We have a sequence of iid Bernoulli variables, $X_n \sim \text{Bernoulli}(\pi)$, with $n \in \mathbb{N}$. Let $S_t$ and $F_t$ be the number of successes and failures by time $t$, respectively.
How would $S_t$ be distributed? I know this is a count process with $\mathbb{E}[S_t] = \pi t$. I don't know what this count process is called though.
More generally, consider the infinitesimal limit, $X_n = \pm \sqrt{dt}$ (as done in the derivation of Brownian motion). $S_t$ would be the time spent increasing up to time $t$. What's the distribution of $S_t$?
Edit: if we let $dt = t/n$, then by the CLT, $\frac{1}{n}\text{Binom}(n,\pi) \simeq \mathcal{N}(\pi, \pi(1-\pi)/n)$. With a step size of $dt$, we have $\frac{1}{n} \sum_{i=1}^n dt X_i \sim \mathcal{N}(t \pi , t^2/n \pi(1-\pi)) $.
It is a Binomial $Bin(t;\pi)$. If you want to prove it consider the MGF (or CF) of $S_t$