Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that $$\int_{-\infty}^{\infty} X^n_t \, \mathrm{d} t =\int_{m}^{M} X_t^n \, \mathrm{d}t < \infty,$$ where $M = \sup_{0 \leq s \leq T} B_s$ and $m = \inf_{0 \leq s \leq T} B_s$, for a standard Brownian motion $B$ and some fixed $T>0$. Let $Y_t$ be a continuous random variable which also satisfies that $$\int_{\infty}^{\infty} Y_t \, \mathrm{d} t = \int_m^M Y_t \, \mathrm{d} t.$$

Is it true (maybe under some additional assumptions) that $$\int_{-x}^x X^n_t \, \mathrm{d} t \xrightarrow{d} \int_{-x}^{x} Y_t \,\mathrm{d}t $$ for any $x>0$ implies the same convergence in distribution with $x$ replaced by $\infty$?

**EDIT**: How about the case where $X_t^n = X_t$ for $n \geq 1$ where $X_t$ is non-negative and continuous? Does the conclusion hold?