📂Probability Theory

Levy's Continuity Theorem in Probability Theory

Theorem ¹

Let a measurable space $\left( \mathbb{R}^{d} , \mathcal{B} \left( \mathbb{R}^{d} \right) \right)$ be given. Denote by $n \in \overline{\mathbb{N}}$ a probability measure with $\mu_{n}$, and the corresponding characteristic function by $\varphi_{n}$. The following are equivalent:

• (a): $\mu_{n}$ weakly converges to $\mu_{\infty}$.
• (b): For all $t \in \mathbb{R}^{d}$, $$\lim_{n \to \infty} \varphi_{n} (t) = \varphi_{\infty} (t)$$

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• $\overline{\mathbb{N}} = \mathbb{N} \cup \left\{ \infty \right\}$ is a set that includes natural numbers and infinity.

Proof

It’s challenging and there are too many lemmas that need to be proven beforehand, so it is omitted for now and remains a project for the future².

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