Stochastic Processes and Financial Mathematics
(part one)

9.2 Stirling’s Approximation $\text{msconly}$

Stirling’s approximation is a fundamental inequality that is used across many different parts of mathematics. It connects the factorial function with the constants $\pi$ and $e$, and is often written informally as $n!\approx \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n$. We mention it here because the distribution of $S_n$, where $(S_n)$ is a random walk, can sometimes be expressed using formulae involving binomial coefficients. Stirling’s approximation is helpful to calculate the limits of such quantities as $n\to \mathrm{\infty }$.

It is also helpful to introduce some notation from analysis that you may not have seen before: if $(a_n)$ and $(b_n)$ are real valued sequences then we write

$$a_n\sim b_n\text{ to mean that }\underset{n\to \mathrm{\infty }}{lim}\frac{a_n}{b_n}=1.$$

We will use this notation throughout the remainder of Chapter 9.

This is a result from real analysis so we won’t include a proof in this course. Using that $e^{1/n}\to 1$, it is straightforward to check that Theorem 9.2.1 implies the weaker result

$$\begin{array}{}\text{(9.7)}& n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n.\end{array}$$

We will apply Stirling’s approximation in this form. Equation (9.7) appears on the formula sheet.