All Questions
6
questions
2
votes
0
answers
33
views
Lower bound of $\frac{\|(\mathbf X \otimes \mathbf X^\top)\theta\|_2^2}{np}$
According to Theorem 7.16 of High-Dimensional Statistics: A Non-Asymptotic Viewpoint (M. Wainwright, 2019), we know that for $\mathbf X\in\mathbb R^{n\times p}, X_{ij}\overset{iid}{\sim}N(0,1),$ there ...
- 63
0
votes
0
answers
54
views
Why is the multivariate normal distribution is $(\Sigma, C)$ sub-gaussian?
The definition of \textit{sub-gaussian} from a book I work with is: $X\in\mathbb{R}^n$ is $(\Sigma,C)$ sub-gaussian if $$\mathbb{P}(\lvert X^\top u\rvert>t)<Ce^{-t^2/(2u^\top\Sigma u)}, \qquad u\...
- 11
4
votes
1
answer
372
views
Exponential bound for tail of standard normal distributed random variable
Let $X\sim N(0,1)$ and $a\geq 0$. I have to show that $$\mathbb{P}(X\geq a)\leq\frac{\exp(\frac{-a^2}{2})}{1+a}$$
I have no problem showing that $\mathbb{P}(X\geq a)\leq \frac{\exp(\frac{-a^2}{2})}{a\...
- 103
0
votes
1
answer
31
views
Lower bound on empirical Gaussian probablity
Consider $\{a_1,a_2,...,a_n\}$ n points in $\mathbb{R}$. Assume their mean is $0$, their standard deviation is then given by $\sigma=\sqrt{\dfrac{1}{n}\sum_ia_i^2}$. Let $p(x)=\dfrac{1}{\sqrt{2\pi}\...
- 700
1
vote
0
answers
131
views
High probability upper bound for linear combination of Gaussian random variables
Suppose that $x_1, \dots, x_n$ are i.i.d. with $x_i \sim N(0,I_k)$. Let $A_1, \dots, A_n$ be matrices with dimension $k \times k$ and $\|A_i\|_2 \leq 1$. Consider the following random vector
$$y = \...
- 1,180
2
votes
0
answers
70
views
Show that $\operatorname{Pr}(Z-X \geq 0)$ converges to one
Suppose that $V_i$, for $i \in \mathbb{N}$, are i.i.d. standard normal random variables and $Y_i = \sum_{k=1}^i V_k$ for $i \in \mathbb{N}$ with $Y_0 = 0$. Let $X_n = (\sum_{i=1}^n V_i Y_{i-1})^2 Y_n^...
- 1,180