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3 votes
0 answers
105 views

Maximal-type inequality for a Borel probability measure supported on a subset of $L^2(\mathbb{R}^d)$

Let $\mu$ be a Borel probability measure on $L^2(\mathbb{R}^d)$ for $d\ge 1$ which is moreover supported on the unit sphere $$S=\{\phi\in L^2(\mathbb{R}^d): \| \phi\|_{ L^2(\mathbb{R}^d)}=1\}.$$ Let ...
Dispersion's user avatar
1 vote
0 answers
87 views

$f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...
vaoy's user avatar
  • 309
1 vote
1 answer
241 views

Integration by parts for indicator of a sphere to indicator of a ball

Broadly speaking, I have a radial distribution on $\mathbb R^n$, i.e., the pdf only depends on the $\ell_2$-norm of the argument. I would like to obtain an expression for the pdf in the form $\int_{w=...
Nicolas Resch's user avatar
2 votes
1 answer
173 views

Radon transform of the function $h(x_1,\ldots,x_n) = x_1 g(x_1,\ldots,x_n)$, where $g$ is the density of multivariate Gaussian $N(\mu,\Sigma)$

Given an absolutely integrable function $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined for every $(w,b) \in (\mathbb R^n \setminus \{0\}) \times \mathbb R$ by $$ R[f](w,b) := ...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
116 views

Finding a square integrable dominating function for function class

problem statement For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$ where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p ...
ato_42's user avatar
  • 11
4 votes
2 answers
274 views

Is it true that the quantile function of an $L^1$ random variable is $L^2(]0,1[)$?

Let $(\Omega, \mathcal A, P)$ be a probability space. Let $X:\Omega \rightarrow \mathbb R$ be an $L^1(\Omega, \mathcal A, P)$ random variable. We define the distribution function of $X$ by $$F(x) = ...
W. Volante's user avatar
0 votes
1 answer
221 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) := \...
Yicong Liang's user avatar
3 votes
2 answers
1k views

Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?

Hello, As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem. My question is whether each real-...
Anand's user avatar
  • 1,649