= 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 $p=\mathbb P(\eta\geq x)>0$. On this event convexity gives $f(Z_j+x)^+\leq f(Z_j)^++f(Z_j+\eta)^+$, 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 $Z_j\leq jM$ 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 $f$.
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