# Large Deviations for Self-Normalized Sums

I am trying to understand the main result (Theorem 1.1) in this paper by Shao, which gives a large deviation bound for the self-normalized sum of iid variables $$\frac{\sum X_i}{\sqrt{n}\sqrt{\sum X_i^2}}$$ without any conditions on the moments of $$X$$. One of the key steps involves applying Cramer's theorem to a sum of the variables $$bX_i -x\left(\frac{X_i^2+b^2}{2}\right)$$ for $$b>0$$, $$x>\mathbb{E}X/\sqrt{\mathbb{E}X^2}$$, and $$\mathbb{E}X\geq 0$$. And, to this end, the author claims without proof that $$\mathbb{E}e^{t\left[bX_i -x\left((X_i^2+b^2)/2\right)\right]}<\infty$$ for all such $$b$$, $$x$$, and $$t\geq 0$$. Perhaps I am missing something obvious, but I don't see how to establish this. I tried using a Taylor expansion as well as the AM-GM inequality to get a bound $$\mathbb{E}e^{t\left[bX_i -x\left((X_i^2+b^2)/2\right)\right]}\leq\min\left\lbrace \mathbb{E}e^{tb(1-x)X},c\mathbb{E}e^{t(1-x)X^2/2} \right\rbrace$$ but I couldn't see how to get the result from either of these methods. Any help with this would be greatly appreciated!

The conditions $$x>EX/\sqrt{EX^2}$$ and $$EX\ge0$$ imply $$x>0$$. So, $$t[bX_i -x(X_i^2+b^2)/2]$$ is a quadratic polynomial in $$X_i$$ whose leading coefficient $$-tx/2$$ is less than $$0$$ (except for the trivial case $$t=0$$). Therefore this polynomial is bounded from above, and hence the random variable $$e^{t[bX_i -x(X_i^2+b^2)/2]}$$ is bounded. So, its expectation is finite.