概率论中的收敛（基本定义与结论）
几乎处处收敛
\( \begin{align*} f_n\overset{\mathrm{~a.e.~}}{\to}f &\iff \exists N,\mu(N)=0, \mathrm{~s.t.~} \omega\in N^\mathrm{c}, f_n(\omega)\to f(\omega)\\ &\iff \mu \left( \bigcap\limits_{n=1}^{\infty} \bigcup\limits_{k=n}^{\infty} \{|f_k-f|\geq\varepsilon\} \right)=0, \forall \varepsilon>0\\ &\iff \lim\limits_{n\to\infty} \mu \left( \bigcup\limits_{k=n}^{\infty} \{|f_k-f|\geq\varepsilon\} \right)=0, \forall \varepsilon>0, \mu<\infty\\ &\iff \lim\limits_{n\to\infty} \mu \left( \sup\limits_{k\geq n} \{|f_k-f|\geq\varepsilon\} \right)=0, \forall \varepsilon>0, \mu<\infty \end{align*} \)
\( f_n\overset{\mathrm{~a.e.~}}{\to}f \implies f\mathrm{~a.e.~}可测 \)
\( f_n\overset{\mathrm{~a.e.~}}{\to}f \implies f_n\overset{\mathrm{~a.un.~}}{\to}f, \mu<\infty \)
几乎必要收敛
\( X_n\overset{\mathrm{~a.s.~}}{\to}X \iff P\{|X_n-X|\geq \varepsilon,\mathrm{~i.o.~}\}, \forall \varepsilon>0 \)
\( \mathbf{X}_n\overset{\mathrm{~a.s.~}}{\to}\mathbf{X} \implies \mathbf{X}_n\overset{P}{\to}\mathbf{X} \)
几乎一致收敛
\( \begin{align*} f_n\overset{\mathrm{~a.un.~}}{\to}f &\iff \forall\delta>0, \exists A\in\mathcal{F}, \mu(A)<0, \mathrm{~s.t.~} \forall \omega\in A^\mathrm{c}, f_n(\omega)\rightrightarrows f(\omega)\\ &\iff \lim\limits_{n\to\infty} \mu \left( \bigcup\limits_{k=n}^{\infty} \{|f_k-f|\geq\varepsilon\} \right)=0, \forall \varepsilon>0 \end{align*} \)
\( f_n\overset{\mathrm{~a.un.~}}{\to}f \implies f_n\overset{\mathrm{~a.e.~}}{\to}f \)
\( f_n\overset{\mathrm{~a.un.~}}{\to}f \implies f_n\overset{\mu}{\to}f \)
依测度收敛收敛
\( f_n\overset{\mu}{\to}f \iff \lim\limits_{n\to\infty}\mu\{|f_n-f|\geq\varepsilon\}=0, \forall \varepsilon>0 \)
依概率收敛收敛
\( \mathbf{X}_n\overset{P}{\to}\mathbf{X} \iff \lim\limits_{n\to\infty}\mu \{||\mathbf{X}_n-\mathbf{X}||\geq\varepsilon\}=0, \forall \varepsilon>0 \)
\( X_n\overset{\mathrm{~a.s.~}}{\to}a,a<b \implies \lim\limits_{n\to\infty}P \left( \bigcap\limits_{n=1}^{\infty} \{X_k\leq b\} \right)=1, \forall \varepsilon>0 \)