1
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1answer
20 views

Verify conditions of the Extension Theorem for a probability measure $\mathbb P$ of the Uniform[0,1] distribution

I am (self-)studying the book by Rosenthal called A first look at rigorous probability theory. My question is on verifying the conditions on a probability measure $\mathbb P$ of the Uniform[0,1] ...
1
vote
1answer
25 views

A question on measures/densities

Let $Q,P,\nu$ be three measures on the same space such that $P\ll Q,P\ll\nu,Q\ll\nu$. Define $p=\frac{dP}{d\nu},q=\frac{dQ}{d\nu}$. Then $$\frac{1}{2}\int|p-q|d\nu=1-\int\min\{p,q\}d\nu$$ This should ...
1
vote
1answer
23 views

In the proof of that Complete convergence is equivalent to convergence a.s. under independence

Complete convergence is equivalent to convergence a.s. under independence In here, it uses the second Borel-cantelli lemma for the converse. But, it is necessary to verify that $(X_n - X)$'s are ...
0
votes
1answer
29 views

Average of IID Cauchy RVs

Suppose that $X_i$'s are iid Cauchy RV's with pdf $f_u (x) = \frac{1}{\pi} \frac{u}{u^2+x^2}$. I am aware that the RV $Y:=\frac{1}{N}\sum_{k=1}^N X_k$ has the same density as the $X_i$'s. I am trying ...
1
vote
0answers
20 views

Sequences of refining partitions of a measurable space

Let $(\Omega,\mathcal F)$ be a measurable space. For $k\in\mathbb N$ let $(\mathcal F_k)_{k\in\mathbb N}$ be a filtration of $\mathcal F$ such that each $\mathcal F_k$ is generated by a finite ...
1
vote
0answers
68 views

Proof that a function is measurable

Suppose $f$ is a joint probability density function of random variables $X$ and $Y$. $Y$ is integrable. I need to prove that the function $g(x) = \int_{\Bbb R} f(x,y)ydy$ is measurable function. I ...
1
vote
1answer
23 views

A basic question on measure

Suppose I have a measure in $B(\Bbb R)$ such that for each real number there is a neighbourhood where the measure is zero. Is that measure be necessarily zero measure ? How to prove it ? I can't take ...
1
vote
1answer
28 views

Expected distance within a distribution is smaller?

consider we have two general distributions $f_1$ and $f_2$, assume they have different support $S_1$ and $S_2$. Is the expected distance btween two points draw from the same distribution smaller than ...
1
vote
0answers
11 views

What is the relationship of the EMD (Earth movers Distance) and total variation (and other probability measures)?

I was trying to understand different methods for comparing probability distribution and saw the following paper/reference: http://arxiv.org/abs/math/0209021 In it it defines and compares and ...
1
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0answers
39 views

Delta Function as a Conditional Distribution

This is problem 20 from chapter 21 of A Modern Approach to Probability Theory by Fristedt and Gray: Suppose that $X$ is a random variable measurable with respect to a $\sigma$-field $\mathcal{G}$. ...
2
votes
1answer
23 views

Usefulness of criterion for weak convergence

I am currently reading the book Convergence of Probability Measures by Patrick Billingsley, and I came across the following theorem: Theorem. Let $(S,\rho)$ be a metric space, and $B(S)$ be the Borel ...
0
votes
1answer
21 views

interacting probabilitys

Find two absolutely continuous probability measures $\mu(x)dx$ and$\nu(x)dx$ with finite second moments. Such that the function $f(t)$ we have that $\dfrac{d^{2}}{dt^{2}}f(t)<0$ where ...
2
votes
1answer
35 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
0
votes
1answer
47 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
0
votes
0answers
28 views

Interchanging limits in stochastic order notation definition

A sequence of real-valued random variables $(X_n)$ is said to be $O_P(b_n)$ where $(b_n)$ is a sequence of positive numbers if $$ \lim_{T\to\infty} \limsup_{n\to\infty}P\{|X_n|>T b_n\}=0$$ Is it ...
3
votes
1answer
43 views

