Sufficient conditions on $F$ such that $F(X)\in\mathcal{L}^{p}$ for all $X\in\mathcal{L}^{p}$

Question

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $\mathcal{L}^{p}$ the space of real-valued random variables with finite $p$-th moment ($p\in[1,\infty]$). Moreover, let $F:\mathbb{R}\rightarrow\mathbb{R}$ be measurable.

What are sufficient conditions on $F$, such that $X\in\mathcal{L}^{p}\Rightarrow F(X)\in\mathcal{L}^{p}$ ?

Example: F is bounded. Then $F(X)\in\mathcal{L}^{\infty}$ and thereby also $F(X)\in\mathcal{L}^{p}$.

Thanks for your time and effort!

Answer 1

I think the usual condition to require would be linear growth: there exists a constant $C$ such that $|F(t)| \le C(1+|t|)$. Suppose so; then if $X \in L^p$ we can write $$\|F(X)\|_p \le C \|(1 + |X|)\|_p \le C(1 + \|X\|_p) < \infty$$ using Minkowski's inequality.

I suspect that in general this is necessary and sufficient, though I don't offhand know how to prove that.

Note that any Lipschitz function satisfies this; in particular, any differentiable function with bounded derivative.

For a space with infinite measure, you have to drop the 1 and require $|F(t)| \le C|t|$. This is satisfied by any Lipschitz function with $F(0)=0$.