復(fù)習(xí)筆記Day108

2023-02-26 20:53 作者:間宮_卓司 0人讀過 | 我要投稿

這幾天在看應(yīng)堅(jiān)鋼的概率論，一開始選這本書的原因倒不是因?yàn)槲抑肋@本書的觀點(diǎn)比較高（相較于我看過的其他概率論課本來說），不過看看倒也無所謂···因?yàn)楦怕收摲旁趶?fù)試的面試部分，所以我不怎么打算做題，主要是基礎(chǔ)知識(shí)要梳理清楚。過幾天如果我能讀的下去的話，我會(huì)在讀完這本書后把這本書的大體框架寫成筆記發(fā)上來，不過應(yīng)該會(huì)跳著讀了，一些完全不可能問到的地方就跳過了。

雖然說不怎么做題，但是因?yàn)樵诹?xí)題里面看到了之前做過的數(shù)分題，所以還是打算試一下

108.1 利用 $%5Ctext%7BBernoulli%7D$ 的大數(shù)定律證明 $%5Ctext%7BWeierstrass%7D$ 定理：設(shè) $f$ 是 $%5B0%2C1%5D$ 上的連續(xù)函數(shù)，定義 $%5Ctext%7BBernstein%7D$ 多項(xiàng)式

$f_n%5Cleft(%20x%20%5Cright)%20%3D%5Csum_%7Bk%3D0%7D%5En%7B%5Cleft(%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09n%5C%5C%0A%09k%5C%5C%0A%5Cend%7Barray%7D%20%5Cright)%20f%5Cleft(%20%5Cfrac%7Bk%7D%7Bn%7D%20%5Cright)%7Dx%5Ek%5Cleft(%201-x%20%5Cright)%20%5E%7Bn-k%7D%2Cx%5Cin%20%5Cleft%5B%200%2C1%20%5Cright%5D%20$

則 $f_n$ 在 $%5B0%2C1%5D$ 上一致收斂于 $f$

首先回顧一下 $%5Ctext%7BBernoulli%7D$ 的大數(shù)定律

此外還可以知道， $%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cxi%20_n%3Dk%20%5Cright)%20%3D%5Cleft(%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09n%5C%5C%0A%09k%5C%5C%0A%5Cend%7Barray%7D%20%5Cright)%20p%5Ek%5Cleft(%201-p%20%5Cright)%20%5E%7Bn-k%7D$ ，為了湊出 $%5Ctext%7BBernstein%7D$ 多項(xiàng)式的形式，先試試看把 $p$ 換成 $x$ 。這個(gè)時(shí)候， $%5Ctext%7BBernstein%7D$ 多項(xiàng)式就變成了

$f_n%5Cleft(%20x%20%5Cright)%20%3D%5Csum_%7Bk%3D0%7D%5En%7B%5Cleft(%20%5Cbegin%7Barray%7D%7Bc%7D%0A%09n%5C%5C%0A%09k%5C%5C%0A%5Cend%7Barray%7D%20%5Cright)%20%5Cfrac%7Bk%7D%7Bn%7D%7Dx%5Ek%5Cleft(%201-x%20%5Cright)%20%5E%7Bn-k%7D%2Cx%5Cin%20%5Cleft%5B%200%2C1%20%5Cright%5D%20$

這就是 $%5Cfrac%7B%5Cmathbb%7BE%7D%20%5Cxi%20_n%7D%7Bn%7D$ ，那么現(xiàn)在來估計(jì)一下 $%5Cleft%7C%20%5Cfrac%7B%5Cmathbb%7BE%7D%20%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C$

