1. Boole’s inequality

In probability theory, Boole’s inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.

布爾不等式（Boole’s inequality），由喬治·布爾提出，指對于全部事件的概率不大于單個事件的概率總和。

Formally, for a countable set of events A1, A2, A3, …, we have

1.1 Proof using induction

1.2 Proof without using induction

1.3 Generalization

Boole’s inequality may be generalized to find upper and lower bounds on the probability of finite unions of events. These bounds are known as Bonferroni inequalities, after Carlo Emilio Bonferroni;

Boole’s inequality is the initial case, k = 1. When k = n, then equality holds and the resulting identity is the inclusion–exclusion principle.

2. Doob’s inequality

for every K>0 and p>1.

2.1 proof of Doob’s inequalities

3. 中心極限定理（Central Limit Theorem）

中心極限定理指的是給定一個任意分布的總體。我每次從這些總體中隨機抽取 n 個抽樣，一共抽 m 次。 然后把這 m 組抽樣分別求出平均值。 這些平均值的分布接近正態分布。

我們先舉個栗子?

現在我們要統計全國的人的體重，看看我國平均體重是多少。當然，我們把全國所有人的體重都調查一遍是不現實的。所以我們打算一共調查1000組，每組50個人。 然后，我們求出第一組的體重平均值、第二組的體重平均值，一直到最后一組的體重平均值。中心極限定理說：這些平均值是呈現正態分布的。并且，隨著組數的增加，效果會越好。 最后，當我們再把1000組算出來的平均值加起來取個平均值，這個平均值會接近全國平均體重。

其中要注意的幾點：

• 總體本身的分布不要求正態分布
  上面的例子中，人的體重是正態分布的。但如果我們的例子是擲一個骰子（平均分布），最后每組的平均值也會組成一個正態分布。（神奇！）

• 樣本每組要足夠大，但也不需要太大
  取樣本的時候，一般認為，每組大于等于30個，即可讓中心極限定理發揮作用。

中心極限定理也就是這么兩句話：
1）任何一個樣本的平均值將會約等于其所在總體的平均值。
2）不管總體是什么分布，任意一個總體的樣本平均值都會圍繞在總體的平均值周圍，并且呈正態分布。

在實際生活當中，我們不能知道我們想要研究的對象的平均值，標準差之類的統計參數。中心極限定理在理論上保證了我們可以用只抽樣一部分的方法，達到推測研究對象統計參數的目的。

3.1 中心極限定理有什么用呢？

1）在沒有辦法得到總體全部數據的情況下，我們可以用樣本來估計總體如果我們掌握了某個正確抽取樣本的平均值和標準差，就能對估計出總體的平均值和標準差。舉個例子，如果你是北京西城區的領導，想要對西城區里的各個學校進行教學質量考核。同時，你并不相信各個學校的的統考成績，因此就有必要對每所學校進行抽樣測試，也就是隨機抽取100名學生參加一場類似統考的測驗。作為主管教育的領導，你覺得僅參考100名學生的成績就對整所學校的教學質量做出判斷是可行的嗎？答案是可行的。中心極限定理告訴我們，一個正確抽取的樣本不會與其所代表的群體產生較大差異。也就是說，樣本結果（隨機抽取的100名學生的考試成績）能夠很好地體現整個群體的情況（某所學校全體學生的測試表現）。當然，這也是民意測驗的運行機制所在。通過一套完善的樣本抽取方案所選取的1200名美國人能夠在很大程度上告訴我們整個國家的人民此刻正在想什么。2）根據總體的平均值和標準差，判斷某個樣本是否屬于總體如果我們掌握了某個總體的具體信息，以及某個樣本的數據，就能推理出該樣本是否就是該群體的樣本之一。通過中心極限定理的正態分布，我們就能計算出某個樣本屬于總體的概率是多少。如果概率非常低，那么我們就能自信滿滿地說該樣本不屬于該群體。

大數定律https://www.zhihu.com/question/19911209/answer/245487255

4. Kolmogorov extension theorem

In mathematics, the Kolmogorov extension theorem (also known as Kolmogorov existence theorem, the Kolmogorov consistency theorem or the Daniell-Kolmogorov theorem) is a theorem that guarantees that a suitably “consistent” collection of finite-dimensional distributions will define a stochastic process. It is credited to the English mathematician Percy John Daniell and the Russian mathematician Andrey Nikolaevich Kolmogorov.

