= Solution
Starting at zero, the next renewal is at time $Y_1$, and there are no intervening renewals because $Y_1\ge1$. Thus the <first return time> obeys $T_0=Y_1$ in distribution, indeed under the original renewal construction. Its <mean recurrence time> is
$$
\boxed{\mathbb E_0T_0=\mathbb EY_1=\mu.}
$$
The chain returns to zero almost surely and has a finite expected return time, so zero is a <positive recurrent state>. The assumption $\mu<\infty$ is what upgrades recurrence to positive recurrence.
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