# Limit of Lp norm

Posted on 28 Apr 2015
Tags: math, probability

While proving a lemma in support of a proposition in support of a theorem in a paper I’m wrapping up, I found myself needing to show that $$\lim_{p \to \infty} ||X||_p \to ||X||_\infty$$ for $$X$$ some random variable. A discussion on Mathematics Stack Exchange describes a proof for the more general situation where $$f$$ is a measurable function on a finite measure space and $$||f||_q < \infty$$ for $$q$$ sufficiently large. The conclusion in this case is that $$\lim_{p \to \infty} ||f||_p = ||f||_\infty$$.

A nice thing about the random variable special case is that it allows for a slightly longer but still interesting proof (a scenic route to the result?). The sketch is first to show that $$||X||_p$$ is nondecreasing in $$p$$ for $$p > 0$$. This can be accomplished by comparing $$||X||_p$$ and $$||X||_q$$ for $$q > p$$ using Jensen’s inequality with the convex function $$g(x) = |x|^{q / p}$$. This implies that $$||X||_p \uparrow L$$ for some $$L \in \mathbb{R} \cup \{ \infty \}$$. It’s easy to show that $$L \leq ||X||_\infty$$. It’s only a bit less easy to show that $$L \geq ||X||_\infty$$, using the standard trick of comparing $$||X||_p \geq ||\, |X| \wedge M||_p$$ for $$M < ||X||_\infty$$, and then carefully taking the limits in $$M$$ and $$p$$.