Laplace's method is a way of approximating integrals and related quantities, like expectations, see https://en.wikipedia.org/wiki/Laplace%27s_method
Questions tagged [laplace-method]
192 questions
22
votes
1 answer
Limit of $\lim_{t \to \infty} \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx}$
Let
\begin{align}
f(t,k,p)= \frac{ \int_0^\infty \cos(x t) e^{-x^k}dx}{\int_0^\infty \cos(x t) e^{-x^p}dx},
\end{align}
My question: How to find the following limit of the function $f(t,k,p)$
\begin{align}
\lim_{t \to \infty}…
Boby
- 6,381
16
votes
3 answers
Asymptotic integral expansion of $\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$ for $x \to \infty$
I'm still having a little trouble applying Laplace's method to find the leading asymptotic behavior of an integral. Could someone help me understand this? How about with an example, like:
$$\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$$ for $x>0$, as…
Alex
- 889
- 5
- 12
12
votes
3 answers
Leading order asymptotic behaviour of the integral $\int^1_0 \cos(xt^3)\tan(t)dt$
I'm trying to get the leading order asymptotic behaviour of the integral:
$$\int^1_0 \cos(xt^3)\tan(t)dt$$
I'm trying to use the Generalised Fourier Integrals and the Stationary Phase Method, but I can't understand how to start this.
THIS IS WHAT I…
bsaoptima
- 541
11
votes
4 answers
Product of two normally distributed random variables: Asymptotic approximation of tail probability
Suppose $X \sim N(1,1)$, $Y \sim N(1,1)$ are iid normal random variables. My research problem is finding out the asymptotics of the tail function of XY (since the explicit formula is too complicated).
If I'm not mistaken, the expression should look…
BigFun
- 69
10
votes
1 answer
Integral asymptotic expansion of $\int_0^{\pi/2} \exp(-xt^3\cos t)dt$ as $x \to \infty$
I have the integral
$$I(x)=\int_0^{\pi/2}\exp(-xt^3\cos t)dt$$
and I want to derive the first two terms in the asymptotic expansion for $x\rightarrow \infty$, which should give me
$$\frac{1}{3x^{1/3}}\Gamma(1/3)+\left(\frac{1}{6}+\frac{8}{\pi^3}…
Alexander
- 1,281
8
votes
1 answer
Using the Saddle point method (or Laplace method) for a multiple integral over a large number of variables
I am trying to understand the saddle point method used in the large N limit of matrix models.
First, for the case of the integral of a single variable I found this notes
There they say that you can approximate the…
physics_teacher
- 103
7
votes
2 answers
How can I improve my proof of Stirling's Theorem?
I'm trying to prove Robbin's inequality:
$$
n! \le \sqrt{2 \pi n}(n/e)^n e^{1/(12n)}.
$$
Step 1: I start from the integral formulation
\begin{align}
n! = \int_0^\infty x^n e^{-x} dx
&=
(n/e)^n\int_0^\infty (x/n)^n e^{n-x}…
Thomas Ahle
- 5,629
7
votes
1 answer
Converse of the Watson's lemma
Watson's lemma basically says
$$
f(t) \sim t^{\alpha} \,\,\,(\text{for small } t) \implies \int_0^{\infty} f(t) e^{-st} dt \sim \frac{\Gamma(\alpha + 1)}{s^{\alpha + 1}} \,\,\,(\text{for large } s).
$$
Under what condition is its converse true? Or…
liott
- 95
7
votes
2 answers
Laplace's method with nontrivial parameter dependency
I need to approximate the following integral using Laplace's method:
$$
\int_0^{\infty} \frac{x^{\lambda} \lambda^{-x}}{(1+x^2)^\lambda} dx \\ =
\int_0^{\infty} \exp\left(\lambda \log(x) - x\log(\lambda)-\lambda \log(1+x^2)\right) dx
$$
as…
freizeit
- 71
- 3
6
votes
2 answers
Large $n$ behavior of $\int_0^{\infty}\int_0^{\infty}e^{-x} e^{-y}y^n \cosh\left(2c\sqrt{xy}\right){dx}{dy}$
I stumbled on a double integral
$$\int_0^{\infty}\int_0^{\infty}e^{-x} e^{-y}y^n \cosh\left(2c\sqrt{xy}\right){dx}{dy},\quad 0
Nikitan
- 801
6
votes
3 answers
Using Laplace Transforms to solve $\int_{0}^{\infty}\frac{\sin(x)\sin(x/3)}{x(x/3)}\:dx$
So, I've come across the following integral (and it's expansion) many times and in my study so far, Complex Residues have been used to evaluate it. I was hoping to find an alternative approach using Laplace Transforms. I believe the method I've…
user150203
6
votes
2 answers
Integral asymptotic expansion of $\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{\sinh t}}dt$ for $x \to \infty$
$$\int_{0}^{\infty} \frac{e^{-x \cosh t}}{\sqrt{\sinh t}}dt$$
I'm trying to use Laplace's method to find the leading asymptotic behavior as $x$ goes to positive infinity, but I'm having some trouble. Could someone help me?
Alex
- 889
- 5
- 12
6
votes
2 answers
Laplace integral and leading order behavior
Consider the integral:
$$
\int_0^{\pi/2}\sqrt{\sin t}e^{-x\sin^4 t} \, dt
$$
I'm trying to use Laplace's method to find its leading asymptotic behavior as $x\rightarrow\infty$, but I'm running into problems because the maximum of $\phi(t)$ (i.e.…
Alex
- 889
- 5
- 12
6
votes
2 answers
Let $X$ be standard normal and $a>b>0$, prove that $\lim\limits_{\epsilon\to 0}\epsilon^2\log P(|\epsilon X -a|
Let $X$ be a standard normal random variable, with $a,b>0$ and $a-b>0$, prove that $$\lim_{\epsilon\to 0}\epsilon^2\log P(|\epsilon X -a|
user223391
5
votes
2 answers
Estimate a sum of binomial coefficients
I should know this by the time, but: can someone tell me how to rigorously compute the leading order (including the constant) of the following sum:
$$\sum_{ 1\leq k \leq n/3 } {2 k \choose k} {n-2k-1 \choose k-1}$$
as $n \to \infty$; I know how to…
Olivier
- 1,355