All Questions
Tagged with pr.probability measure-theory
823 questions
8
votes
1
answer
726
views
continuity of the Boltzmann entropy in the Wasserstein metric
For Lebesgue-absolutely continuous probability measures $\rho\ll \mathcal{L}^d$ in the whole space $\mathbb{R}^d$ with finite second moments (i-e $\rho\in \mathcal{P}^2_{ac}(\mathbb{R}^d)$), let
$$
\...
- 23.2k
0
votes
1
answer
374
views
Approximation of general measurable maps by simple functions [closed]
Let $f : (\Omega, \mathcal F) \to (\mathbb R, \mathcal B(\mathbb R)$ be a measurable map, then it is well-known that $f$ could be approximated by a sequence $(f_n)$ of simple measurable functions, ...
- 206
3
votes
1
answer
802
views
Sub-$\sigma$-algebras and conditional expectation
Is it true that any sub-$\sigma$-algebra of a Rokhlin-Lebesgue space is induced (up to completion) by a measurable map into another Rokhlin-Lebesgue space?
In other words, is it true that conditional ...
- 17k
2
votes
1
answer
328
views
Criterion for weak convergence of probability measures on S' or D'
Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...
CommunityBot
- 1
1
vote
0
answers
166
views
Finitely additive measure over integers [duplicate]
We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).
...
- 19
1
vote
1
answer
720
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Question about uniform continuity under Skorokhod Metric
Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see:
http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
- 1,872
4
votes
1
answer
565
views
convergence of integral for each bounded function in probability
Let $\mu, \mu_1, \mu_2, \dots$ be random measures on
a Polish space (separable completely metrizable topological space) $(S, {\mathcal S})$.
Suppose I know that
$$\int f d \mu_n \to \int f d\mu$$
...
- 544
4
votes
2
answers
714
views
Polish by compact is Polish?
Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?
I have a specific space in mind, so if the ...
- 14k
-1
votes
1
answer
148
views
Continuity of function mapping $\mathcal{P}(\mathcal{P}(X))$ to $\mathcal{P}(X)$ [closed]
Given a topological space $Y$, let $\mathcal{P}(Y)$ be the set of all probability measures on $Y$, endowed with the weak* topology.
Let $X$ be a topological space (for convenience, it might be Polish ...
- 2,381
21
votes
2
answers
3k
views
A measure on the space of probability measures
This question was originaly posted in the stackexchange https://math.stackexchange.com/questions/1226701/a-measure-on-the-space-of-probability-measures but since it only got a comment I decided to ...
- 7,197
7
votes
2
answers
816
views
Rademacher average based Hoeffding Inequality
I am following these lecture notes:
Given the i.i.d. $\mathcal{Z}$-valued random variables $Z_1,\dotsc,Z_m$ and $\mathcal{G}$ is a set of bounded functions $g\colon \mathcal{Z}\to[a,b]$.
Corollary 2....
- 619
9
votes
1
answer
483
views
Defining functions pointwise vs. almost everywhere (w.r.t. uncountably many mutually singular measures)
My question is motivated by a general measure-theoretic problem that one frequently encounters in probability: the need to work with uncountably many mutually singular measures at once, and with ...
- 984
1
vote
0
answers
260
views
Generating the sigma algebras on the set of probability measures
I was wondering if somebody could help me see/provide a reference to the following fact: Let $X$ be a metrizable set, $\mathcal{F}$ the corresponding Borel sigma-algebra on $X$, and $\triangle\left(X,\...
- 11
1
vote
1
answer
258
views
Conversion between condtional expection conditioned on $\sigma$-algebra and on r.v
Let $(\Omega, \mathcal F, P)$ be a probability space, and let $\mathcal G \subseteq \mathcal F$ be a sub-$\sigma$-algebra of $\mathcal F$ and $X : \Omega \to \mathbb R$ a random variable. Then the ...
- 41.1k
0
votes
0
answers
145
views
Discrete measures and discrete kernels
This is a cross-post from math.stack. Let $d\in\mathbb N$ and $\mu$ be the probability measure on $\mathbb R^d$ defined by $\mu=\sum_{k=1}^\infty 2^{-k}\delta_{x_k}$ for some sequence $(x_k)_{k\in\...
CommunityBot
- 1
2
votes
0
answers
160
views
Is it possible to improve the order of convergence of averages of random variables if they are not identically distributed?
Let $X_n$ be a sequence of independent random variables (but not necessarily identically distributed)
taking values in $[-1,1]$ that have the following property:
1) The average $A_n := \frac{(X_1+ \...
- 3,245
7
votes
1
answer
444
views
Is an infinite-dimensional "Lebesgue measure" uniquely determined by a set of positive finite measure?
Let $\mu$ be a probability measure on a subset $C \subset \mathbb{R}^\infty$ of the space of sequences, and assume, for simplicity, that $C$ is closed and convex.
We say that $\mu$ admits shifts if ...
