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3 votes
1 answer
84 views

What (continuous) stochastic processes have path measures that are absolutely continuous w.r.t. Wiener measure?

Suppose I have a stochastic process $\{Z_t\}_{t \in T}$ for which I know the sample paths to be a.s. continuous (we can also assume some usual stuff, such as $T$ a compact metric space, $Z$ having ...
1 vote
0 answers
87 views

Supremum of sums of functions in $L^1$ taking random signs

Consider the Banach space $X=L^1([0,1])$, and let $n\gg1$ and $x_1, ..., x_n$ be any points in the unit sphere of $X$. Is there any reasonable lower bound for $$\sup_{(\epsilon_i)_{i=1}^n \in \{-1,+1\}...
2 votes
0 answers
88 views

Dependence and $L^2$ projections of functions

tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function? Let $w$ be a density on $\...
15 votes
2 answers
734 views

On sums of independent random variables in Banach spaces

Let $(\xi_n)_{n\ge 1}$, $(\eta_n)_{n\ge 1}$ be independent mean-zero random variables with values in a Banach space $X$ such that $$\sum_n\mathbb P(\xi_n\in A)\le\sum_n\mathbb P(\eta_n\in A)$$for any ...
2 votes
1 answer
93 views

Why do distributional isomorphisms preserve joint distribution?

Let $(\Omega,\mathcal{A},\mu)$ and $(\Omega',\mathcal{A}',\mu')$ be probability spaces and $$f_1,\ldots,f_n:\Omega\to\mathbb R,\; f_1',\cdots, f_n':\Omega'\to\mathbb{R}$$ be integrable random ...
1 vote
1 answer
161 views

The space of linear operators between Hilbert spaces has martingale type 2

I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
12 votes
0 answers
196 views

UMD constant of finite dimensional spaces

For a Banach space $B$, its one-sided Unconditional Martingale Difference (UMD) constant $C^-_p$ (for $p \in (1,\infty)$) is the smallest value such that for all $B$-valued martingale difference ...
5 votes
2 answers
245 views

Differentiability of the map $x\mapsto \delta_x$ in the Arens-Eells/Lipschitz-free space

$\DeclareMathOperator\AE{AE}\DeclareMathOperator\Lip{Lip}$Let $\AE(X)$ denote the Arens-Eells space on a Banach space $X$. Consider the map: $$ \begin{aligned} \delta: X & \rightarrow \AE(X) \\ x&...
4 votes
2 answers
2k views

Convergence of Gaussian measures

Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
30 votes
1 answer
1k views

Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
15 votes
3 answers
2k views

Disintegrations are measurable measures - when are they continuous?

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
0 votes
0 answers
302 views

Convergence of characteristic functions vs. weak convergence of measures and the Ito-Nisio theorem

In section 2.6 of Linde's Probability in Banach Spaces: Stable and Infinitely Divisible Distributions the author is pointing out that in infinite-dimensional Banach spaces the convergence of ...
0 votes
0 answers
68 views

Convex optimization under asymmetric loss in infinite dimensional space

The following problem is common in financial economics $$ \min_{m \in L^2} \mathbb{E}[ \phi(y(\theta)-m)] \quad \text{s.t. } \mathbb{E}[ mx ]= q $$ That is, given a random variable $y(\theta)$ ($\...
2 votes
1 answer
183 views

Does set of finitely additive probability measures embed linearly into a strictly convex dual Banach space?

