Consider a simple random walk $$\mathcal{X}_t= \sum_{n<t} X_n,$$ where $P(X_n=1)= P(X_n=-1)= 1/2.$

If I put an extra condition that excludes cases with more than 5 consecutive +1, or -1 in the sum:

For every $n$, between 1 and t-4:

$$|X_n+ X_{n+1}+ X_{n+2}+ X_{n+3}+ X_{n+4}|< 5.$$

Can I still expect that $\mathcal{X}_t \ll \sqrt{t}$ almost surley?

If yes, how can we prove this?