Difference between density and distribution [in formal mathematical terms]

A similar question has been already asked but its not in mathematical framework and therefore seems to be different. According to definitions from the book that I am reading, a random variable and a ...
1
vote
1answer
43 views

Weak*-convergence of probability measures

Let $(\Omega,\mathcal F)$ be a measurable space and $X:\Omega\rightarrow\mathbb R^d$ a $\mathcal F$-measurable, bounded map. Let $(\mathbb Q_n)_{n\in\mathbb N}$ be a sequence of probability measures ...
1
vote
1answer
29 views

Borel-Cantelli Lemma “Corollary” in Royden and Fitzpatrick

The Borel-Cantelli Lemma in Royden and Fitzpatrick's "Real Analysis" seems to be a sort of "corollary" of the non-probabilistic ones I see online. It says: "Let $(E_k)_{k=1}^{\infty}$ be a countable ...
1
vote
0answers
24 views

How to evaluate this expectation value?

How to evaluate this expected value: $$\mathbb{E} \left( \smash{\displaystyle\max_{I\in\mathbb{M}}\sum_{i\in I} \xi_i^2 } \right)\le ?,$$ where $\xi_i\overset{ind}{\sim} N(0,1), ...
1
vote
0answers
39 views

Expectation of $p$-norm under a Gaussian on the Hilbert space $L^2(S^1)$

Let $\mu$ be a centered Gaussian measure with (nondegenerate) covariance $Q$ on the Hilbert space $L^2(S^1;\mathbb R)$ where $S^1$ is the circle. We can take for example the covariance ...
0
votes
3answers
18 views

A question on limit of weak-* convergence of probability measures

Let $(X,\mu)$ be a measure space. Assume $X$ is compact. It is well-known that the space $\mathcal{P}(X)$ of probability measures on $X$ is compact in weak-* topology. Let's consider a sequence of ...
0
votes
1answer
27 views

Scale invariant measures must have power law densities

If $\mu$ is a scale-invariant measure(say on $\mathbb{R}^{+}$), i.e. for any set $A$, $\mu\left(\frac{A}{c}\right)=g\left(c\right)\mu\left(A\right)$ where $c>0$, then is it necessary that $g$ must ...
1
vote
1answer
33 views

$f \le 1 \Rightarrow f =1 $ a.s.

I know the title doesn't say much, but I hope you'll help me nonetheless. Here's my problem. Let $P, Q$ be two probabilistic measures, $P$ is atomless and the measures have the same independent ...
0
votes
2answers
38 views

Motivation behind the proeprties of sigma algebra

What is the motivation behind the class $B$ of all measurable sets to satisfy the following properties : 1) $A_1, A_2 \in B$ implies $A_1 \cup A_2 \in B$ 2) $\{A_n\} \in B $ and $\{A_n\}$ is ...
2
votes
1answer
68 views

Can a countably generated $\sigma$-algebra be “approximated” by a $\sigma$-algebra generated by a countable partition?

My question is a bit vague, hopefully someone can still clarify. Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space and assume that $\mathcal F$ is countably generated. My question is, does ...
2
votes
0answers
27 views

Measurability of a particular map

Let $X$ and $Y$ be standard Borel spaces and let $\mathcal P(X)$ denote the space of probability measures over $X$ endowed with the topology of weak convergence. Consider a map $f:X\times \mathcal ...
3
votes
1answer
33 views

A Problem in Convergence of Sequences of Random Variables

Let $\left( X_n \right)$ be a sequence of independent random variables on the measure space $(\Omega, \xi,\mathbb{P})$ with $$ \mathbb{P} \left( X_n=1 \right)= p_n \text{ and } \ \mathbb{P} \left( ...
1
vote
1answer
89 views

Prove an assertion on infinite quadratic variation

If $f$ is a continuous function defined on $[0,1]$ which has the following property: $\forall M >0$, $\forall p \in Q\cap[0,1)$, $\exists q \in Q\cap[0,1]$ and $q > p$ such that $|f(p) - f(q)| ...
0
votes
0answers
18 views

