Quotient of two Laplace integrals (2) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T01:40:28Z http://mathoverflow.net/feeds/question/50671 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/50671/quotient-of-two-laplace-integrals-2 Quotient of two Laplace integrals (2) LI Yutian 2010-12-29T18:40:46Z 2010-12-29T20:44:45Z <p>In one attempt to prove a probability theorem (of K.L. Chung and P. ErdÅ‘s, 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 $$\label{eq3} \lim_{n\to\infty}\frac{\int_a^b\varphi(x)[f(x)]^ndx}{\int_a^b\psi(x)[f(x)]^ndx}=\frac{\varphi(\xi)}{\psi(\xi)}.$$</p> <p>Remark 1: This is true for $f(x)\in C^2$, by Laplace's method. </p> <p>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: <a href="http://mathoverflow.net/questions/48290" rel="nofollow">http://mathoverflow.net/questions/48290</a> </p> <p>This problem is still open.</p> <p>Thank you.</p>