All Questions
11 questions
3
votes
1
answer
136
views
Concentration of sample median for iid Gaussians
Let $X_1, \dots, X_n$ be iid according to $\mathcal{N}(0, 1)$, and let $M_n$ be the median of the $X_1, \dots, X_n$. I recall reading a concentration inequality for $M_n$ that was (roughly) as follows:...
- 75
2
votes
0
answers
75
views
References for a class of Banach space-valued Gaussian processes
Let $E$ be a separable Banach space, consider a centered $E$-valued Gaussian process $\{x_t,t\ge 0\}$ that satisfies
\begin{equation}
\mathbb{E}\phi(x_s)\psi(x_t)=R(s,t)K(\phi,\psi),\quad \phi,\psi\in ...
- 121
0
votes
1
answer
102
views
Lower bounds for truncated moments of Gaussian measures on Hilbert space
Let $\mu_C$ be a centered Gaussian probability Borel measure on a real separable Hilbert space $\mathcal{H}$ with covariance operator $C$. Denote the ball with radius $r$ in $\mathcal{H}$ centered at ...
- 505
4
votes
1
answer
311
views
Examples of Borel probability measures on the Schwartz function space?
Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...
- 3,477
3
votes
2
answers
102
views
Reference for Wiener type measure on $C(T)$ when $T$ is open
I'm considering Gaussian process on open domain $T$ in $\mathbb{R}^n$ and I tried to follow the abstract Wiener space construction of Gross. Since my sample paths are meant to be continuous, I thought ...
- 163
0
votes
0
answers
71
views
References on estimates for suprema of uncentered Gaussian processes?
Let $X_t, t \in T$ denote a centered Gaussian process. Let $d(t, s) = \sqrt{\mathbb{E} (X_t - X_s)^2}$.
Consider a mean function $t \mapsto \mu_t$.
Define the expected supremum
$$
S(T, \mu) = \mathbb{...
- 420
5
votes
1
answer
942
views
Moments of maximum of independent Gaussian random variables
Let $X = (X_1, \ldots, X_d) \in \mathbb{R}^d$ be a mean-zero Gaussian random vector with identity covariance matrix. Are there upper bounds for
$$E \left(\|X\|_{\infty}^k \right)$$ for $k=1, \ldots, ...
- 519
5
votes
0
answers
857
views
Anti-concentration inequality for Gaussian random vector
I am trying to obtain an explicit expression for $C$ in terms of $b$ in the following inequality.
Suppose that $Y$ is a centred Gaussian random vector in $\mathbb R^p$ such that $\operatorname EY_j^...
- 423
1
vote
1
answer
140
views
Reference request: Cover times, Mixing Times and DGFF applied in statistics?
I am trying to find if in active research in statistics, there is interest in mixing times, cover times of graphs, and/or the discrete Gaussian free field?
I haven't found anything so far for the ...
- 11
2
votes
1
answer
95
views
Bounding exceedance probabilities for correlated normal variables
Suppose $y\sim N(0,\Sigma)$ is an $n-$dimensional vector. I'm interested in an upper bound for $\Pr(\max_{1\leq i\leq n} y_i > k)$ for $k$ large. I know a little about $\Sigma$: $\sigma_{ii}=\...
- 269
1
vote
1
answer
115
views
Supremum of centered jointly generalized chi-square random variables
Let $\zeta_n$ be a sequence of centered jointly generalized chi-square random variables, i.e. $\zeta_n = \sum_{k=1}^{m_n} a_{k,n}(\xi_{k,n}^2 - E[\xi_{k,n}^2])$, and $\xi_{k,n}$ are centered jointly ...
- 1,533