convergence of normal sum r.v. tail

Question

Let $(Z_n)_{n\ge 1}$ be a sequence of i.i.d standard normal r.v.'s. I am trying to evaluate the limit $$\lim_n \frac{\log P(Z_1+...+Z_n>n) }{ n}.$$ The limit evaluates to -1/2 using some very strange order expansions (wolfram alpha). I am looking for a simpler way to solve this.

Answer 1

The sum $S_n=Z_1+Z_2+\ldots+Z_n$ is distributed as $N(0,n)$, so $P(S_n>n)=Q(\sqrt n)$ where $Q$ is the upper tail cdf of the normal, $Q(x)=1-\Phi(x)$ in the usual notation. It is a well-known fact about the gaussian Mill's ratio $Q(x)/\phi(x)$ that $$\lim_{x\to\infty} \frac{xQ(x)}{\phi(x)} = 1,$$ where $\phi(x)$ is the density function for the standard normal. Hence $$\lim_{n\to\infty} \log\left(\sqrt n Q(\sqrt n)/ \phi(\sqrt n)\right) = 0$$

and $$\lim_{n\to\infty} \left( \frac{\log n}2 +\log Q(\sqrt n) +\frac{\log(2\pi)}2+n/2\right) = 0 .$$ This implies $\lim_{x\to\infty} \frac{\log Q(\sqrt n)} n = -\frac 1 2$.