All Questions
Tagged with pr.probability fa.functional-analysis
616 questions
1
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Rate of convergence in narrow convergence
Does anyone help me in the following question?
I have a sequence of probability measures $\mu_n$ and know that $\mu_n$ converges narrowly to a probability measure $\mu$. Is there any way to estimate ...
16
votes
3
answers
918
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What is the minimal $C_k$, such that every $f\colon \{-1,1\}^n\to \mathbb{R}$ of degree at most $k$ satisfies $\|f\|_2\le C_k\|f\|_1$
Every $f\colon\{-1,1\}^n\to \mathbb{R}$ can be repsenented as a multilinean polynomial of the form $$f(x_1,x_2,\ldots ,x_n)=\sum _{S\subseteq [n]} \hat{f}(S)\prod_{i\in S} x_i $$ The degree of the ...
- 311
2
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0
answers
238
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Examples for Markov generators with pure point spectrum
I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
- 235
5
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0
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161
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$L^p$ estimates for Ornstein-Uhlenbeck: what is known beyond hypercontractivity?
Consider an infinite-dimensional Gaussian random vector $X$, and a positive random variable $f(X) \in L^p, p > 1$. Let $f(X) \sim \sum_n f_n(X)$ be its (formal) chaos expansion. Let $(U_\rho, \rho \...
- 4,010
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1
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195
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Existence of bounded $n-$th derivative of the solution of differential equation
This question is the copy from mat.stackexchange.com here. I requestioned here due to the very limited responses there.
Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)...
- 133
4
votes
1
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1k
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For what nonnegative measures $\mu$ does $\mu*e^{-|\cdot|}\in L^{\infty}$?
I am trying to characterize all measures on $\mathbb{R}$ such that
$$
\sup_{x\in\mathbb{R}} \: (\mu*f)(x)<+\infty,
$$
where $f(x)$ is some specific integrable functions, such as $f(x)=e^{-|x|}$, ...
- 1,649
0
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1
answer
222
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Behavior of the integral of products of probability densities
Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition
$$
T(x_1,\ldots,x_n) := \...
- 65
6
votes
3
answers
2k
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Estimating the variance of a discrete normal distribution
Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...
- 167
3
votes
1
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99
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Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$
Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| \,...
- 235
4
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1
answer
1k
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Can't figure out "standard application" of the Garsia-Rodemich-Rumsey Lemma
I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...
- 617
0
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0
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184
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Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
- 5,380
3
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0
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217
views
Small rectangle probability
Let $H$ be a Hilbert space and $\mu$ be a centered Gaussian measure on it. Also, let the eigenpair corresponding to $\mu$ be $(i^{-\alpha} , e_i)$ with $\alpha > 1$. Assume we have a ball of radius ...
- 33
2
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1
answer
139
views
Reynolds operator from the potential theoretic point of view
In the book "Conditional Measures and Applications", it was pointed out that "Reynolds operators have not yet been studied from the potential theoretic point of view ." Have there been any research ...
- 21
40
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5
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5k
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"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
4
votes
1
answer
444
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PDE-Based Triangle Inequality for Optimal Transportation
Suppose $\Omega$ is a suitably regular domain in $\mathbb{R}^n$ and $\rho_0,\rho_1\in\textrm{Prob}(\Omega)$. Benamou and Brenier showed that the $L_2$ transportation distance between $\rho_0$ and $\...
- 705
5
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1
answer
275
views
Is there a good notion of "random bounded linear map" on a separable Hilbert space?
Let $H$ be a separable Hilbert space and let $\{e_i\}$ be an orthonormal basis. My first question is:
Is there a probability measure on $B(H)$ such that for $T$ chosen uniformly randomly the ...
- 29.2k
1
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0
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57
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Looking for CDFs that I can integrate a particular transformation of
I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate
$$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
- 11
2
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0
answers
64
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Almost-Monotone Kernels - Examples and/or Covering Theorems
I am looking for examples (or, if it exists, a theory) of almost-monotone kernels. First, a bit of notation.
Recall that if $(\leq, \Omega)$ is a partially ordered set, then the set of measures $\...
- 81
1
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0
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147
views
Bounding Rayleigh quotient for stochastic matrix
Suppose you have an irreducible, stochastic matrix $A$ with left Perron-Frobenius eigenvector $v$ (corresponding to the eigenvalue $1$), and suppose the next largest eigenvalue for $A$ is $\lambda$. ...
- 121
1
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1
answer
720
views
Question about uniform continuity under Skorokhod Metric
Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
- 1,835
3
votes
0
answers
324
views
Equivalence of Gaussian measures on Hilbert space
Suppose we have 2 nondegenerate Gaussian measures given by N(0,T) and N(0,S) supported on a separable Hilbert space H. T and S are such that eigenbasis of S lies in the cameron martin space of N(0,T)....
- 93
-1
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1
answer
75
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Finiteness of "novel variance" from a kernel on a compact space [closed]
Let $c(i,i')$ be a kernel function on a reasonable index space $I$. Choose a dense sequence of points $\{i_1, i_2, \cdots \} \subseteq I$, and define the one-point kernel functions $k_n := c(\cdot, ...
- 8,512
5
votes
1
answer
472
views
Measures which exhibit the "uncorrelated implies independent" property
Let $X$ be a topological linear space, and let $X^*$ be its dual space. Suppose that $X$ is complete and Hausdorff, and $X^*$ separates points. Let $Y$ be another such space, and let $f : X \to Y$ be ...
- 8,512
1
vote
0
answers
74
views
Closed for the motion of an interacting particle system
I am dealing with interacting particle systems approximately in the sense of http://www.math.vu.nl/~rmeester/onderwijs/Interacting_Particle_Systems/liggett.pdf p. 5 except I am reading a book by the ...
