Let Z∈Gaussian(0,1) be a D211: Standard gaussian random real number.
Let t∈R be a D993: Real number.
Let t∈R be a D993: Real number.
Then
E(eitZ)=e−t2/2
Result R3680
on D2325: Random euclidean real number characteristic function
Subresult of R2665: Characteristic function of gaussian random real number
Subresult of R2665: Characteristic function of gaussian random real number
Characteristic function of standard gaussian random real number
Formulation 1
Let Z∈Gaussian(0,1) be a D211: Standard gaussian random real number.
Let t∈R be a D993: Real number.
Let t∈R be a D993: Real number.
Then
FZ(t)=exp(−t22)
Proofs
Let Z∈Gaussian(0,1) be a D211: Standard gaussian random real number.
Let t∈R be a D993: Real number.
Let t∈R be a D993: Real number.
By definition, we have
Zd=lim
where \xi_1, \xi_2, \xi_3, \ldots are random integers with
\begin{equation}
\mathbb{P}(\xi_n = -1) = \mathbb{P}(\xi_n = 1) = \frac{1}{2}
\end{equation}
for all positive integers n \geq 1. Using result R2405: Characteristic function uniquely identifies the distribution of a random real number, it is thus sufficient to show that the characteristic function of \lim_{N \to \infty} \sum_{n = 1}^N \frac{\xi_n}{\sqrt{N}} is given by t \mapsto e^{- t^2 / 2}.
To this end, denote S_N : = \sum_{n = 1}^N \xi_n. Results
imply that N^{-1/2} S_N has characteristic function \begin{equation} \mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right) = \left( \cos \frac{t}{\sqrt{N}} \right)^N \end{equation} By definition of D1927: Standard real cosine function, we have \begin{equation} \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots \end{equation} Thus \begin{equation} \cos \frac{t}{\sqrt{N}} = 1 - \frac{t^2}{2! N} + \frac{t^4}{4! N^2} - \frac{t^6}{6! N^3} + \dots \end{equation} so that we have the approximation \begin{equation} \cos \frac{t}{\sqrt{N}} = 1 - \frac{t^2}{2 N} + o_{t \to 0}(1) \end{equation} Therefore, result R3304: Approximating sequence for the natural exponential function shows that the limit for (\cos \frac{t}{\sqrt{N}})^N exists as N \to \infty and \begin{equation} \begin{split} \mathbb{E} \left( e^{i t Z} \right) = \mathbb{E} \left( e^{i t \lim_{N \to \infty} N^{-1/2} S_N } \right) & = \mathbb{E} \left( \lim_{N \to \infty} e^{i \frac{t}{\sqrt{N}} S_N} \right) \\ & = \lim_{N \to \infty} \mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right) \\ & = \lim_{N \to \infty} \left( \cos \frac{t}{\sqrt{N}} \right)^N \\ & = \lim_{N \to \infty} \left( 1 - \frac{t^2}{2 N} + o(1) \right)^N \\ & = \lim_{N \to \infty} \left( 1 + \frac{- t^2 / 2}{N} + o(1) \right)^N = e^{-t^2 / 2} \end{split} \end{equation} \square
To this end, denote S_N : = \sum_{n = 1}^N \xi_n. Results
| (i) | R3912: Characteristic function of rademacher random integer |
| (ii) | R3913: Characteristic function for I.I.D. sum of random basic real numbers |
imply that N^{-1/2} S_N has characteristic function \begin{equation} \mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right) = \left( \cos \frac{t}{\sqrt{N}} \right)^N \end{equation} By definition of D1927: Standard real cosine function, we have \begin{equation} \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \dots \end{equation} Thus \begin{equation} \cos \frac{t}{\sqrt{N}} = 1 - \frac{t^2}{2! N} + \frac{t^4}{4! N^2} - \frac{t^6}{6! N^3} + \dots \end{equation} so that we have the approximation \begin{equation} \cos \frac{t}{\sqrt{N}} = 1 - \frac{t^2}{2 N} + o_{t \to 0}(1) \end{equation} Therefore, result R3304: Approximating sequence for the natural exponential function shows that the limit for (\cos \frac{t}{\sqrt{N}})^N exists as N \to \infty and \begin{equation} \begin{split} \mathbb{E} \left( e^{i t Z} \right) = \mathbb{E} \left( e^{i t \lim_{N \to \infty} N^{-1/2} S_N } \right) & = \mathbb{E} \left( \lim_{N \to \infty} e^{i \frac{t}{\sqrt{N}} S_N} \right) \\ & = \lim_{N \to \infty} \mathbb{E} \left( e^{i \frac{t}{\sqrt{N}} S_N} \right) \\ & = \lim_{N \to \infty} \left( \cos \frac{t}{\sqrt{N}} \right)^N \\ & = \lim_{N \to \infty} \left( 1 - \frac{t^2}{2 N} + o(1) \right)^N \\ & = \lim_{N \to \infty} \left( 1 + \frac{- t^2 / 2}{N} + o(1) \right)^N = e^{-t^2 / 2} \end{split} \end{equation} \square