3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.