2
votes
0answers
41 views

Representation of Stochastic Integrals as Lebesgue/Bochner Integrals

Just as the Riemann–Stieltjes integral can be equivalently defined as a Lebesgue integral with the corresponding Lebesgue–Stieltjes measure, I am looking for the corresponding results for the ...
2
votes
0answers
55 views

Understanding a theorem from “Probability theory of Banach Spaces ” book.

I don't understand the proof after "The hypothesis of the theorem indicate...... , can someone kindly explain it for me . Thanks :)
1
vote
0answers
50 views

Banach space :space of all adapted processes continuous equipped wih specific norm is complete

Let $\mathbb{B}$ be space of all adapted processes continuous equipped with the norm $\lVert Y\rVert_{\mathbb{B}}^2=E\left[\sup_{t\in [0,T]} |Y_{t}|^{2}\right] < \infty $, ...
1
vote
0answers
41 views

canonical form of dyadic martingales

Let $(X_k)_{1\leq k \leq n}$ be a Walsh-Paley $L^p$-martingale (a dyadic martingale) with values in a Banach space $X$. Why does there exist a dyadic martingale $(Y_k)_{1\leq k \leq n}$ with the ...
4
votes
1answer
115 views

Random variables with the same distributions

Let $K$ and $L$ be locally compact Hausdorff spaces. Also, let $P$ be a Radon probability measure on $K$ so that $(K,P)$ is a probability space. I want to know, whether two random variables ...
0
votes
1answer
69 views

reference for conditional expectation

Suppose $1\leq p<\infty$. Let $E$ be a Banach space. Consider a filtration $F_n$ on some probability space $\Omega$. Let $X\in L^p(\Omega,E)$ where $L^p(\Omega,E)$ denote the Bochner space. In ...
5
votes
1answer
825 views

Is $p$-norm decreasing in $p$?

I could show that $\|\cdot\|_p$ is decreasing in $p$ for $p\in (0,\infty)$ in $\mathbb{R}^n$. Following are the details. Let $0<p<q$. We need to show that $\|x\|_p\ge \|x\|_q$, where $x\in ...
5
votes
2answers
765 views

Dual space of the space of finite measures

Since I am reading some stuff about weak convergence of probability measures, I started to wonder what is the dual space of the space consisting of all the finite (signed) measures (which is well ...
5
votes
1answer
158 views

Comparison between Rademacher average and random average

Let $X$ be a Banach space. We let $j=e^{2i\pi/3}$. Let $(\epsilon_i)$ be a sequence of independent Rademacher variables on a fixed probability space $\Omega$. Let $(\varphi_i)$ be a sequence of ...