All Questions
32 questions
2
votes
1
answer
89
views
Upper bound on the Levy-Prokhorov distance between the distributions of continuous Gaussian processes in terms of their covariances
Denote by $d$ the supremum metric on the space $C[0,T]$ of continuous real-valued functions on $[0,T]$:
$$
d(f,g) = \sup_{t \in [0,T]} |f(t)-g(t)|.
$$
Let $\rho$ be the Levy-Prokhorov metric on the ...
- 177
2
votes
0
answers
74
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
1
vote
0
answers
133
views
A question about one Malliavin derivative calculation
Recently, I've asked here a question. While trying to find an answer on my own, I found an idea which I now will briefly describe below. I am not familiar enough with the Malliavin calculus, so my ...
- 620
0
votes
0
answers
101
views
Integration with respect to $B_H(t) B_H(s) - \mathbb{E} \{ B_H ( t ) \, B_H ( s) \}$
The time-derivative $\frac{dB_H}{dt}$ of the fractional Brownian motion may be interpreted as a random Schwartz distribution acting on a test function by
$$
\left\langle \frac{dB_H}{dt}, f \right\...
- 620
8
votes
1
answer
421
views
Is there an infinite dimensional Stein's lemma?
Classical Stein's lemma says that if $\mathbf{X}$ is a centered Gaussian random vector and $g$ is a function which is nice enough, we have
$$
\mathbb{E} \, X_i \, g ( \mathbf{X} )
= \sum_k \...
- 620
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
0
votes
1
answer
92
views
Lower bounding the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w_i)t_i\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w_i$ are iid standard gaussian variables, $s$ is a scalar denoting the strength of Gaussian noise.
How ...
- 405
0
votes
2
answers
118
views
the infimum of a random process
Let $X_{t}=\sum_{i=1}^n(1+s\cdot w)\sin(t_i)$ where $t\in T=[-\pi/2,\pi/2]^n/\{\vec 0\}$, $w\sim\mathbb{N}(0,1)$, $s$ is a scalar denoting the strength of Gaussian noise. How to find the condition on $...
- 405
4
votes
1
answer
238
views
Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange.
Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel.
My ...
- 203
3
votes
2
answers
184
views
Maximizing expectation of gaussian process over covariance matrix with fixed trace
Let $\mathcal{A} = \{\Sigma \in PSD_{n\times n}(\mathbb{R}), \wedge \forall i,\Sigma_{ii}=1\}$. Then $\mathcal{A} \subset M_{n\times n}(\mathbb{R})$ is convex, closed, and bounded.
For each $\Sigma \...
- 143
3
votes
2
answers
256
views
Local nondeterminism
I'm trying to understand Berman's classic paper on the subject ("Local Nondeterminism and Local Times of Gaussian Processes"). In order to define local nondeterminism, he considers the ratio
...
- 205
1
vote
0
answers
100
views
Ito formula for fractional BM + drift and supremum bound
Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a ...
- 111
3
votes
1
answer
501
views
Regularity of Gaussian process sample paths
Consider a Gaussian process on $[0,1]$ given by a kernel function $K: [0,1]^2\to\mathbb{R}$. Under what conditions can we conclude that the sample paths are $C^k$ with probability 1?
This question is ...
- 505
1
vote
1
answer
178
views
Tail bound on the RKHS norm of a zero-mean Gaussian process
Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
- 1,127
2
votes
1
answer
124
views
Limiting behavior of $k^{th}$ order statistics of n non-i.i.d chi square random variables
This is related to one of my previous questions here.
Let $(Z_1, Z_2, \ldots, Z_n)\sim N(0, \Omega)$, where $\Omega = (1-\mu) I_{n\times n} + \mu \boldsymbol{1}_n\boldsymbol{1}_n^\top $. Here $\...
- 399
1
vote
1
answer
107
views
Law of OU process with time-dependent dynamics
Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
- 77
0
votes
0
answers
86
views
A non trivial example of a Gaussian semi-Markov process?
Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov ...
- 143
1
vote
1
answer
512
views
Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
- 5,405
0
votes
1
answer
102
views
Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
- 536
1
vote
1
answer
66
views
Comparing noisy truncated RV with noisy regular RV
For some reason, I'm having difficulties proving something that is intuitively simple.
Assuming I have two a random variable, $x$ and $x^{truncated}$, where $x^{truncated}$ is the truncated version of ...
- 183
2
votes
1
answer
244
views
Reference: hitting time of Gaussian process
Let $X_t$ be an OU process and $Y_t$ be the Gaussian process defined by
$$
Y_t = y+\int_0^t X_s ds + W_t,
$$
for some Brownian motion independent of $X_t$. Let $y,a>0$; is there a large deviation ...
- 5,405
1
vote
0
answers
62
views
Distances between up and down crosses in Gaussian Processes
Given a gaussian process $g := \mathcal{GP}\left(\mu, \Sigma \right)$,
where $\mu$ is the mean and $\Sigma$ is the covariance function, I am interested in estimating the mean value $L_m$ of the ...
- 91
1
vote
1
answer
301
views
Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality
In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...
- 281
1
vote
1
answer
287
views
Random matrix and spherical spin-glass
The Hamiltonian of the p-spherical spin glass model is
$$H_{N,p}(\sigma)=\frac{1}{N^{\frac{p-1}{2}}} \sum_{i_1,...,i_p=1}^N X_{i_1,...,i_p} \sigma_{i_1}\cdot...\cdot \sigma_{i_p}$$
where $\sigma \in ...
- 536
2
votes
2
answers
548
views
Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors
Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
- 325
1
vote
1
answer
2k
views
Autocovariance of time integrated Ornstein–Uhlenbeck process
$\newcommand{\Cov}{\operatorname{Cov}}\newcommand{\Var}{\operatorname{Var}}$if $X(t)$ is the Ornstein–Uhlenbeck process and $Y(t)$ the time integrated OU process I am trying to calculate the ...
- 141
6
votes
2
answers
662
views
Infimum of Gaussian process
Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about g(...
- 373
2
votes
2
answers
492
views
Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?
I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
6
votes
1
answer
398
views
Does a Gaussian process shrink under a contraction map
Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...
- 757
6
votes
4
answers
3k
views
Calculating the probability of an event defined by a condition on a Gaussian random process
Although the question itself can be expressed succinctly, I couldn't come up with a nice self-explanatory title - suggestions are welcome.
Motivation/Background
I was investigating whether it would ...
6
votes
2
answers
2k
views
Are Gaussian Processes more important than other stochastic processes?
I am doing a course at university and it deals with Gaussian Processes mainly. We use them for fitting data and prediction, machine learning, regression, classification. Is there any particular reason ...
- 87
5
votes
3
answers
898
views
Lower bound for Gaussian random vector with negative correlation
Let $X = (X_1,\ldots,X_n) \in \mathbb{R}^n$ be jointly Gaussian with mean $0$, covariance matrix: $Var(X_i) = 1$, $Cov(X_i, X_{i+1}) = -1/2$, and $Cov(X_i, X_j) = 0$ else.
Let $\zeta \in \mathbb{R}^...
- 1,010