All Questions
Tagged with gaussian pr.probability
220 questions
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
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
1
answer
281
views
Hermite polynomial and Gaussian random variable
The following formula is well known: $E[H_k(X,E[X])H_q(Y,E[Y])]=\delta_{kq}E[XY]^k$ for a joint Gaussian r.v. $(X, Y),$ $H_k$ are Hermite polynomiale.
Is there a generalization for this to a joint ...
- 573
2
votes
1
answer
1k
views
measure of a degenerate Gaussian distribution
I want to do computations with a degenerate Gaussian measure, but I do not know how to represent it
in a close form.
After starting with a Gaussian random variable and restricting it to a condition, I ...
- 23
2
votes
1
answer
334
views
Gaussian sum VS Brownian motion
Given independent Gaussian $d$ dimensional vectors $G_i$,
Let $ \sigma^2_n=\mathbb{E}(\sum_{i \le n} G_i) \cdot (\sum_{i \le n} G_i)^T$. $||\sigma_n^2||$ is norm of $\sigma_n^2$.
Is there a $d$-...
- 553
2
votes
1
answer
64
views
Maximum Number of modes of $V=U+Z$ where $Z$ standard normal and $|U|\le a$
Let $f_V$ be a pdf of random variable $V$ where
\begin{align}
V=U+Z
\end{align}
and where $U$ and $Z$ are independent and $Z$ is Gaussian. Moreover, suppose that $|U| \le A$.
Can we find the upper ...
- 671
2
votes
2
answers
486
views
The correlation between a Gaussian random variable and its multiplication with another random variable
Suppose $X$ is a multivariate Gaussian random variable $X\sim \mathcal{N}\left(0,H\right)$ and we define a new random variable $\eta$ by its multiplication with some other random variable $Y$, i.e., $...
- 21
2
votes
1
answer
199
views
Gaussian Poincare inequality in $1$ dimensions together with localization issue
Let $d\mu$ be a Gaussian measure on $\mathbb{R}$ with the center $a \in \mathbb{R}$ and variance $1$.
Let $B(a,r) \subset \mathbb{R}$ be the interval $[a-r,a+r]$.
Then, for any smooth mapping $f : \...
- 3,477
2
votes
1
answer
94
views
Mixture of gaussian density agree with another gaussian on positive measure
I noticed this post. But still I'd like to follow up with a specific case I have in mind. Say $p(x| \theta)$ is the density of a Gaussian distribution on $\mathbb{R}^n$ with mean $\theta$ and known ...
- 23
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
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
2
votes
3
answers
999
views
Sum of Square of the Eigenvalues of Wishart Matrix
Let $A\in\mathbb{R}^{m\times d}$ matrix with iid standard normal entries, and $m\geqslant d$, and define $S=A^T A$.
I want to have a tight upper bound for $\sum_{k=1}^d \lambda_k^2$, where $\...
- 1,096
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 $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...
2
votes
1
answer
2k
views
Overall covariance of Mixture of Gaussian
I have a Mixture of Gaussians to model an arbitrary distribution. I would like to model a distribution derived from this GMM with:
Mean = Weighted average mean of GMM means.
I am not sure about how ...
- 121
2
votes
0
answers
97
views
+100
Inequalities for norm of centered Gaussian and uncentered Gaussian
Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm.
Let $x \in \mathbb{R}^n$ and define
$$
F(x) = \mathbb{E}[\|x + g\| - \|g\|].
$$
I am wondering if it is possible to ...
- 420
2
votes
0
answers
56
views
Sum of independent Wisharts
Suppose random vectors $y_1,y_2,\ldots,y_m$ are independent and the distribution of each $y_i$ is a $d$-dimensional complex Gaussian with mean $0$ and covariance $\Gamma_i$, that is $y_i \sim \mathcal{...
- 193
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
2
votes
1
answer
119
views
Deriving the distribution of standardized variables with empirical mean and standard deviation
I'm working with a set of independent and identically distributed random variables $\{ x_i \}_{i=1}^N$, where each $x_i$ follows a Gaussian distribution $P_X(x) = \mathcal{N}(x; \mu, \sigma^2)$. This ...
- 305
2
votes
0
answers
85
views
Faster Convergence in CLT for sums and convolutions of Gaussians?
Let $n\in\mathbb{N}$ and $\sigma>0$ be fixed.
I have a certain class $\mathcal{C}$ of random variables I am interested in analyzing.
This contains
$\vec X\sim \mathcal{N}(0, \sigma^2I_n)$
Sums of (...
- 2,183
2
votes
0
answers
66
views
Iterated chaos expansion
Using the notations from Normal Approximations with Malliavin Calculus, Chapter 2
random variables $F$ in the probability space generated by an iso-normal Gaussian family $X(h)$,
$$E[X(h)X(g)] = \...
