连续（Continuity）- 有界（Bounded） - 收敛（Convergence）
连续（Continuity）
弱 ----> 强
-------------------------------------------------------------------------------------------------------------
continuity -> uniform continuity -> absolute continutiy -> Lipschitz Continuity
连续函数 一致连续 绝对连续 李普希兹连续
$1/x$ $\sqrt x$
【uniform continutity】
a function f is uniformly continuous if it is possible to guarantee that f(x) and f(y) be as close to each other as we please by requiring only that x and y are sufficiently close to each other; unlike ordinary continuity, where the maximum distance between f(x) and f(y) may depend on x and y themselves.
https://en.wikipedia.org/wiki/Uniform_continuity
【absolute continutiy】
| f(x) | is uniformly continuous
https://en.wikipedia.org/wiki/Absolute_continuity
【Lipschitz continuity】
函数的变化率是有限的（如果函数可导，则导数有界）
简单来说，Lipschitz连续就类似，一块地不仅没有河流什么的玩意儿阻隔，而且这块地上没有特别陡的坡。其中最陡的地方有多陡呢？这就是所谓的李普希兹常数
https://en.wikipedia.org/wiki/Lipschitz_continuity
有界（Bounded）
bounded -> Uniform boundedness
the sequence of functions $\{ f_n | f_n(x) = sin(nx) \}$ is uniformly bounded
the sequence of functions $\{ g_n | g_n(x) = nsin(x) \}$ is not uniformly bounded
https://en.wikipedia.org/wiki/Uniform_boundedness
收敛（Convergence）
逐点收敛(pointwise convergence) -> 一致收敛(uniform convergence)
【pointwise convergence】
The sequence $f_n(x)$ converges pointwise to the function $f$, iff
for every $x$, $\lim_{x \to +\infty} f_n=f(x)$
【uniform convergence】
the sequence functions ${ S_n(x) }$ is uniformly convergent: if for every $\epsilon>0$, there exists a number N, such that for all $n>N$, $|f_n(x)-f(x)|<\epsilon$
https://en.wikipedia.org/wiki/Uniform_convergence
随机变量的收敛
研究一列随机变量是否会收敛到某个极限随机变量
https://en.wikipedia.org/wiki/Convergence_of_random_variables
【convergence in distribution】
- the weakest form of convergence
- related to central limit theorem
Definition:
A sequence $X_1$, $X_2$, ... of random variables is said to converge in distribution to a random variable X if
$\lim\limits_{n \to \infty} F_n(x)=F(x)$ for every $x\in\mathbb{R}$ at which $F$ is continues. (仅仅考虑$F(x)$连续的地方的分布函数值)
$X_n \overset{d}{\to} X$
【Convergence in probability】
- related to the weak law of large numbers
- related to the consistent estimator
meanning：the probability of an “unusual” outcome becomes smaller and smaller as the sequence progresses.
Definition: A sequence $\{X_n\}$ of random variables converges in probability towards the random variable $X$ if for all $\epsilon > 0$, $\lim\limits_{n \to \infty}{Pr(|X_n-X|>\epsilon)}=0$
$X_n \overset{p}{\to} X$
【Almost sure convergence】
类似于函数列收敛中pointwise convergence，
Definition：To say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X means that,
$Pr(\lim\limits_{n \to \infty}{X_n=X})=1$
$X_n \overset{a.s.}{\to} X$
