I'm confused with the property of Brownian motion that it is of infinite variation. Consider the sequence $$\sum_{i=0}^{t-1} a^{t-i} (B_{i+1} - B_i)$$ where $a \in [0,1)$ and $B_t$ is Brownian motion.
Would this sequence converge for $t \rightarrow \infty$ like the geometric series? Or does it not converge due to infinite variation of Brownian motion?
The variables $$S_m= \sum_{i=0}^{m-1} a^{m-i} (B_{i+1} - B_i)$$ converge in distribution to the law of $$\sum_{j=1}^\infty a^j Z_j \,,$$ where $Z_j$ are i.i.d. standard normal variables.
The variables $ S_m$ do not converge a.s. because $E[S_m S_{m+k}] \to 0$ as $k \to \infty$ for each $m$.