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An agent $A$ is performing a random walk on the number line. Let $X_t$ be his position at time $t$. $X_{t+1}$ is calculated according to the following rules:- $ X_{t+1} =$ \begin{cases} ...

Let $S_n$ denote a simple random walk with i.i.d. increments $X_i$ such that $P(X_1 = 0) = P(X_1=1) = 1/2$, i.e. $$S_0 = 0, \ S_n = X_1 + \dots + X_n.$$ The behavior of $S_n$ as $n \to \infty$ is ...