8.2

8.2.5 Weak convergence

Definition 8.33

• \(\{P_{n}\}\),
  • Borel probablity measures on $\mathbb{R}^{n}$,
• $P$,
  • Borel probablity measures on $\mathbb{R}^{n}$,
• \(\{F_{n}\}\),
  • CDF of \(\{P_{n}\}\),
• $F$,
  • Borel probablity measures on $\mathbb{R}^{n}$,

\(\{P_{n}\}\) is said to converge wealy to $P$ if

\[F_{n}(x) \rightarrow F(x)\]

where $x$ is all continuous points of $F$.

■

Theorem 8.34

• $X_{n}$,
• $X$,

If $X_{n} \overset{p}{\rightarrow} X$, \(X_{n} \overset{w}{\rightarrow} X\).

proof

$\Box$