (in the Levy-Prokhorov metric)?

I know that similar results hold for the spaces $C[0,1]$, $D[0,1]$ and the Frechet space $C(\mathbb{R})$, but I just don't know how to deal with this one.

Consider two continuous functions $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$ and another real random variable $X$. Then $T_n :=f(X_n,\cdot)$ and $V :=g(X,\cdot)$ can be interpreted as $\mathcal{L}^2$-valued random variables. Denote the distribution of $T_n$ by $\mathbb{P}_n$ and the distribution of $V$ by $\mathbb{P}$ (then $\mathbb{P}_n$ and $\mathbb{P}$ are defined on the Borel-$\sigma$-field on $\mathcal{L}^2$).

Now if we know that $(\mathbb{P}_n)_{n \in \mathbb{N}}$ is tight and that $(f(X_n,t_1), \ldots ,f(X_n,t_k)) \longrightarrow (g(X,t_1), \ldots ,g(X,t_k))$ in distribution (as $\mathbb{R}^k$-valued random variables) for all $t_1,\ldots,t_k \in \mathbb{R},k \in \mathbb{N}$, can we conclude that $\mathbb{P}_n$ is converging weakly to $\mathbb{P}$ (in the Levy-Prokhorov metric)?

I know that similar results hold for the spaces $C[0,1]$, $D[0,1]$ and the Frechet space $C[0,\infty)$, but I just don't know how to deal with this one.

Consider two continuous function $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$ and another real random variable $X$. Then $T_n :=f(X_n,\cdot)$ and $V :=g(X,\cdot)$ can be interpreted as $\mathcal{L}^2$-valued random variables. Denote the distribution of $T_n$ by $\mathbb{P}_n$ (then $\mathbb{P}_n$ is defined on the Borel-$\sigma$-field on $\mathcal{L}^2$).

Now if we know that $(\mathbb{P}_n)_{n \in \mathbb{N}}$ is tight and that $(f(X_n,t_1), \ldots ,f(X_n,t_k)) \longrightarrow (g(X,t_1), \ldots ,g(X,t_k))$ in distribution (as $\mathbb{R}^k$-valued random variables) for all $t_1,\ldots,t_k \in \mathbb{R},k \in \mathbb{N}$, can we conclude that $T_n \rightarrow V$ in distribution (as $\mathcal{L}^2$-valued random variables)?

I know that similar results hold for the spaces $C[0,1]$, $D[0,1]$ and the Frechet space $C(\mathbb{R})$, but I just don't know how to deal with this one.

Consider two continuous function $f,g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ with $f(x,\cdot), g(x,\cdot) \in \mathcal{L}^2(\mathbb{R},\mathcal{B},\lambda)$ for all $x \in \mathbb{R}$ and a sequence of real random variables $(X_n)_{n \in \mathbb{N}}$ and another real random variable $X$. Then $T_n :=f(X_n,\cdot)$ and $V :=g(X,\cdot)$ can be interpreted as $\mathcal{L}^2$-valued random variables. Denote the distribution of $T_n$ by $\mathbb{P}_n$ and the distribution of $V$ by $\mathbb{P}$ (then $\mathbb{P}_n$ and $\mathbb{P}$ are defined on the Borel-$\sigma$-field on $\mathcal{L}^2$).

Now if we know that $(\mathbb{P}_n)_{n \in \mathbb{N}}$ is tight and that $(f(X_n,t_1), \ldots ,f(X_n,t_k)) \longrightarrow (g(X,t_1), \ldots ,g(X,t_k))$ in distribution (as $\mathbb{R}^k$-valued random variables) for all $t_1,\ldots,t_k \in \mathbb{R},k \in \mathbb{N}$, can we conclude that $\mathbb{P}_n$ is converging weakly to $\mathbb{P}$ (in the Levy-Prokhorov metric)?

I know that similar results hold for the spaces $C[0,1]$, $D[0,1]$ and the Frechet space $C(\mathbb{R})$, but I just don't know how to deal with this one.