- 623
2
votes
0
answers
164
views
Fractional Brownian motion covariance with a twist
Let $H \in (0, 1)$, $D \in \mathbb{R}$ and assume that the following function
$$
r ( t, s ) = \frac{1}{2} \, \Big[ t^{2H} + s^{2H} - | t - s |^{2H} \Big] + D \, t^H s^H,
\quad t, \, s \geq 0
$$
is ...
- 620
2
votes
0
answers
57
views
Regularity on $\mathbb{T}^3$ of the "functional average" of a map $S : C^\infty(\mathbb{T}^3, \mathbb{R}) \to L^2(\mathbb{T}^3, \mathbb{R})$
For simplicity, let $C^\infty(\mathbb{T}^3, \mathbb{R})$ be the real Frechet space of periodic smooth functions on $\mathbb{R}^3$. Here, $\mathbb{T}^3$ is the $3$-dimensional torus.
For a fixed ...
- 3,477
2
votes
0
answers
61
views
Approximate logarithmic bound on expected maximum via central limit theorem
If $Z_i$ are standard normal, possibly dependent, one can show that
$$E\left[\max_{i=1,...,M} Z_i^2\right]\leq 3\ln M + 1.$$
I'm looking for a similar (asymptotic) bound for asymptotically normal ...
- 203
2
votes
0
answers
174
views
Slepian's Lemma for Range?
Let $\vec{x}$ and $\vec{y}$ be zero mean $n$-variate Gaussian variables with covariances $\Sigma^x, \Sigma^y$. Suppose they have identical marginals ($\sigma_{i,i}^x = \sigma_{i,i}^y$ for all $i$), ...
- 620
2
votes
2
answers
548
views
Quantifying the effect of noise on the posterior variance in Gaussian processes / multivariate Gaussian vectors
Consider a real-valued Gaussian process $f$ on some compact domain $\mathcal{X}$ with mean zero and covariance function $k(x,x') \in [0,1]$ (also known as the kernel function). This question concerns ...
- 325
2
votes
0
answers
247
views
Moments of a Normal-Wishart distribution
Do known expressions exist for the moments of a gaussian-wishart (aka normal wishart) distribution?
$$NW(\mu,K\mid\mu_0,\lambda_0, v, W) =
\frac{|\lambda_0K|^{1/2}}{(2\pi)^{d/2}}e^{-0.5([\mu - \mu_0]...
- 121
2
votes
0
answers
207
views
Gaussian integrals and Showing $ \int f({\vec {x}})e^{\left(-{\frac {1}{2}}\sum \limits _{i,j=1}^{n}A_{ij}x_{i}x_{j}\right)}d^{n}x=e^{D}f|_{x=0}$
This is related to my other question on tackling a gaussian integral for $f(w,u)=\frac{1}{w-u}$.
Q1 Suggestions on evaluating gaussian integrals with "nice" functions (not necessarily polynomials)
...
- 5,474
2
votes
0
answers
86
views
when is the average of a function with Gaussian inputs bounded away from zero
Define a function $\phi(x):\mathbb{R}\rightarrow\mathbb{R}$. Consider the expected value function defined as follows
\begin{align*}
\mu(\beta)=E[g\phi
(\beta g)]\quad with \quad g\sim\mathcal{N}(0,1)\...
- 363
2
votes
0
answers
386
views
What is the concentration of measure for Gaussian random variables which are independent, but are transformed?
This might be a too easy question for Mathoverflow, but Googling led to similar questions and answers here (though not the one I was looking for).
The question is split into two:
I have a matrix $X \...
- 131
2
votes
0
answers
526
views
Gaussian measure on Banach spaces
Given any separable Banach space $B$ and a centered Gaussian measure $Q$ on it with Cameron-Martin space $H$, does there exist a Hilbert space $G$ and a Gaussian measure $W$ on it such that following ...
- 21
1
vote
1
answer
797
views
Which distributions of $X$ and $Y$ yield a Gaussian $Z=XY$?
Let $Z=XY$ where $X$, $Y$ are random variables with support of non-trivial measure. For what distributions of $X$ and $Y$ can $Z$ be guaranteed to be Gaussian?
- 409
1
vote
2
answers
331
views
Anti-concentration of gaussian variable
Let $X$ be $\mathcal{N}(\mu,\sigma^2)$ gaussian. Its expectation $\mu$ is positive. Can we derive a lower bound on
$$\mathbb{P}(X\geq\epsilon)\geq g(\epsilon,\mu,\sigma) \text{ where } \epsilon\leq\mu$...
- 405
1
vote
1
answer
113
views
How to upper bound the difference between these two Gaussian-like densities?
$
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\cov}{cov}
\DeclareMathOperator*{\supp}{supp}
\DeclareMathOperator*{\dom}{dom}
\newcommand{\...
- 657
1
vote
1
answer
301
views
Obtaining a lower bound on the expectation using the Sudakov-Fernique inequality
In my work I wish to obtain a lower bound for the term below. Here the expectation is taken over $h$, a standard random Gaussian vector of length $n$. The minimum is taken over all $\{i_1,\dots,i_L\} \...