I am trying to better understand a condition that appears in Theorem 1 of this paper. Let $K$ be a convex and compact subset of a locally convex tvs. The condition is: $K$ embeds linearly into a ...
0 votes
1 answer
194 views

Gaussian integral $\int_X \|x\|_X^2 \mu(dx)$ in Banach space

For a centered Gaussian measure $\mu$ on a Hilbert space $X$, it is known that $$\int_X \|x\|^2 \mu(dx) = tr(Q)$$ where $Q$ is the covariance operator. Is there a similar version for Gaussian measures ...
2 votes
0 answers
520 views

Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$. If $X$ is reflexive, ...
0 votes
1 answer
102 views

Law of a step function and its generalization to two dimensions on an appropriate spaces

Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively: A step function: $u_1(x)=\begin{cases} u_{L}, x<c_1, \\[2ex] u_{R}, x>c_1, \end{cases}$ A "...
0 votes
0 answers
58 views

Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$

Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$,...
8 votes
0 answers
182 views

Distribution domination for sums of independent random variables in Banach spaces

Let $X$ be a Banach space and let $(\xi_n)$ and $(\eta_n)$ be independent mean-zero random variables with values in $X$ satisfying $$ \sum_n \mathbb P(\xi_n \in A) \leq \sum_n \mathbb P(\eta_n \in A), ...
5 votes
1 answer
652 views

Proof of Pinelis (1992) - Banach space inequalities

I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3: Let $(f_n)$ be a martingale in a separable ...
5 votes
1 answer
358 views

Pisier's property $(\alpha)$

Let $\Omega$ be a probability space. Suppose $(\epsilon_i)_{1\leq i\leq n}$ is a sequence of i.i.d. Bernoulli random variables on $\Omega,$ i.e. $(\epsilon_i)_{1\leq i\leq n}$ are independent and $P(\...
2 votes
1 answer
117 views

Size of the orbit of a dense set

This question is a follow-up to: this post. Let $X$ be a separable Banach space, $\phi\in C(X;X)$ be an injective continuous non-affine map, and $A$ be a dense $G_{\delta}$ subset of $X$. How big ...
5 votes
1 answer
224 views

Conditional expectation of random vectors

$\newcommand{\E}{\mathsf{E}}$ $\newcommand{\P}{\mathsf{P}}$ The following additional question was asked in a comment by user Oleg: Suppose that $(\Omega,\mathcal F,\P)$ is a probability space, $B$ ...
4 votes
2 answers
378 views

Basic properties of expectation in non-separable Banach spaces

$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $B$ be a (maybe nonseparable) Banach space equipped with the Borel $\sigma$-algebra $\mathscr{B}(B)$. Let $R:B\to \mathbb{R}$ be a bounded linear ...
6 votes
1 answer
261 views

Area of $n$-sphere contained outside $\ell_1$ ball

For a given $r>1$, what is the surface area of $\mathbb S^{n-1}$ (the sphere of radius 1 in $\mathbb R^n$) which is contained outside of the $\ell_1$ ball of radius $r$? Or equivalently, if $X\sim ...
0 votes
1 answer
249 views

Central limit theorem in Banach space in scheme of series

I wonder whether Theorem 2 from the paper J. Zinn, Annals of Probability, 1977, vol. 5, 283-286 can be extended to the CLT for a scheme of series. (The paper is available in the web.) Let $G$ be ...
4 votes
1 answer
193 views

A bound on the square distance of a random walk on undirected graph

Fact: Let $G$ be an $n$-vertex undirected graph and $(X_s)_{s\in \mathbb N}$ a stationary random walk on $G$. Then for every $s\in \mathbb{N}$, $ \mathbb{E}[d_G(X_s,X_0)^2] \le C s \log n $, for some ...
7 votes
4 answers
946 views

On operator ranges in Hilbert & Banach spaces

Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent: (1) ran($A$) $\subset$ ...
17 votes
5 answers
4k views

Brownian motion and spheres

Consider a Brownian motion on $[0;1]$. A (finite) discrete approximation of this Brownian motion consists of $N$ iid Gaussian random variables $\Delta W_i$ of variance $\frac{1}{N}$: $$ W\left(\frac{k}...
3 votes
1 answer
232 views

Is there a canonical uniform probability measure on compact subsets of Banach spaces?

