 Integrability of translated convex random-walk rewards

Let $Z_j$ be the sum of $j$ independent identically distributed real increments. For a finite convex function $f$, integrability of $f(Z_j)$ and $f(Z_{j+1})$ implies integrability of $f(Z_j+x)$ for every fixed $x$. Its negative part is controlled by negative part of a finite convex function is Lipschitz. For $x>0$, if the increment law is unbounded above, take an independent increment $\eta$ with $p=\mathbb{P}(\eta \ge x)>0$. On this event convexity gives $f(Z_j+x{)}^{+}\le f(Z_j{)}^{+}+f(Z_j+\eta {)}^{+}$, so the expectation is bounded by $\mathbb{E}f(Z_j{)}^{+}+p^{-1}\mathbb{E}f(Z_{j+1}{)}^{+}$. If increments are bounded above by $M$, use $Z_j\le jM$ and bound the positive part by $f(Z_j{)}^{+}+f(jM+x{)}^{+}$. For $x<0$, use the symmetric lower-tail argument. This supplies translated integrability without assuming integrable increments or a growth bound on $f$.