In one attempt to prove a probability theorem (of K.L. Chung and P. Erdős, 1951) using analytic argument, I try to prove the following Let $\varphi(x)$ and $\psi(x)$ be two complex-valued continuous functions on $[a,b]$, and let $f(x)$ be a complex-valued continuously differentiable function on $[a,b]$. Suppose that $|f(x)|$ has an absolute maximum at an interior point, say $\xi$, of the interval, and $f'(\xi)=0$. Then \begin{equation}\label{eq3} \lim_{n\to\infty}\frac{\int_a^b\varphi(x)[f(x)]^ndx}{\int_a^b\psi(x)[f(x)]^ndx}=\frac{\varphi(\xi)}{\psi(\xi)}. \end{equation}

Remark 1: This is true for $f(x)\in C^2$, by Laplace's method.

Remark 2: Michael has given a counter example without the assumption $f'(\xi)=0$. This is a good example. Please see in the origin version of the problem: Quotient of two Laplace integrals

This problem is still open.

Thank you.