Let the sequence of random variables $X_{n}(s)$ consist of independent equiprobable Bernoulli random variables, that, $P[X_n{(}s) = 0] = 0.5 = P[X_n{(}s) = 1]$
From my perspective, this sequence does not converge almost everywhere, and neither is converges with probability. But I am not clear whether it converge in distribution.
It converges in distribution but not in probability or almost surely. Any sequence $(X_n)$ which is identically distributed (not necessarily independent) converges in distribution. In fact $$P(|X_n-X_m|>\epsilon)$$ $$ =P(X_n=1,X_m=0)+P(X_n=0,X_m=1)=\frac 1 2$$ for any $\epsilon \in (0,1)$ so $(X_n)$ cannot converge in probability.
It does converge in distribution to a random variable with bernoulli distribution with probability $1/2$. In fact, it also converges in probability to that random variable. Recall that a RV can converge in probability to a RV and not to a single value.
