Stirling's Approximation

Jay Zhou Lv1
  2023-06-20 11:47 2023-06-20 11:47 Created   2023-06-30 01:07:14 2023-06-30 01:07:14 Updated    

Gamma Function

Property

Proof

Taylor Approximation

Gaussian Integral

Proof

Also

Laplace's Method

Suppose that is a twice continuously differentiable function on , Then

Proof

Lower Bound

Given is continuous

We have Given that this holds for

Upper Bound

For small enough Also We have that for finite and Given that can be arbitrarily small

and approaches

Sufficient assumption: We have

Combining

Given the lower bound and upper bound, we conclude that

Putting it Together

Laplace's Method

We have

Continue

  • Title: Stirling's Approximation
  • Author: Jay Zhou
  • Created at : 2023-06-20 11:00:47
  • Updated at : 2023-06-30 01:07:14
  • Link: https://ja1zhou.github.io/2023/06/20/Stirling-s-Approximation/
  • License: This work is licensed under CC BY-NC-SA 4.0.
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Stirling's Approximation
  1. Gamma Function
    1. Property
    2. Proof
  2. Taylor Approximation
  3. Gaussian Integral
    1. Proof
  4. Laplace's Method
    1. Proof
  5. Putting it Together
    2. Continue