4.1 Statement of the theorem

In fact, it is always possible to take as the underlying probability space $\Omega =({\mathbb{R}}^n{)}^T$ and to take for $X$ the canonical process $X:(t,Y)?Y_t$. Therefore, an alternative way of stating Kolmogorov’s extension theorem is that, provided that the above consistency conditions hold, there exists a (unique) measure $\nu$ on $({\mathbb{R}}^n{)}^T$ with marginals${\nu }_{t_1\dots t_k}$ for any finite collection of times $t_1\dots t_k$. Kolmogorov’s extension theorem applies when $T$ is uncountable, but the price to pay for this level of generality is that the measure $\nu$ is only defined on the product σ-algebra of $({\mathbb{R}}^n{)}^T$, which is not very rich.

4.2 Explanation of the conditions

4.3 Implications of the theorem

Since the two conditions are trivially satisfied for any stochastic process, the power of the theorem is that no other conditions are required: For any reasonable (i.e., consistent) family of finite-dimensional distributions, there exists a stochastic process with these distributions.

The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space. However, in many applications the starting point is really the finite-dimensional distributions of the stochastic process. The theorem says that provided the finite-dimensional distributions satisfy the obvious consistency requirements, one can always identify a probability space to match the purpose. In many situations, this means that one does not have to be explicit about what the probability space is. Many texts on stochastic processes do, indeed, assume a probability space but never state explicitly what it is.

The theorem is used in one of the standard proofs of existence of a Brownian motion, by specifying the finite dimensional distributions to be Gaussian random variables, satisfying the consistency conditions above. As in most of the definitions of Brownian motion it is required that the sample paths are continuous almost surely, and one then uses the Kolmogorov continuity theorem to construct a continuous modification of the process constructed by the Kolmogorov extension theorem.

4.4 General form of the theorem

The Kolmogorov extension theorem gives us conditions for a collection of measures on Euclidean spaces to be the finite-dimensional distributions of some ${\mathbb{R}}^n$-valued stochastic process, but the assumption that the state space be ${\mathbb{R}}^n$ is unnecessary. In fact, any collection of measurable spaces together with a collection of inner regular measures defined on the finite products of these spaces would suffice, provided that these measures satisfy a certain compatibility relation. The formal statement of the general theorem is as follows.

This theorem has many far-reaching consequences; for example it can be used to prove the existence of the following, among others:

• Brownian motion, i.e., the Wiener process,
• a Markov chain taking values in a given state space with a given transition matrix,
• infinite products of (inner-regular) probability spaces.

4.5 History

According to John Aldrich, the theorem was independently discovered by British mathematician Percy John Daniell in the slightly different setting of integration theory.

5. Lebesgue’s dominated convergence theorem

In measure theory, Lebesgue’s dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the L1 norm. Its power and utility are two of the primary theoretical advantages of Lebesgue integration over Riemann integration.

In addition to its frequent appearance in mathematical analysis and partial differential equations, it is widely used in probability theory, since it gives a sufficient condition for the convergence of expected values of random variables.

5.1 Statement

5.2 Proof

Without loss of generality, one can assume that f is real, because one can split f into its real and imaginary parts (remember that a sequence of complex numbers converges if and only if both its real and imaginary counterparts converge) and apply the triangle inequality at the end.

Lebesgue’s dominated convergence theorem is a special case of the Fatou–Lebesgue theorem. Below, however, is a direct proof that uses Fatou’s lemma as the essential tool.

https://en.wikipedia.org/wiki/Boole%27s_inequality
https://planetmath.org/alphabetical.html
https://zhuanlan.zhihu.com/p/25241653
https://www.zhihu.com/question/22913867
https://en.wikipedia.org/wiki/Kolmogorov_extension_theorem
https://blog.csdn.net/weixin_44207974/article/details/111503988
https://blog.csdn.net/weixin_44207974/article/details/111602960
https://en.wikipedia.org/wiki/Dominated_convergence_theorem

總結

以上是生活随笔為你收集整理的Boole‘s，Doob‘s inequality，中心极限定理Central Limit Theorem，Kolmogorov extension theorem, Lebesgue‘s domin的全部內容，希望文章能夠幫你解決所遇到的問題。

如果覺得生活随笔網站內容還不錯，歡迎將生活随笔推薦給好友。