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da537ef4a57236c206a800001be49018b03161c8: [[Pasted Image 20230417205625_574.jpg]]
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echo allow-loopback-pinentry >> ~/.gnupg/gpg-agent.conf~/.___echo -n "#\!/usr/bin/bash \ngpg --pinentry-mode loopback --batch --yes --passphrase-fd 1 --passphrase-file ~/.___ $@" >> ~/.gpg-pinentry-loopbackchmod +x ~/.gpg-pinentry-loopbackgit config --global gpg.program $HOME/.gpg-pinentry-loopbackecho -n "default-cache-ttl 34560000\n max-cache-ttl 34560000" >> ~/.gnupg/gpg-agent.confFor all $t_1, \ldots, t_k \in T, k \in \mathbf{N}$ let $\nu_{t_1, \ldots, t_k}$ be probability measures on $\mathbf{R}^{n k}$ s.t.
$$
\nu_{t_{\sigma(1)}, \cdots, t_{\sigma(k)}}\left(F_1 \times \cdots \times F_k\right)=\nu_{t_1, \cdots, t_k}\left(F_{\sigma^{-1}(1)} \times \cdots \times F_{\sigma^{-1}(k)}\right)
$$
for all permutations $\sigma$ on ${1,2, \ldots, k}$ and
$$
\nu_{t_1, \ldots, t_k}\left(F_1 \times \cdots \times F_k\right)=\nu_{t_1, \ldots, t_k, t_{k+1}, \ldots, t_{k+m}}\left(F_1 \times \cdots \times F_k \times \mathbf{R}^n \times \cdots \times \mathbf{R}^n\right)
$$
for all $m \in \mathbf{N}$, where (of course) the set on the right hand side has a total of $k+m$ factors.
Then there exists a probability space $(\Omega, \mathcal{F}, P)$ and a stochastic process $\left{X_t\right}$ on $\Omega, X_t: \Omega \rightarrow \mathbf{R}^n$, s.t.
$$
\nu_{t_1, \ldots, t_k}\left(F_1 \times \cdots \times F_k\right)=P\left[X_{t_1} \in F_1, \cdots, X_{t_k} \in F_k\right],
$$
for all $t_i \in T, k \in \mathbf{N}$ and all Borel sets $F_i$.
Kolmogorov的延拓定理解决了在给定各个坐标空间上的概率测度的情况下，如何构造一个概率测度来描述乘积空间的问题。
简单来说，Kolmogorov的延拓定理允许我们为定义在不同空间上的一组随机变量构造一个联合概率测度。
它提供了一种将概率测度从有限维空间延拓到无限维乘积空间的方法。
Kolmogorov的延拓定理要求满足以下条件：完全一致性条件（完全可测性）：给定一组可测空间和其上的概率测度，这些测度必须满足完全一致性条件。
即，对于任意的有限个可测集合，它们的乘积集合在每个坐标空间上的投影集合的概率测度必须与已知的概率测度一致。
Kohlrausch条件（无信息传递性）：该条件要求当从一个坐标空间的可测集合推断到另一个坐标空间时，概率测度是一致的。
换句话说，如果两个可测集合的投影集合在某个坐标空间上的概率已知，则这两个可测集合的乘积集合在其他坐标空间上的概率也应该是一致的。
满足这些条件，就可以使用Kolmogorov的延拓定理构造一个满足一致性要求的概率测度。
Let $T$ denote some Interval (thought of as “time”), and let $n \in \mathbb{N}$. For each $k \in \mathbb{N}$ and finite sequence of distinct times $t_{1}, \dots, t_{k} \in T$, let $\nu_{t_{1} \dots t_{k}}$ be a probability measure on $(\mathbb{R}^{n})^{k}$. Suppose that these measures satisfy two consistency conditions:
Then there exists a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and a stochastic process $X : T \times \Omega \to \mathbb{R}^{n}$ such that
:$\nu_{t_{1} \dots t_{k}} \left( F_{1} \times \dots \times F_{k} \right) = \mathbb{P} \left( X_{t_{1}} \in F_{1}, \dots, X_{t_{k}} \in F_{k} \right)$
for all $t_{i} \in T$, $k \in \mathbb{N}$ and measurable sets $F_{i} \subseteq \mathbb{R}^{n}$, i.e. $X$ has $\nu_{t_{1} \dots t_{k}}$ as its finite-dimensional distributions relative to times $t_{1} \dots t_{k}$.
