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Results tagged with pr.probability
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user 99157
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
3
votes
3
answers
199
views
$\mathbf{y}=f(\mathbf{x},\mathbf{z})=g(\mathbf{x})$ if $\mathbf{z}\perp \!\!\! \perp \{\math...
Let $\mathbf{y},\mathbf{x},\mathbf{z}$ are real-valued random vectors with possibly different dimensions. Assume $\mathbf{y}=f(\mathbf{x},\mathbf{z})$ for some function $f$.
If $\mathbf{z} \perp\!\!\! …
- 193
-2
votes
1
answer
42
views
$E(\mathbf{y}|\mathbf{x}+\mathbf{z})=g(\mathbf{x})$ almost surely, if $\mathbf{z}\perp \!\!\...
Let $\mathbf{y},\mathbf{x}$ and $\mathbf{z}$ be real-valued random vectors with possibly different dimensions.
If $\mathbf{z} \perp\!\!\!\perp \{\mathbf{y},\mathbf{x}\}$ (i.e., $\mathbf{z}$ is statist …
- 193
0
votes
0
answers
198
views
$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?
For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold?
$$
\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)
$$
I know t …
- 193
3
votes
1
answer
242
views
Independent input feature z can be removed: if y=f(x+z,z), then y=g(x)?
Let $y\in \mathbb{R}$ and $\mathbf{x},\mathbf{z}\in\mathbb{R}^p$ be random variable and random vectors. Assume $y=f(\mathbf{x}+\mathbf{z},\mathbf{z})$ for some function $f$.
Is the following statement …
- 193
1
vote
0
answers
199
views
Rank of cross-covariance matrix
Let $\boldsymbol{X}=(X_1,\dots,X_p)^T$ and $\boldsymbol{Y}=(Y_1,\dots,Y_q)^T$ be two random vectors. Denote $r_x=\text{rank}(\text{Cov}(\boldsymbol{X})),r_y=\text{rank}(\text{Cov}(\boldsymbol{Y})), r_ …
- 193
4
votes
1
answer
293
views
Finding high-dimensional correlation matrices that are both sparse and low-rank
Let $\boldsymbol{R}$ be the correlation matrix of $X_i,i=1,\dots,p$ with a large $p\gg q=\text{rank}(\boldsymbol{R})$. Is that reasonable to assume that $\boldsymbol{R}$ is both (approximately) sparse …
- 193