Solution
= Solution
Observe the lazy walk on $\mathbb Z_{2^k}$ only after every second genuine jump. Two independent nearest-neighbor signs have sum $-2,0,2$ with probabilities $1/4,1/2,1/4$. Division by two therefore produces one step of the lazy simple random walk on $\mathbb Z_{2^{k-1}}$. The <Strong Markov property> at successive jump times proves
$$
\left(\frac{X_{T_j^{(k)}}^{(k)}}2\right)_{j\geq0}
\overset d=
\left(X_j^{(k-1)}\right)_{j\geq0}.
$$