- 281
1
vote
1
answer
281
views
A uniqueness proposition involving Erf, the error function
This is a generalization of a previous MO question, "Reducing system of equations involving Erf, Error Function".
Consider the system of equations:
$$1/2 + {\rm Erf}(x) - \alpha {\rm Erf}(\frac{x+y}{...
- 123
1
vote
1
answer
208
views
Extreme confusion with the exact meaning of Gaussian measure with "translation-invariant" covariance
In physics literature, the covariance of a Gaussian measure $\mu$ on a function space is denoted as $C(x,y)$. Moreover, they say that if the covariance is translation-invariant, then actually $C(x,y)=\...
- 3,477
1
vote
1
answer
195
views
Reference request: Inequalities involving convex sets and Gaussian variables stated in a paper by Talagrand
I'm looking for references for two facts that are stated without proof in the paper:
Talagrand, M., Are all sets of
positive measure essentially convex?, Lindenstrauss, J. (ed.) et al.,
Geometric ...
- 484
1
vote
2
answers
163
views
Integral with linear function, Normal PDF, Normal CDF
I am trying to calculate the following integral:
$$\int_a^\infty x \Phi(cx+d) \phi\left(\frac{x-\mu}{\sigma}\right) dx,$$
where $\Phi$, $\phi$ denote the CDF and PDF of the standard Normal $N(0,1)$.
I ...
- 49
1
vote
1
answer
118
views
Comparison of Rademacher and Gaussian moments under linear transformations
Let $X$ be an $n$ dimensional standard Gaussian and let $U$ be an $n \times n$ orthogonal matrix. Then, the random vector $Z = U^\top X$ is also distributed as a standard Gaussian in $R^n$ and we have ...
1
vote
1
answer
124
views
Multiplying a log-concave function to a Gaussian probability density reduces its variance
Let $X$ be a random Gaussian vector with probability density $p_X(x)$. Let $Y$ be the random variable with density proportional to $p_X(x)e^{-g(x)}$ for some convex function $g$. Does it hold that
$$
...
- 11
1
vote
1
answer
178
views
Tail bound on the RKHS norm of a zero-mean Gaussian process
Let $f \sim \mathcal{GP}(0, K)$ be a zero-mean Gaussian process defined on a compact set $\mathcal{D} \subset \mathbb{R}^d$, where $K \colon \mathcal{D} \times \mathcal{D} \rightarrow \mathbb{R} $ is ...
- 1,127
1
vote
1
answer
107
views
Law of OU process with time-dependent dynamics
Fix a non-negative integer $k$ and let $M^1:\mathbb{R}^n\rightarrow \mathbb{R}^n$ and $M^2,\Sigma:\mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ be $k$-times continuously differentiable functions, ...
- 77
1
vote
1
answer
512
views
Conditions for Gaussianity of SDE
Fix $T>0$, $x \in \mathbb{R}^n$, and let $\mu$ and $\sigma_1,\dots,\sigma_m$ be (globally) Lipschitz-continuous functions from $[0,T]\times \mathbb{R}^n$ to $\mathbb{R}^n$. Thus, for every $0\leq ...
- 5,405
1
vote
1
answer
2k
views
Convolution of two Gaussian mixture model
Suppose I have two independent random variables $X$, $Y$, each modeled by the Gaussian mixture model (GMM). That is,
$$
f(x)=\sum _{k=1}^K \pi _k \mathcal{N}\left(x|\mu _k,\sigma _k\right)
$$
$$
g(y)=\...
- 41
1
vote
1
answer
82
views
Expectation value of multilinear forms over independent Gaussian vectors
Let $A$ be a symmetric multilinear form on $\left(\mathbb{R}^d\right)^{\otimes n}\times \left(\mathbb{R}^d\right)^{\otimes n}$ and consider the random variable:
\begin{align*}
X=A(g_1,\ldots,g_n,g_1,\...
- 111
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
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
1
vote
2
answers
388
views
Average Multivariate Gaussian
Suppose we have a (possibly infinite) collection k-variate gaussian distributions $\{(\mathcal{N}(\mu_{\lambda}, \Sigma_{\lambda}))\}$ ($\lambda$ is just a label), and for each distribution $\mu \in [-...
- 11
1
vote
3
answers
293
views
Lipschitz continuous maps from $\mathbb R^n$ to $\mathbb R^n$ that preserve Gaussian measure?
The only ones I can think of are linear maps like rotations and permutations. Is there a more general characterization?
- 41
1
vote
1
answer
227
views
Does the Gaussian Poincare inequality hold for $p=1$ as well as $p=2$?
Let $X$ be a real-valued standard normal variable. Then, for any differentiable function $f: \mathbb{R} \to \mathbb{R}$ such that $E[f(X)^2] < \infty$ and $E[\bigl( f'(X) \bigr)^2] < \infty$, it ...
- 3,477