# Tagged Questions

**0**

votes

**1**answer

74 views

### 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) := ...

**2**

votes

**3**answers

135 views

### 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 ...

**2**

votes

**1**answer

73 views

### 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| ...

**2**

votes

**1**answer

133 views

### 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 ...

**0**

votes

**0**answers

79 views

### 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 ...

**3**

votes

**0**answers

210 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 ...

**2**

votes

**1**answer

101 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 ...

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votes

**3**answers

588 views

### “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
$$ ...

**3**

votes

**1**answer

170 views

### 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 ...

**5**

votes

**1**answer

188 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 ...

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vote

**0**answers

41 views

### 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 ...

**2**

votes

**0**answers

54 views

### 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 ...

**1**

vote

**0**answers

44 views

### Bounding Rayleigh quotioent 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 $\lambda$. ...

**2**

votes

**0**answers

162 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 ...

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votes

**1**answer

55 views

### 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, ...

**5**

votes

**1**answer

180 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 ...

**0**

votes

**0**answers

47 views

### Hermite coefficients of a positive density

It is well known that a necessary condition for a function in $L_2$ to be a.e. positive is that its fourier transform is positive-definite (in fact, due to Bochner's theorem, this is also a sufficient ...

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votes

**0**answers

41 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 ...

**3**

votes

**1**answer

157 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 ~|~ ...

**2**

votes

**1**answer

517 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 ...

**2**

votes

**1**answer

96 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 $\{-1,1\}$. ...

**4**

votes

**0**answers

155 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 ...

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vote

**2**answers

152 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?

**4**

votes

**0**answers

110 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}
...

**3**

votes

**1**answer

128 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,
...

**2**

votes

**0**answers

62 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 ...

**5**

votes

**2**answers

151 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 ...

**3**

votes

**3**answers

335 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 ...

**9**

votes

**0**answers

168 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, ...

**2**

votes

**0**answers

131 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 ...

**2**

votes

**1**answer

124 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 $\tau_u : X \to X$ ...

**3**

votes

**1**answer

206 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 ...

**2**

votes

**0**answers

76 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 ...

**7**

votes

**3**answers

301 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 ...

**1**

vote

**1**answer

128 views

### 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 ...

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vote

**0**answers

75 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 $\phi(x):=1+|x|$,
$$
...

**4**

votes

**1**answer

103 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 ...

**0**

votes

**1**answer

103 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
...

**2**

votes

**0**answers

149 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 ...

**2**

votes

**1**answer

194 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 ...

**2**

votes

**0**answers

139 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 ...

**1**

vote

**0**answers

152 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

567 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 ...

**4**

votes

**1**answer

148 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 ...

**7**

votes

**2**answers

303 views

### Measure theory in nuclear spaces

Much of the literature on measure theory in linear spaces focuses on the case of normed linear spaces (e.g., the outstanding book by Vakhania, or its sequel). However, nuclear linear spaces "as far ...

**6**

votes

**1**answer

231 views

### Why aren't operator semigroups studied from a dynamical perspective?

Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying ...

**2**

votes

**2**answers

226 views

### Logarithmic mean

Logarithmic mean of two positive real numbers is well defined in the literature, it has also been extended to more than two arguments in various papers. Is there any notion of logarithmic mean of ...

**4**

votes

**0**answers

111 views

### Pettis Integrability and Laws of Large Numbers

Let $(\Omega, \mathcal F, \mathbb P)$ be a probability space, and let $V$ be a topological vector space with a dual space that separates points. Let $v_n : \Omega \to V$ be a sequence of Pettis ...

**2**

votes

**1**answer

96 views

### Linear combination of i.i.d. $Z_i$ distributed as $Z_1$

A classical property of the Gaussian distribution is that, if $\{Z_i\}_{1 \leq i \leq n}$ are i.i.d. standardised Gaussian distributions (i.e. $Z_i \sim N(0,1)$) and $S = \sum_{i=1}^n a_i Z_i$ where ...

**11**

votes

**2**answers

519 views

### Intuitive explanation of Dvoretzky's theorem

I am wondering if anyone has an enlightening explanation of why Dvoretzky's theorem (which says that a high-dimensional convex body has an almost round central section) is true -- there are a number ...