A variable with all moments finite that is not sub-Gaussian

ProofIntermediate⏱ 30 minHigh-Dim Probabilitycounterexamplesub-exponentialmoments

Theory: Sub-Gaussian and Sub-Exponential Random Variables →

Problem

Show that $X\sim\text{Exp}(1)$ has all moments finite but is not sub-Gaussian, and compute $\|X\|_{\psi_1}$ . Then show that $Z^2$ for $Z\sim\Ncal(0,1)$ is also sub-exponential but not sub-Gaussian.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

Hints — 0 of 2 revealed

Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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