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 $X\colon (t,Y) \mapsto Y_t$. 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.
A stochastic process $\left{B_t\right}_{t \geq 0}$ is called a Brownian motion if it can be constructed as follows:
Fix $x \in \mathbb{R}^n$ and define
$$
p(t, x, y) = \frac{1}{(2 \pi t)^{n/2}} \exp \left(-\frac{|x-y|^2}{2t}\right)
$$
Then if $0 \geq t_1 < t_2 < \cdots < t_k$ define a measure $\mu_{t_1, \cdots, t_k}$ on $\mathbb{R}^{nk}$ by
$$
\mu_{t_1, \cdots, t_k} (F_1 \times \cdots \times F_k) = \int_{F_1 \times \cdots \times F_k} p(t_1, x_0, x_1) p(t_2 - t_1, x_1, x_2) \cdots p(t_k - t_{k-1}, x_{k-1}, x_k) dx_1 \cdots dx_k
$$
where $F_i \in \mathcal{B}(\mathbb{R}^n)$.
Then $\mu_{t_1, \cdots, t_k}$ is a probability measure on $\mathbb{R}^{nk}$ and the finite dimensional distributions of $\left{B_t\right}_{t \geq 0}$ are given by
$$
P^x(B_{t_1} \in F_1, \cdots, B_{t_k} \in F_k) = \mu_{t_1, \cdots, t_k} (F_1 \times \cdots \times F_k).
$$
We state some basic properties of Brownian motion:
![[Pasted image 20230722114843.png]]
git config --global user.name "wzh4464"git config --global user.email "32484940+wzh4464@users.noreply.github.com"git config --global user.signingkey ****git config --global commit.gpgsign true-s means --signoff and -S means --gpg-sign1 | gpg --export-secret-keys -a **** > private.key |
gpg --batch --import private.keygpg --edit-key ****trust -> 5 -> y -> quitplugin name: obsidian git
不知道有没有用的:
curl -L https://aka.ms/gcm/linux-install-source.sh | shgit-credential-manager configure使用gpg来验证身份
git config --global credential.credentialStore gpginstall pass
pass init ****
git config --global core.editor "vim"timedatectl set-local-rtc 1
Advance Option on the right above, then choose Download onlyip link show and copy the mac address.https://ww2.mathworks.cn/licensecenter/licenses/40746419/5495716/activations?bypass=true./install -inputfile installer_input.txt -tmpdir /home/jeff/zihan/tmpThis plugin provides a few utilities to make it more enjoyable on macOS (previously named OSX).
To start using it, add the macos plugin to your plugins array in ~/.zshrc:
1 | plugins=(... macos) |
Original author: Sorin Ionescu
| Command | Description |
|---|---|
tab |
Open the current directory in a new tab |
split_tab |
Split the current terminal tab horizontally |
vsplit_tab |
Split the current terminal tab vertically |
ofd |
Open the current directory in a Finder window |
pfd |
Return the path of the frontmost Finder window |
pfs |
Return the current Finder selection |
cdf |
cd to the current Finder directory |
pushdf |
pushd to the current Finder directory |
pxd |
Return the current Xcode project directory |
cdx |
cd to the current Xcode project directory |
quick-look |
Quick-Look a specified file |
man-preview |
Open man pages in Preview app |
showfiles |
Show hidden files in Finder |
hidefiles |
Hide the hidden files in Finder |
itunes |
DEPRECATED. Use music from macOS Catalina on |
music |
Control Apple Music. Use music -h for usage details |
spotify |
Control Spotify and search by artist, album, track… |
rmdsstore |
Remove .DS_Store files recursively in a directory |
btrestart |
Restart the Bluetooth daemon |
freespace |
Erases purgeable disk space with 0s on the selected disk |
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