$%5Cbegin%7Baligned%7D%0A%09%26%5Cleft%7C%20%5Cfrac%7B%5Cmathbb%7BE%7D%20%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%5Cle%20%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3B%5Cleft%5C%7B%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%5Cle%20%5Cvarepsilon%20%5Cright%5C%7D%20%5Cright)%20%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%5Cle%20%5Cvarepsilon%20%5Cright)%20%0A%5C%5C%26%2B%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3B%5Cleft%5C%7B%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3E%5Cvarepsilon%20%5Cright%5C%7D%20%5Cright)%20%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3E%5Cvarepsilon%20%5Cright)%5C%5C%0A%09%26%5Cle%20%5Cvarepsilon%20%2B%5Cfrac%7BM%7D%7B4%5Cvarepsilon%20%5E2n%7D%5C%5C%0A%5Cend%7Baligned%7D$

這里的 $%5Cmathbb%7BE%7D(X%3BA)$ 代表把樣本空間限制在 $A$ 上對(duì)隨機(jī)變量 $X$ 去求期望

其中的 $M%3D%5Cmax%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%20%5Cright)%20$ ，分母的估計(jì)來源于 $%5Ctext%7BBernoulli%7D$ 大數(shù)定律的證明，從上面的估計(jì)可以看出，對(duì)于與 $x$ 無關(guān)的充分大的 $n$ ，會(huì)成立 $%5Cleft%7C%20%5Cfrac%7B%5Cmathbb%7BE%7D%20%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%5Cle%202%5Cvarepsilon%20$ ，也就是 $%5Cleft%7C%20f_n%5Cleft(%20x%20%5Cright)%20-x%20%5Cright%7C%3C2%5Cvarepsilon%20$

對(duì)于一般的情況來說，從上面的證明可以看出來，實(shí)際上就是要估計(jì)

$%5Cleft%7C%20%5Cmathbb%7BE%7D%20%5Cleft(%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C$

為了估計(jì)這個(gè)數(shù)，試著把它往上面的式子的形式上湊。因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=f(x)" alt="f(x)">在 $%5B0%2C1%5D$ 上連續(xù)，所以它在 $%5B0%2C1%5D$ 上一致連續(xù)，也就是說 $%5Cforall%20%5Cvarepsilon%20%3E0$ ， $%5Cexists%20%5Cdelta%20%3E0$ ， $%5Cforall%20%5Cleft%20%7Cx-y%20%5Cright%7C%3C%5Cdelta$ 且 $x%2Cy%5Cin%5B0%2C1%5D$ ，都成立 $%5Cleft%7C%20f%5Cleft(%20x%20%5Cright)%20-f%5Cleft(%20y%20%5Cright)%20%5Cright%7C%3C%5Cvarepsilon%20$ ，那么

$%5Cbegin%7Baligned%7D%0A%09%26%5Cleft%7C%20%5Cmathbb%7BE%7D%20%5Cleft(%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3D%5Cleft%7C%20%5Cmathbb%7BE%7D%20%5Cleft(%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright)%20%5Cright%7C%5C%5C%0A%09%26%5Cle%20%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%20%5Cright)%5C%5C%0A%09%26%3D%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3B%5Cleft%5C%7B%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%5Cle%20%5Cvarepsilon%20%5Cright%5C%7D%20%5Cright)%20%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%5Cle%20%5Cvarepsilon%20%5Cright)%5C%5C%0A%09%26%2B%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3B%5Cleft%5C%7B%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3E%5Cvarepsilon%20%5Cright%5C%7D%20%5Cright)%20%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3E%5Cvarepsilon%20%5Cright)%5C%5C%0A%09%26%5Cle%20%5Cvarepsilon%20%2B%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3B%5Cleft%5C%7B%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3E%5Cdelta%20%5Cright%5C%7D%20%5Cright)%20%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3E%5Cdelta%20%5Cright)%5C%5C%0A%09%26%5Cle%20%5Cvarepsilon%20%2B%5Cfrac%7BM%7D%7B4%5Cdelta%20%5E2n%7D%5C%5C%0A%5Cend%7Baligned%7D$

然后和上面一樣可以導(dǎo)出結(jié)論，其中第二個(gè)不等號(hào)是因?yàn)?/p>

$%5Cleft%5C%7B%20%5Cleft%7C%20f%5Cleft(%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D%20%5Cright)%20-f%5Cleft(%20x%20%5Cright)%20%5Cright%7C%3E%5Cvarepsilon%20%5Cright%5C%7D%20%5Csubset%20%5Cleft(%20%5Cleft%7C%20%5Cfrac%7B%5Cxi%20_n%7D%7Bn%7D-x%20%5Cright%7C%3E%5Cdelta%20%5Cright)%20$

