Stochastic Processes and Financial Mathematics
(part one)

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9.2 Stirling’s Approximation $\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 $n! \approx \sqrt {2\pi n}\l (\frac {n}{e}\r )^n$. 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 \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

\[a_n\sim b_n \quad \quad \text { to mean that } \quad \quad \lim _{n\to \infty } \frac {a_n}{b_n}=1. \]

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

Theorem 9.2.1 For all $n\in \N $ it holds that

\[\sqrt {2\pi n}\l (\frac {n}{e}\r )^ne^{1/(12n+1)} \;<\; n! \;<\; \sqrt {2\pi n}\l (\frac {n}{e}\r )^ne^{1/(12n)}.\]

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{equation} \label {eq:stirling} n! \sim \sqrt {2\pi n}\l (\frac {n}{e}\r )^n. \end{equation}

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

Stochastic Processes and Financial Mathematics (part one)

9.2 Stirling’s Approximation \(\msconly \)

Stochastic Processes and Financial Mathematics
(part one)