- 4,010
2
votes
2
answers
557
views
trivial map on $\sigma-$algebra $\mod{}0$ is trivial
Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
- 41.1k
3
votes
0
answers
119
views
Reasoning about dependent and independent quantities by "degrees of freedom"
In his classic textbook Foundations of the Theory of Probability Kolmogorov defines Independence a little bit differenent then it is usually done today. He denotes a probability space by $(E, \mathcal ...
- 798
1
vote
1
answer
119
views
Extend product sigma-algebra to cross-constant sets
We have two probability spaces $(\Omega_1,\mathcal{F_1},P_1)$ and $(\Omega_2,\mathcal{F_2},P_2)$. Is it possible to construct probability space $(\Omega=\Omega_1\times\Omega_2,\mathcal{F},P)$ such ...
- 13
0
votes
0
answers
133
views
What is the sigma field of the derivative of a process?
When $t\to X_t$ is an absolutely continuous process ($X_t= X_0+ \int_0^t Y_s dt$ for some measurable process $Y_t$) we have for all $t$ $$\sigma(Y_t) \subset \cap_{\epsilon >0}\sigma(X_{s}, s\in [t,...
- 19
1
vote
0
answers
417
views
Defining density of a random function using Radon-Nikodym Theorem
Let $(\Omega,\mathbb{F},P)$ be a probability space and $E$ be an infinite dimensional Banach space and $\mathbb{B}$ be the $\sigma$-algebra of Borel subset of $E$.
Let $X$ be random function defined ...
- 213
1
vote
0
answers
90
views
Measurability of solution of diffusion equation in sub sigma algebra
I want to solve the following problem:
Get $\omega \in \Omega \subset \mathbb{R}$, $x \in D \subset \mathbb{R}^2$ and $0<a_i\leq a(.,.)\leq a_x<\infty$.
Let $a( x;. )$ and $f(x;.)$ be $\...
CommunityBot
- 1
2
votes
0
answers
98
views
Finding a general form of the density function when we have a four dimensional random variable
Consider a subject having time of the specific event $T_i$, which is a single sample from a
distribution $F_i$ with density $f_i$ and support
$[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
- 35
2
votes
1
answer
474
views
How does Azuma's Inequality result from Pinelis Inequality?
According to [1]
Let $(\mathcal{X},||\cdot||)$ be a separable Banach space and let
$S(\mathcal{X})$ denote the class of all sequences
$f=(f_j)=(f_0,f_1,...)$ of Bochner-integrable random ...
- 3,968
3
votes
0
answers
112
views
McDiarmid's inequality on normed spaces
McDiarmid's inequality says if a function $f: \mathcal{X}^n\to\mathbb{R}$ has the property that
$$
\sup_{x_1,\dotsc,x_n,x'_i\in\mathcal{X}} |f(x_1,\dotsc,x_n)-f(x_1,\dotsc,x_{i-1},x'_i,x_{i+1},\dotsc,...
- 619
5
votes
0
answers
137
views
Large Deviations: Exponential decay in normed spaces
Let $(X_1,X_2,\cdots)$ be a sequence of independent and identically distributed random variables taking values in some general normed space $(V,||\cdot||)$. Denote $\mu=E[X_1]$ and $S_n=\frac{1}{n}[...
- 619
9
votes
0
answers
223
views
Cramer's theorem in Hilbert spaces
I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:
Let $X_1,X_2,\cdots$, be ...
- 619
0
votes
2
answers
435
views
conditional expectation under convex combinaison of probability measures(II)
Let $(\Omega,\mathcal{F})$ denote some measurable space. Let $P_1$ and $P_2$ denote respectively two probability measures. Now let $\mathcal{G}$ be some sub sigma-algebra of $\mathcal{F}$. Given a ...
- 26
3
votes
0
answers
465
views
How to define Product of Conditional Measures?
I have been wondering how to define the product of conditional measures as defined by Renyi-Popper. I spell the details below.
If $(X,\Sigma)$ is a measurable space, then the function $\mu : \Sigma\...
- 191
4
votes
0
answers
1k
views
Total variation and Hellinger distance inequality between truncated Gaussians
We know that the total variation distance, $d_{TV}(P,Q) = \frac{1}{2}\left|\left|P-Q\right|\right|_1$, between any two distributions $P$ and $Q$ is lower bounded by their squared Hellinger distance, $...
- 41
0
votes
0
answers
454
views
Reference: Bochner Integral`
What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
- 5,405
7
votes
1
answer
453
views
Do the terms of an iid sequence whose law has infinite expected value necessarily exceed the partial sums of the sequence infinitely often?
Let $\mu$ be a probability measure on $(0,\infty)$, and let $(\mathbf X_n)_1^\infty$ be a sequence of independent $\mu$-distributed random variables. Fix $\kappa > 0$, and consider
A) $\int x \; d\...
