Expectation of the power of sum of two random variables whose probabilities are individualy bounded

Question

I have been inspired by this question: Expectation of the sum of two random variables whose probabilities are individualy bounded I am wondering if solution can be generalized for $$ E(A+B)^p, $$ where $p>1$.

Answer 1

You can apply almost exactly the same procedure by just using that $$ \mathbb{P}(|A+B|^p>x) \; = \; \mathbb{P}(|A+B|>x^{1/p})$$ since the $p$-th root is increasing on $[0,\infty)$.