What is Gaussian vector? — Mathematics

Definition 23.10 University Mathematics — Year 3 · Chapter 23 — Characteristic Functions and the Central Limit Theorem

A random vector $X = (X_1, \dots, X_d)$ is Gaussian if every linear combination $\langle t, X\rangle = \sum t_iX_i$ is a (possibly degenerate) real Gaussian variable. Its law is determined by the mean vector $m = (\E X_i)$ and the covariance matrix $\Sigma = \bigl(\operatorname{Cov} (X_i, X_j)\bigr)$ : indeed the characteristic function of the vector, $\varphi_X(t) = \E\eu^{\iu\langle t, X\rangle}$ , is the value at $1$ of the cf of $\langle t, X\rangle$ :

\varphi_X(t) = \exp\Bigl(\iu\langle t, m\rangle - \tfrac12\,t^{\mathsf T}\Sigma\,t\Bigr),

and $d$ -dimensional characteristic functions are injective (same smoothing proof as Theorem 23.3, coordinatewise Gaussians).

Examples

Example 23.9 (Confidence intervals, honestly derived)

Poll $n$ independent voters; $\hat p_n = S_n/n$ estimates the true $p$ , with $\sigma^2 = p(1-p) \leq \frac14$ . The CLT gives, for large $n$ ,

\P\Bigl(\abs{\hat p_n - p} \leq \frac{z}{2\sqrt n}\Bigr) \;\geq\; \P\Bigl(\Bigl|\frac{S_n - np}{\sigma\sqrt n}\Bigr| \leq z\Bigr) \longrightarrow \Phi(z) - \Phi(-z),

where $\Phi$ is the standard Gaussian distribution function. With $z = 1.96$ : asymptotic confidence $95\%$ , and margin $\frac{1.96}{2\sqrt n} \leq 3\%$ requires $n \geq \bigl(\frac{1.96}{0.06}\bigr)^2 \approx 1068$ — the number behind every “ $\pm3$ points, $95\%$ ” one reads; compare Chebyshev’s $5556$ (Exercise 22.7). The $\sqrt n$ is universal: to halve the error, quadruple the sample — the same law that fixes Monte Carlo’s cost (Exercise 23.7).

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