Let $x_i$, $i = 1,2 ,\cdots$ be a time series, and consider the random elements $$ W_n(t) = \frac{1}{\sqrt{n}}\sum_{i}^{[nt]} x_i $$ on the Skorohod space $D[0,1]$.
Standard argumentfor showing $W_n$ converges weakly to, say, Brownian motion $W$ goes as follows:
Show the finite dimensional distributions of $W_n$ converges to that of $W$.
Show the sequence $W_n$, $n = 1, 2, \cdots$, is tight.
It is well known that convergence in finite dimensional distribution need not imply weak convergence. But I have never seen an example where Step 1 holds but Step 2 fails. Anyone know such an example?
It need not be restricted to Brownian motion---similar question can be asked about e.g. fractional Brownian motion or the Rosenblatt process.
Let $(e_i)_{i\geqslant 1}$ be an i.i.d. sequence where $e_1$ is centered and has variance one. Let $x_i=e_i+y_i-y_{i+1}$, where $(y_i)_{i\geqslant 1}$ is a sequence of independent identically distributed random variables. Then $$ W_n(t)=\frac 1{\sqrt n}\sum_{i=1}^{[nt]}e_i+\frac 1{\sqrt n}\left(y_1-y_{[nt]+1}\right) $$ hence the convergence of the finite dimensional distributions holds. However, tightness of $(W_n)$ would hold only if $\max_{1\leqslant i\leqslant n}\left\lvert y_i\right\rvert/\sqrt n\to 0$ in probability, which may not be the case when $y_1$ does not have a finite moment of order $2$.