Let $X_n$ be a sequence of iid $N(0,1)$ random variables. Find the approximate value of $P(X_1^2+X_2^2+\dots+X_n^2 >= 2n)$ when $n$ is large. Use large deviation theory.
I have tried but failed to solve this, how to find the moment generating function of square of random variable, I have got tuck there.
Cramér's large deviation upper bound reads $$P(X_1^2+\cdots+X_n^2\geqslant2n)\leqslant R(2)^n,$$ where, for every $x\geqslant1$, $$R(x)=\inf_{t\gt0}\mathrm e^{-xt}E(\mathrm e^{tX_1^2}).$$ A standard computation yields $$E(\mathrm e^{tX_1^2})=\int_\mathbb R\frac1{\sqrt{2\pi}}\mathrm e^{-(1-2t)u^2/2}\mathrm du=\frac1{\sqrt{1-2t}},$$ for every $t\lt\frac12$ hence the quantity in the infimum defining $R(x)$ is minimal at $$t=\frac{x-1}{2x},$$ and $$R(x)=\sqrt{x\mathrm e^{1-x}}.$$ In particular, $$R(2)=\sqrt{2/\mathrm e}\approx.86.$$