All Questions
32
questions
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 $\...
- 21
2
votes
1
answer
92
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 ...
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 ...
- 408
5
votes
2
answers
241
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&...
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 ...
- 6,824
0
votes
0
answers
282
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 ...
- 157
2
votes
1
answer
174
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 ...
- 839
2
votes
0
answers
515
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, ...
- 199
0
votes
1
answer
98
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 "...
- 647
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$,...
- 6,824
8
votes
0
answers
177
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),
...
- 131
5
votes
1
answer
208
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, $B$ ...
- 121k
4
votes
2
answers
367
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 ...
- 931
4
votes
1
answer
186
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 ...
- 401
5
votes
1
answer
354
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(\...
- 455