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]
172 questions
20
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
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16
votes
2 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
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- 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
- 501
- 9
9
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
$$\frac{1}{3x^{1/3}}\Gamma(1/3)+\left(\frac{1}{6}+\frac{8}{\pi^3}…
Alexander
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8
votes
1 answer
Using the Saddle point method (or Laplace method) for a multiple integral over a large number of variables
I am trying to understand the saddle point method used in the large N limit of matrix models.
First, for the case of the integral of a single variable I found this notes
There they say that you can approximate the…
physics_teacher
- 103
- 6
7
votes
2 answers
How can I improve my proof of Stirling's Theorem?
I'm trying to prove Robbin's inequality:
$$
n! \le \sqrt{2 \pi n}(n/e)^n e^{1/(12n)}.
$$
Step 1: I start from the integral formulation
\begin{align}
n! = \int_0^\infty x^n e^{-x} dx
&=
(n/e)^n\int_0^\infty (x/n)^n e^{n-x}…
Thomas Ahle
- 4,348
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7
votes
2 answers
Laplace's method with nontrivial parameter dependency
I need to approximate the following integral using Laplace's method:
$$
\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
$$
as…
freizeit
- 71
- 3
6
votes
3 answers
Using Laplace Transforms to solve $\int_{0}^{\infty}\frac{\sin(x)\sin(x/3)}{x(x/3)}\:dx$
So, I've come across the following integral (and it's expansion) many times and in my study so far, Complex Residues have been used to evaluate it. I was hoping to find an alternative approach using Laplace Transforms. I believe the method I've…
user150203
6
votes
2 answers
Let $X$ be standard normal and $a>b>0$, prove that $\lim\limits_{\epsilon\to 0}\epsilon^2\log P(|\epsilon X -a|
Let $X$ be a standard normal random variable, with $a,b>0$ and $a-b>0$, prove that $$\lim_{\epsilon\to 0}\epsilon^2\log P(|\epsilon X -a|
user223391
6
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
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5
votes
1 answer
How to find path for method of steepest descent
We have integral:
$$\int_0^1\exp\left(n\left(\frac{itz}{\sqrt{a(1-a)n}}+a\ln(z)+(1-a)\ln(1-z)\right)\right)dz=\int_0^1\exp(nf(z))dz,$$
where $0
CROCO
- 51
- 2
5
votes
1 answer
Laplace functional of cluster process
Consider the simple cluster process:
$$\sum_n \xi_n \epsilon_{X_n}$$ where $\{X_n\}$ are Poisson points independent of the iid non-negative integer sequence $\{\xi_n\}$. How do I find the Laplace functional? I am a bit confused reading about it…
bilbo
- 319
- 1
- 8
5
votes
1 answer
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
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5
votes
3 answers
Applying Laplace Method for asymptotic approximations
I am trying to verify the following asymptotic approximation as $x \rightarrow \infty$:
$$\int^{1}_{0}t^{-\frac{1}{2}} \cos(t) e^{-xt^{\frac{1}{2}}} \, dt \sim \frac{2}{x}$$
This method is such that $$\int^\beta_\alpha g(t)e^{xh(t)} \, dt \sim g(a)…
Evan
- 663
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- 16
5
votes
0 answers
Approximation of a Riemann sum (not really) by a Laplacian integral
I have a sum of the form:
$$S_n = \frac{1}{n} \sum_{i=0}^n \mathrm{e}^{n f(i/n)} g(i/n)$$
where $f(x)$ and $g(x)$ are smooth functions defined for $0\le x \le 1$. I am interested in the Asymptotic behaviour of $S_n$ as $n\rightarrow +\infty$.
What I…
a06e
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