No, this isn't true, and to construct a counterexample, take any of the stable distributions with nice scaling properties that have first moments but not second moments, that would be in the usual notation, ( https://en.wikipedia.org/wiki/Stable_distribution) $\alpha \in (1,2)$, and base the random walk on X-1. $$P(S_n - n > 0) = P( \frac {S_n}{n^{\frac 1 \alpha}} - n^{1-\frac 1 \alpha} > 0) = P(X_1 > n^{\frac {\alpha -1} \alpha})$$$$$$ The wikipedia article mentions that this last is order of $\frac 1 {n^{\alpha -1}}$, which sums to infinity. The random walk, though, is just a negative mean r.w. and so it is not >0 i.o.

No, this isn't true, and to construct a counterexample, take any of the stable distributions with nice scaling properties that have first moments but not second moments, that would be in the usual notation, ( https://en.wikipedia.org/wiki/Stable_distribution) $\alpha \in (1,2)$, and base the random walk on X-1. $$P(S_n - n > 0) = P( \frac {S_n}{n^{\frac 1 \alpha}} - n^{1-\frac 1 \alpha} > 0) = P(X_1 > n^{\frac {\alpha -1} \alpha})$$$$$$ The wikipedia article mentions that this last is order of $\frac 1 {n^{\alpha -1}}$, which sums to infinity. The random walk, though, is just a negative mean r.w. and so it is not >0 i.o. This, https://en.wikipedia.org/wiki/Hsu–Robbins–Erdős_theorem,shows that this behavior is typical of r.w.s without 2nd moments.

No, this isn't true, and to construct a counterexample, take any of the stable distributions that have first moments but not second moments, that would be in the usual notation, ( https://en.wikipedia.org/wiki/Stable_distribution) $\alpha \in (1,2)$, and base the random walk on X-1. $$P(S_n - n > 0) = P( \frac {S_n}{n^{\frac 1 \alpha}} - n^{1-\frac 1 \alpha} > 0) = P(X_1 > n^{\frac {\alpha -1} \alpha})$$$$$$ The wikipedia article mentions that this last is order of $\frac 1 {n^{\alpha -1}}$, which sums to infinity. The random walk, though, is just a negative mean r.w. and so it is not >0 i.o.

No, this isn't true, and to construct a counterexample, take any of the stable distributions with nice scaling properties that have first moments but not second moments, that would be in the usual notation, ( https://en.wikipedia.org/wiki/Stable_distribution) $\alpha \in (1,2)$, and base the random walk on X-1. $$P(S_n - n > 0) = P( \frac {S_n}{n^{\frac 1 \alpha}} - n^{1-\frac 1 \alpha} > 0) = P(X_1 > n^{\frac {\alpha -1} \alpha})$$$$$$ The wikipedia article mentions that this last is order of $\frac 1 {n^{\alpha -1}}$, which sums to infinity. The random walk, though, is just a negative mean r.w. and so it is not >0 i.o.