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1 vote

### Dudley's inequality: Sending $\delta$ to $0$

To deduce $\mathbb{E}[\sup_{t, s\in T\colon d(t, s)<\delta}X_t - X_s]\to 0$ as $\delta\to 0$, I think separability for the process suffices. Recall that we say a process is separable if there ...
• 90
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• 41.5k
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### Calculating Probability in Disjoint and Independent Events

No, it's not true that $\ P(A'\cap C'\,|\,B)=\frac{P(A'\cap\,C')}{P(B)}\ .$ By definition $$P(A'\cap C'\,|B)=\frac{P((A'\cap C')\color{red}{\cap B})}{P(B)}\ ,\tag{1}\label{e1}$$ and since \begin{...
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If $X$ is discrete and $P(X>0)>0$ then there exists some $x_i>0$ such that $P(X=x_i)>0$. As $P(X\geq 0)=1$, $X$ cannot take on a negative value with a non-zero probability, because the sum ...
• 4,376
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### PDF of $Y=g(X)$ when $X\sim N(0,1)$. $g(X)$ is a piecewise function where each part is constant.

Your expression for the CDF of $Y$ is close, but should be $$F_{Y}\left(y\right)=\begin{cases} 0, & y<-1\\ 1/2, & -1\leq y<1\\ 1, & y\geq1 \end{cases}.$$ Since $Y$ is discrete, it ...
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• 8,842
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### Sub-Gaussian $X_t$, prove $\mathbb{E}\left[\sup_{t\in T}X_t \right] \leq 2 \mathbb{E}\left[\sup_{\rho(t,s)\leq \delta}(X_t-X_s) \right]+J(\delta,T)$

For this problem, bounding the expectation of the supremum is crucial. The following inequality can help in this case. Proposition. Let $\{Z_i\}_{i=1}^{N}$ be $\sigma^2$-sub-Gaussian random variables. ...
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• 14.1k
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### function of random variables - change of variables

\begin{align} \int f(z)p_Z(z)\,dz&=\mathbb E[f(Z)]\\ &=\mathbb E[f(g(X,Y))]\\ &=\int f(g(x,y))p_{X,Y}(x,y)\,dx\,dy\\ &=\int\int f(g(x,y))p_X(x)p_Y(y)\,dx\,dy, \end{align} where I ...
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### If $X, Y$ independent, and $Y$ has same distribution as $Z$, are $X, Z$ independent?

This is not true. Take $X$ and $Y$ i.i.d. and $Z=X$.
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• 90

It sounds like you are trying to keep the same $(\Omega, \mathcal{F})$, so the functions $X_i:\Omega\rightarrow\mathbb{R}$ and $Z_i:\Omega\rightarrow\mathbb{R}$ are always the same, but just change $P:... • 24k 2 votes ### The almost sure event in the law of the iterated logarithm for the Brownian motion: what it looks like The event you have captured in more explicit terms is not$\{\limsup_{t\to 0}W_t/h(t)=1\}$, but rather$\{\limsup_{t\to 0}|W_t/h(t)-1|=0\}$. The placement of the absolute value is crucial! • 25.8k 4 votes ### Proof of a martingale condition regarding martingale transform$\$ \begin{align} C_n(\mathbb E[X_n\vert\mathcal F_{n-1}]-X_{n-1})&=\mathbb E[C_n(X_n-X_{n-1})\vert\mathcal F_{n-1}]\\ &=\mathbb E[(C\bullet X)_n-(C\bullet X)_{n-1}\vert\mathcal F_{n-1}]\\ &...
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