One can construct a finite measure on a compact metric space $(X,d)$ by the following procedure: Fix a non-negative sequence $\{\epsilon_n\}$, $\epsilon_n \to 0$. Let $Y_{\epsilon_n}$ be the minimal ...
4 votes
2 answers
543 views

Gaussian measure on Banach space

Assume we have a Gaussian measure $\mu$ supported on a Banach space $X$. Can we always find a Hilbert space $H$ embedded in $X$ sch that $\mu$ is also supported on $H$?
2 votes
1 answer
5k views

Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$

Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
3 votes
1 answer
1k views

If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?

Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$. Let $H^*$ denote the space ...
6 votes
2 answers
3k views

Dense inclusions of Banach spaces and their duals

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ is a dense, ...
5 votes
2 answers
898 views

Density of Gaussian measures on Banach spaces

I am trying to get my head around this question and was reading (1) which states the same a little bit more general: Let $X$ be a separable Banach space and $X^*$ the dual space. The mean value $...
10 votes
1 answer
253 views

Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be the corresponding Cameron-Martin Hilbert space (also known as ...
6 votes
3 answers
1k views

Is the set of the absolutely continuous functions a Borel set of the space of the continuous functions?

Does anyone knows whether the set of the absolutely continous functions $F :[0,1]\to \mathbb{R}^d$ of the form $$F(t)= a + \int_0^tf(s) ds$$ where $f$ is an integrable function is a Borel set of the ...
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=...
4 votes
1 answer
280 views

Approximation of an integral over the unit ball of L_1

For every $\varepsilon>0$ find a piecewise continuous function $q:[0,1]\rightarrow \mathbb{R}$ such that $\int_0^1 q(x)dx=1$ and $$\int_{0}^1 \int_{0}^{s} \left|\frac{q(s)q(t/s)}{s}- \frac{q(t)q((s-...
1 vote
1 answer
353 views

Agreement of two topologies on a linear space

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide. Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
6 votes
0 answers
262 views

Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
4 votes
1 answer
327 views

Gaussian Valued Random Variables in Geometry of Banach Spaces

Why are Gaussian valued random variables so important in the Geometry of Banach spaces? I am reading the monograph by Pisier - "Probabilistic Methods in the Geometry of Banach Spaces" and in the very ...
4 votes
1 answer
174 views

Set of unitaries with "spread-like" properties

I'm interested in finding two sets of $N$ unitary $N \times N$ matrices $U_{1}, \ldots, U_{N}$, $V_{1}, \ldots, V_{N}$ such that: $ \sup\limits_{X, Y}\sum\limits_{j,k = 1}^{N} |\mathrm{Tr}(YU_{j}XV_{...
5 votes
1 answer
515 views

Derandomizing random matrices

My question is rather general - what is known about derandomization of results in random matrix theory, high-dimensional geometry, Banach spaces etc. using probabilistic constructions (like estimates ...
7 votes
1 answer
423 views

Best constant in comparison between Rademacher and gaussian averages?

Let $(g_k)$ be a sequence of independent standard gaussians variables on a fixed probability space $\Omega$. Let $(\epsilon_k)$ be a sequence of independent rademacher variables. What is the best ...
3 votes
1 answer
1k views

Cyl(E) = Borel(E) for E non-reflexive Grothendieck Banach space

This is sort of a follow-up to Borel(X) = \sigma(X') for X non-separable PROBLEM: Given a Banach space $E$ over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$ that has the Grothendieck property. ...
7 votes
1 answer
1k views

If $H$ is a separable Hilbert space, is $L^2(H)$ separable?

Let $H$ be a separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$. Is the Hilbert space $L^2(H,\gamma)$ separable?
6 votes
1 answer
453 views

The typical size of a random element in a Banach space

Let $X$ be a separable Banach space, and let $\mathbb P$ be a Radon probability measure on $X$ with zero mean and covariance operator $K : X^* \to X$. Let $x$ be an $X$-valued random variable with ...