STIRLING’S FORMULA: AN APPLICATION OF THE

CENTRAL LIMIT THEOREM

K. L. D. Gunawardena

University of Wisconsin – Oshkosh

The large factorials are approximated through the use of Stirling’s formula:

n! ≃ \sqrt{2π} n^{n+1/2}e^{−n}.

The proof of the Stirling’s formula can be found in many texts, such as [1], [2], and [3].

In this short note Stirling’s formula is derived as an application of the Central Limit Theorem. Thus, this proof can be introduced in a mathematical statistics course.

Let X_{1}, X_{2}, …, X_{n} be a random sample from an exponential distribution with mean 1.

By the Central Limit Theorem the limiting distribution of

Z_{n} = \frac{∑_{i=1}^{n} X_{i} − n}{\sqrt{n}}

is standard normal.

That is,

Z_{n} ⟶^{d} Z ∼ N(0, 1) as n → ∞.

Thus, for every x,

P (Z_{n} ≤ x) → P (Z ≤ x) as n → ∞.

Since X_{i} ∼ exp(1), i = 1, 2, …, n, and all independent, ∑_{i=1}^{n} X_{i} has a Gamma distribu-

tion with probability density function

f(t) = \frac{t^{n−1}e^{−t}}{(n − 1)!} t ≥ 0, and zero otherwise.

Hence,

P (Z_{n} ≤ x) = P (∑_{i=1}^{n} X_{i} ≤ n + x\sqrt{n}) = ∫_{0}^{n+x\sqrt{n}} \frac{t^{n−1}e^{−t}}{(n − 1)!} dt.

Thus,

(1)  ∫_{0}^{n+x\sqrt{n}} \frac{t^{n−1}e^{−t}}{(n − 1)!} dt ≃ ∫_{−∞}^{x} \frac{1}{\sqrt{2π}} e^{−t^{2}/2} dt.

Differentiating both sides of (1) with respect to x, we have

(2)  \sqrt{n}\frac{(n + x\sqrt{n})^{n−1}e^{−(n+x\sqrt{n})}}{(n − 1)!} ≃ \frac{1}{\sqrt{2π}} e^{−x^{2}/2}.

Since (2) is true for all x, taking x = 0 in (2) we get

\frac{\sqrt{n} n^{n−1}e^{−n}}{(n − 1)!} ≃ \frac{1}{\sqrt{2π}}

which gives the desired result.

\underline{References}

1. T. M. Apostol, Calculus, Vol. II, Blaisdell Publishing Co., 1962.

2. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., John Wiley & Sons, 1968.

3. K. R. Stromberg, An Introduction to Classical Real Analysis, Wadsworth Inc., 1981.