Show that $X \sim \chi^2(1)$, the chi-squared random variable with one degree of freedom, is a sub-exponential $(4,4)$ random variable.

For this purpose, I first want to show that if $|\lambda|\leq 1/2$ we have $$E[\exp(\lambda(X-1))] \leq (1-2\lambda)^{-1/2}\exp(-\lambda).$$ After that, I want to show that in particular for $|\lambda| \leq 1/4$ we have: $$(1-2\lambda)^{-1/2}\exp(-\lambda)\leq\exp(2\lambda^2)$$

And then, finally I want to show that $X$ is not sub-gaussian.

I believe that should be my progression of logic. However, I am having trouble executing the details. Any help would be appreciated.

  • $\begingroup$ Do you know about moment-generating functions a.k.a MGFs? and if the answer is Yes, do you know that some MGFs exist only for restricted ranges of the argument (such as $|\lambda|<\frac 12$? $\endgroup$ – Dilip Sarwate Oct 11 '18 at 16:16

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