這個(gè)問題的數(shù)分證法見陳紀(jì)修，比這個(gè)復(fù)雜很多

108.2 設(shè)隨機(jī)變量 $%5Cxi$ 是標(biāo)準(zhǔn)化的，證明：

對(duì) $x%3E0$ ， $%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cxi%20%5Cge%20x%20%5Cright)%20%5Cle%20%5Cfrac%7B1%7D%7B1%2Bx%5E2%7D$ 。

此不等式不能再改進(jìn)，即存在標(biāo)準(zhǔn)化的隨機(jī)變量 $%5Cxi$ ，使得 $%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cxi%20%5Cge%20x%20%5Cright)%20%3D%5Cfrac%7B1%7D%7B1%2Bx%5E2%7D$

（提示：先利用 $%5Ctext%7BChebyshev%7D$ 的方法找到適當(dāng)?shù)暮瘮?shù)證明 $%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cxi%20%3Ex%20%5Cright)%20%3D%5Cfrac%7B1%2Ba%5E2%7D%7B%5Cleft(%20x%2Ba%20%5Cright)%20%5E2%7D$ ，然后取適當(dāng)?shù)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=a" alt="a">）

按照提示探索了一段時(shí)間后，可知

$%5Cmathbb%7BP%7D%20%5Cleft(%20%5Cxi%20%3Ex%20%5Cright)%20%3D%5Cmathbb%7BE%7D%20%5Cleft(%201%3B%5Cleft%5C%7B%20%5Cxi%20%3Ex%20%5Cright%5C%7D%20%5Cright)%20%5Cle%20%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20%5Cfrac%7B%5Cxi%20%2Ba%7D%7Bx%2Ba%7D%20%5Cright)%20%5E2%3B%5Cleft%5C%7B%20%5Cxi%20%3Ex%20%5Cright%5C%7D%20%5Cright)%20%5Cle%20%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20%5Cfrac%7B%5Cxi%20%2Ba%7D%7Bx%2Ba%7D%20%5Cright)%20%5E2%20%5Cright)%20$

計(jì)算可知 $%5Cmathbb%7BE%7D%20%5Cleft(%20%5Cleft(%20%5Cfrac%7B%5Cxi%20%2Ba%7D%7Bx%2Ba%7D%20%5Cright)%20%5E2%20%5Cright)%20%3D%5Cfrac%7B1%2Ba%5E2%7D%7B%5Cleft(%20x%2Ba%20%5Cright)%20%5E2%7D$ （這里之前一直把標(biāo)準(zhǔn)化的隨機(jī)變量的期望當(dāng)成1了，結(jié)果半天出不來這個(gè)結(jié)果）

接下來，因?yàn)?img type="latex" class="latex" src="http://api.bilibili.com/x/web-frontend/mathjax/tex?formula=%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20a%7D%5Cln%20%5Cleft(%20%5Cfrac%7B1%2Ba%5E2%7D%7B%5Cleft(%20x%2Ba%20%5Cright)%20%5E2%7D%20%5Cright)%20%3D%5Cfrac%7B2a%7D%7B1%2Ba%5E2%7D-%5Cfrac%7B2%7D%7Bx%2Ba%7D" alt="%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20a%7D%5Cln%20%5Cleft(%20%5Cfrac%7B1%2Ba%5E2%7D%7B%5Cleft(%20x%2Ba%20%5Cright)%20%5E2%7D%20%5Cright)%20%3D%5Cfrac%7B2a%7D%7B1%2Ba%5E2%7D-%5Cfrac%7B2%7D%7Bx%2Ba%7D">，易知 $xa%3D1$ 時(shí)這個(gè)關(guān)于 $a$ 的函數(shù)有最小值，帶入可得