- 1,136
4
votes
1
answer
877
views
Kolmogorov doesn't show existence of Dirichlet process for arbitrary measurable spaces. Why?
I'm trying to understand the problem arising when using Kolmogorov's extension theorem to prove the existence of the Dirichlet process on an arbitrary measurable space $\left(\mathcal{X},\mathcal{A}\...
- 76
0
votes
0
answers
161
views
question about the tightness of probability measures for a general topological space
Let $(E,\mathcal{X})$ be a topological space and denote by $\mathcal{F}$ its collection of Borel subsets referred to $\mathcal{X}$. Now let $\mathcal{P}$ be the set of all probabilities on $(E,\...
- 1,835
3
votes
0
answers
910
views
Girsanov theorem with Geometric Brownian Motion
I am not a student in mathematics, but I am trying to use the following Theorem 8.6.6 (Girsanov theorem II) of Oksendal's SDE with geometric Brownian motion $S_{t}$ instead of the standard Brownian ...
- 24.8k
3
votes
1
answer
688
views
Is it possible to construct any random variable on the Euclidean Probability space?
Let $(\Omega,\mathscr A,P)$ be an arbitrary probability space,
and let $X:\Omega\to\mathbb R$ be a random variable.
Then,
one can generate a random variable $Y$ from the probability space $\big([0,1],\...
- 14.4k
0
votes
0
answers
232
views
Morphisms associated to measured spaces [duplicate]
In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
CommunityBot
- 1
4
votes
1
answer
2k
views
Uncountable family of random variables
Let $\{ \xi _a \}_{a \in [0;1]}$ be a family of independent uniformly distributed on $[0;1]$ random variables on some probability space
$(\Omega, \mathscr{F},P)$, indexed by a continuous parameter. ...
CommunityBot
- 1
8
votes
3
answers
834
views
Do regular conditional distributions almost surely assign trivial measure to all members of the conditioning $\sigma$-algebra?
Let $(X,\Sigma)$ be a standard measurable space, let $\rho$ be a probability measure on $(X,\Sigma)$, and let $\mathcal{E}$ be a sub-$\sigma$-algebra of $\Sigma$. We will say that a stochastic kernel $...
CommunityBot
- 1
3
votes
2
answers
375
views
Infima of conditional densities after disintegration
Consider the measurable partition of the open unit square $(0,1)\times(0,1)$ into horizontal intervals $L_y=(0,1)\times\{y\}$. Let $\mu$ be a Borel probability measure with the disintegration
$$
\mu(...
- 708
0
votes
0
answers
184
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 ...
- 5,380
3
votes
0
answers
206
views
Embedding probability spaces in the completion of $[0,1]^K$
Question: Can every probability space $(X,\scr F,\mu)$ be $\sigma$-embedded in the completion of the space $[0,1]^K$ (equipped with a product of Lebesgue measure) for some set $K$?
Here, $f:\scr F\to ...
- 2,555
5
votes
0
answers
361
views
Existence of an universally measurable pullback
Let $X,Y$ and $Z$ be standard Borel spaces:
topological spaces homeomorphic to Borel subsets of complete separable metric spaces.
Let $K\subseteq X\times Y$ be analytic. Assume that $K_x$ is not ...
- 1,655
2
votes
0
answers
60
views
Stronger version of linearity for functions of measures
Let $X$ be a standard Borel space, and $P(X)$ be space of Borel probability measures on $X$. It is also a standard Borel space if endowed with the topology of weak convergence, so we can integrate ...
CommunityBot
- 1
4
votes
1
answer
1k
views
Quotients of standard Borel spaces
Let $X$ and $Y$ be standard Borel spaces: topological spaces homeomorphic to Borel subsets of complete metric spaces. Given a surjective Borel map $f:X\to Y$, we get an equivalence relation $\sim_f\...
- 1,655
3
votes
1
answer
611
views
Inverse of a Borel surjection
Let $X$ and $Y$ be standard Borel spaces, and let $f:X\to Y$ be a surjective Borel map. Does there exist a Borel inverse of $f$, that is a Borel map $g:Y\to X$ such that $f\circ g = \mathrm{id}_Y$.
...
- 1,655
4
votes
1
answer
152
views
Analytic enlargement of an analytic set
Let $X,Y$ be Borel spaces and $A\subseteq X\times Y$ be an analytic set. Let $\pi:X\times Y \to X$ denote the projection map onto $X$. Does there always exist a set $B$ such that $\pi(B) = X\setminus \...
- 236k
9
votes
3
answers
654
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 $p_1,\ldots,p_n:Y\...
CommunityBot
- 1
1
vote
0
answers
112
views
Make this marginalization statement rigorous
Intuition tells me that
$$ p(x\,|\,y) = \int p(x,\theta\,|\,y) \; d\theta$$
by the "law of marginalization", pretty much for any object $\theta$.
I would like to make this statement rigorous,...
CommunityBot
- 1