- 277
3
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1
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171
views
Characterization of a set in $\mathbb{R}^d$
Let $X= (X_1,\dots, X_d)$ be a fixed vector of random variables on the space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following set.
\begin{equation}\label{main12}
C= \{x\in \mathbb{R}^d ~|~ ...
- 57
2
votes
1
answer
850
views
Absolute convergence of logarithm of polynomial with positive coefficient ($\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} $)
Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is
$$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right|...
3
votes
1
answer
870
views
Karhunen-Loeve expansion for discrete-time process
Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...
user24451
4
votes
0
answers
309
views
Conditional expectation with respect to random closed sets
Short question
If $X$ is a random closed set, and $Y$ is an integrable random variable, I would like a definition of the "conditional expectation" $$\mathbf{E}[Y \mid X\ni x].$$ Has this been worked ...
- 6,287
1
vote
2
answers
226
views
Smooth but non-analytic kernel functions
Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
- 8,512
4
votes
0
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289
views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
user24451
5
votes
2
answers
332
views
General additive function of probability
Let $H$ be a function of finite sequences of probabilities (non-negative numbers summing up to 1) into real numbers, such that:
$H$ is continuous,
$H$ is symmetric w.r.t. the order of its arguments,
$...
- 1,612
3
votes
0
answers
126
views
Are there pathological examples of log-concave measures that admit no shifts?
Does there exist a random vector $X$ in, say, the space $\mathbb{R}^\infty$ of sequences that has the following properties?
The distribution of $X$ is log-concave, i.e. for every $n$ the joint ...
- 4,010
6
votes
2
answers
622
views
If Gaussian measures on a Hilbert space converge weakly to 0, how do their covariance operators converge?
Suppose we have a sequence of Gaussian measures $N(0, S(n))$ supported on a Hilbert space $H$ and we know that the sequence converges weakly to the delta measure at $0$, what are the necessary and ...
- 93
7
votes
4
answers
1k
views
Is the space of tempered distribution second countable?
Let $\mathcal S '(\mathbb R^d)$ be the space of Schwartz tempered distributions equipped with the weak-* topology. I need to know if this space is second countable, i.e. if this topology has a ...
- 630
9
votes
0
answers
305
views
Convergence in $L^2$ of iterated expectations
Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...
- 1,068
1
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0
answers
168
views
Estimates on gradients of diffusion semigroups
Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form
$$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
- 11
5
votes
1
answer
758
views
Cameron-Martin theorem for non-Gaussian measures
Let $X$ be a locally convex topological linear space, and $\mathbb P$ be a probability measure on $X$. Denote the mean vector $m \in X$ and covariance operator $k : X^* \to X$. Let $\tau_u : X \to X$ ...
- 8,512
4
votes
1
answer
580
views
Density of linear functionals in $L^2$
Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $...
- 8,512
3
votes
0
answers
236
views
Is every covariance operator the covariance of a measure?
Let $X$ be a topological linear space over $\mathbb R$ which is complete and Hausdorff with a dual space that separates points. Let $k : X^* \to X$ be an arbitrary covariance operator. i.e., any ...
- 8,512
9
votes
3
answers
868
views
Rosenthal like inequality for weak $\mathbb L^p$-norms
Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's Martingale limit theory and its applications, Theorem 2.10) that there exists a constant $C_p$ such that if $(X_i)_{i=...
- 3,968
2
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1
answer
197
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a question about the proof of identification of dual space
I have a question about the proof below:
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded ...
- 1,835
2
votes
0
answers
136
views
equivalence of topologies defined on $M_1$(a subspace of bounded measures on $\mathbb{R}$)
Let $\mathcal{C}:=\mathcal{C}(\mathbb{R})$ be the space of continuous functions on $\mathbb{R}$ and $\mathcal{C}_b$ its subspace consisting of bounded elements. Define for $\phi(x):=1+|x|$,
$$
\...
- 1,835
4
votes
1
answer
189
views
Weak ergodicity of nonhomogenous products of 0-1 matrices
Here is a question which probably has a negative answer, but I couldn't find any literature directly on it.
Let $(A_n)$ be a sequence of rectangular 0-1 matrices (that is, the entries are restricted ...
- 4,679
0
votes
1
answer
162
views
Extracting moments from a special Z-transform
Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that
\begin{equation}\frac{X_n-n\mu}{\sqrt{n}\sigma}...
- 41
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
2
votes
1
answer
245
views
Probability measures on $L^p$
Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...
- 8,512
2
votes
0
answers
238
views
Fractional Derivatives [closed]
How far these Theories of "Fractional Derivatives" be rigorized ? I have few books and references on Fractional Differential Equations etc (mainly they stress on Applied Mathematics parts and similar ...
user20510
0
votes
0
answers
537
views
matrix Khintchine inequality
The usual Khintchine inequality says that if $\{\epsilon_n\}_{n = 1}^N$ are i.i.d. random variables with $\mathbb{P}(\epsilon_n = \pm 1) = \frac{1}{2}$ for each $n$ then
\begin{equation*}
\left( \...
9
votes
3
answers
654
views
measure with given push-forwards
Let $X,Y$ be locally compact spaces (in my specific case, they are locally compact groups). Suppose that we are given a measure $\mu$ on $X$ and a finite number of quotient maps $p_1,\ldots,p_n:Y\...
4
votes
1
answer
234
views
Statistical models in terms of families of random variables
A statistical model is a function $P : \Theta \to \Delta(X)$, where $\Theta$ is a parameter space, and $\Delta(X)$ is the set of probability measures on a state space $X$.
Suppose that $\Theta$ and $...
- 8,512