As Christian Remgling's example $\mu_n:=\delta_{e_n}$ shows, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness.

It's worth pointing out that a sequence of characteristic functions can converge pointwise to a continuous positive definite function which is not the characteristic function of a random variable. Indeed, consider the map $x\mapsto \exp\left(-\lVert x\rVert^2/2\right)$. Let $(\eta_j)_{j\geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j)_{j\geqslant 1}$ is an orthonormal basis of $H$. By uniqueness theorem in separable Hilbert spaces, $\phi$ would be the characteristic function of $X$. But the sequence $\left(\sum_{j=1}^n\eta_je_j\right)_{n\geqslant 1}$ is not tight.

However, there exists a characterization of tightness of a family of measures on a Hilbert space which involve the characteristic functional and Hilbert Schmidt operators. In Araujo and Giné's book The central limit theorem in Banach spaces, we encounter the following result:

Theorem 1.4.17. Let $H$ be a separable Hilbert space, and $\Gamma$ a set of probability measures on the Borel $\sigma$-algebra of $H$. The set $\Gamma$ has a compact closure for the weak-$^*$ topology if and only if for all $\varepsilon>0$, we can find a family of $\{A_{\mu}^\varepsilon\}_{\mu\in\Gamma}$ Hilbert-Schmidt operators on $H$ such that for a Hilbert basis $\{e_j\}$, the following properties hold:

 

1. $\displaystyle\sup_{\mu\in\Gamma}\sum_{j=1}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2<\infty$;
2. $\displaystyle\lim_{N\to +\infty}\sup_{\mu\in\Gamma}\sum_{j=N}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2=0$;
3. for all $v\in H$, $\mu\in\Gamma$, $$\left|1-\int_He^{i\langle v,x\rangle}d\mu(x)\right|\leqslant \lVert A\mu^{\varepsilon}(v)\rVert+\varepsilon.$$

As Christian Remgling's example $\mu_n:=\delta_{e_n}$ shows, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness.

It's worth pointing out that a sequence of characteristic functions can converge pointwise to a continuous positive definite function which is not the characteristic function of a random variable. Indeed, consider the map $x\mapsto \exp\left(-\lVert x\rVert^2/2\right)$. Let $(\eta_j)_{j\geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j)_{j\geqslant 1}$ is an orthonormal basis of $H$. By uniqueness theorem in separable Hilbert spaces, $\phi$ would be the characteristic function of $X$. But the sequence $\left(\sum_{j=1}^n\eta_je_j\right)_{n\geqslant 1}$ is not tight.

However, there exists a characterization of tightness of a family of measures on a Hilbert space which involve the characteristic functional and Hilbert Schmidt operators. In Araujo and Giné's book The central limit theorem in Banach spaces, we encounter the following result:

Theorem 1.4.17. Let $H$ be a separable Hilbert space, and $\Gamma$ a set of probability measures on the Borel $\sigma$-algebra of $H$. The set $\Gamma$ has a compact closure for the weak-$^*$ topology if and only if for all $\varepsilon>0$, we can find a family of $\{A_{\mu}^\varepsilon\}_{\mu\in\Gamma}$ Hilbert-Schmidt operators on $H$ such that for a Hilbert basis $\{e_j\}$, the following properties hold:

1. $\displaystyle\sup_{\mu\in\Gamma}\sum_{j=1}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2<\infty$;
2. $\displaystyle\lim_{N\to +\infty}\sup_{\mu\in\Gamma}\sum_{j=N}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2=0$;
3. for all $v\in H$, $\mu\in\Gamma$, $$\left|1-\int_He^{i\langle v,x\rangle}d\mu(x)\right|\leqslant \lVert A\mu^{\varepsilon}(v)\rVert+\varepsilon.$$

As Christian Remgling's example $\mu_n:=\delta_{e_n}$, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness.

It's worth pointing out that a sequence of characteristic functions can converge pointwise to a continuous positive definite function which is not the characteristic function of a random variable. Indeed, consider the map $x\mapsto \exp\left(-\lVert x\rVert^2/2\right)$. Let $(\eta_j)_{j\geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j)_{j\geqslant 1}$ is an orthonormal basis of $H$, we have, by uniqueness theorem in separable Hilbert spaces that $\phi$ would be the characteristic function of $X$. However, the sequence $\left(\sum_{j=1}^n\eta_je_j\right)_{n\geqslant 1}$ is not tight.

However, there exists a characterization of tightness of a family of measures on a Hilbert space which involve the characteristic functional and Hilbert Schmidt operators. In Araujo and Giné's book The central limit theorem in Banach spaces, we encounter the following result:

Theorem 1.4.17. Let $H$ be a separable Hilbert space, and $\Gamma$ a set of probability measures on the Borel $\sigma$-algebra of $H$. The set $\Gamma$ has a compact closure for the weak-$^*$ topology if and only if for all $\varepsilon>0$, we can find a family of $\{A_{\mu}^\varepsilon\}_{\mu\in\Gamma}$ Hilbert-Schmidt operators on $H$ such that for a Hilbert basis $\{e_j\}$, the following properties hold:

1. $\displaystyle\sup_{\mu\in\Gamma}\sum_{j=1}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2<\infty$;
2. $\displaystyle\lim_{N\to +\infty}\sup_{\mu\in\Gamma}\sum_{j=N}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2=0$;
3. for all $v\in H$, $\mu\in\Gamma$, $$\left|1-\int_He^{i\langle v,x\rangle}d\mu(x)\right|\leqslant \lVert A\mu^{\varepsilon}(v)\rVert+\varepsilon.$$

As Christian Remgling's example $\mu_n:=\delta_{e_n}$ shows, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness.

It's worth pointing out that a sequence of characteristic functions can converge pointwise to a continuous positive definite function which is not the characteristic function of a random variable. Indeed, consider the map $x\mapsto \exp\left(-\lVert x\rVert^2/2\right)$. Let $(\eta_j)_{j\geqslant 1}$ be a sequence of i.i.d. standard Gaussian random variables and $X:=\sum_{j\geqslant 1}\eta_je_j$, where $(e_j)_{j\geqslant 1}$ is an orthonormal basis of $H$. By uniqueness theorem in separable Hilbert spaces, $\phi$ would be the characteristic function of $X$. But the sequence $\left(\sum_{j=1}^n\eta_je_j\right)_{n\geqslant 1}$ is not tight.

However, there exists a characterization of tightness of a family of measures on a Hilbert space which involve the characteristic functional and Hilbert Schmidt operators. In Araujo and Giné's book The central limit theorem in Banach spaces, we encounter the following result:

Theorem 1.4.17. Let $H$ be a separable Hilbert space, and $\Gamma$ a set of probability measures on the Borel $\sigma$-algebra of $H$. The set $\Gamma$ has a compact closure for the weak-$^*$ topology if and only if for all $\varepsilon>0$, we can find a family of $\{A_{\mu}^\varepsilon\}_{\mu\in\Gamma}$ Hilbert-Schmidt operators on $H$ such that for a Hilbert basis $\{e_j\}$, the following properties hold:

1. $\displaystyle\sup_{\mu\in\Gamma}\sum_{j=1}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2<\infty$;
2. $\displaystyle\lim_{N\to +\infty}\sup_{\mu\in\Gamma}\sum_{j=N}^{+\infty}\lVert A_\mu^\varepsilon(e_j)\rVert^2=0$;
3. for all $v\in H$, $\mu\in\Gamma$, $$\left|1-\int_He^{i\langle v,x\rangle}d\mu(x)\right|\leqslant \lVert A\mu^{\varepsilon}(v)\rVert+\varepsilon.$$