Two definitions of Convergence in distribution (random vector version)

I have seen several posts around about the two definitions for "convergence in distribution", but I was not sure whether the equivalence holds in the random vector version. The two definitions are: ...
1
vote
1answer
31 views

Sample mean converges almost surely

Given an arbitrary sequence of random variables $X_n$ such that $X_n$ converges to $0$ almost surely, how do I show that $S_n/n$ converges to $0$ almost surely? I'm already suspicious of the result: ...
2
votes
1answer
21 views

Measurability of level sets of measures

Let $X$ be a standard Borel space, and $\mathcal P(X)$ be the set of Borel probability measures on $X$ with a topology of weak convergence. It is known that $\{p:p(B) = 1\}$ is a Borel subset of ...
1
vote
1answer
34 views

Can we infer a set has positive measure from the conditional probability formula?

First question: Suppose that I know $P(x\in A|\;y)>0$ where $y$ is a realization of some random variable. Then, in general can I infer using the conditional probability formula $P(x\in A|\; ...
3
votes
1answer
122 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
1
vote
0answers
19 views

The “size” of a continuous uniform selection of points in the unit square

Let $\{X_r\}_{r\in[0,1]}$ be i.i.d. random variables, each distributed uniformly on $[0,1]$. Let $S\subseteq[0,1]^2$ be the random set defined as follows: $$S=\{(r,X_r)\mid r\in[0,1]\}$$ How would ...
1
vote
0answers
17 views

Hellinger Integral properties

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
2
votes
1answer
28 views

Absolutely Continuous measures and Hellinger integral

Let $\mu , \nu$ be two probability measures on $(\Omega , \mathcal{F}).$ Suppose we have a probability measure $\lambda$ such that both $\mu , \nu$ are absolutely continuous with respect to ...
4
votes
0answers
206 views

Weak and Probability convergences

I have a question about this book, "Topics in Random Matrices Theory" of Terence Tao. He claims that if $$\int_{\mathbb{R}}\varphi \ ...
0
votes
0answers
28 views

Change of variables in Lebesgue-Stieltjes integral

Suppose $F$ is a probability distribution function with corresponding measure $\mu_F$ on $\mathbb{R}$. Suppose moreover that $f:\mathbb{R} \rightarrow\mathbb{R}$ is a continuous, increasing function. ...
3
votes
0answers
100 views

Question about Feller's book on the Central Limit Theorem

My question concerns the proof of Theorem 1, section VIII.4, in Vol II of Feller's book 'An Introduction to Probability Theory and its Applications'. Theorem 1 proves the Central Limit Theorem in the ...
3
votes
0answers
64 views

Measurability of one set of measures

Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ ...
0
votes
1answer
27 views

Scaling the Lebesgue-Stieltjes integral

Suppose that $F$ is a distribution function. Denote by $\mu_F$ the measure on $\mathbb{R}$ induced by $F$. Suppose that $a>0$. Define a new distribution function $F_a$ by $F_a(x):= F(ax)$, and ...
1
vote
1answer
23 views

Expectations of martingales

Consider a martingale $(M_n)_{n \geq 0}$ adapted to a filtration $(\mathcal F)_{n \geq 0}$ on a probability space $(\Omega, \mathcal F, P)$. Prove that, for each $k \leq n$; $$E(M_n M_k) = E(M_k^2)$$ ...
0
votes
0answers
41 views

The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
0
votes
0answers
30 views

An outer meassure not being probability measure

I have to prove that an outer measure is not necessary a probability measure. I have this example: Let $\Omega $ be infinity, for every $A \subset P(\Omega)$ $$ \mu^{*}(A) = \left\{ \begin{array}{l ...
3
votes
0answers
59 views

Lebesgue density theorem for compact metric spaces.

Let $X$ be a compact metric space (with balls $B_{\varepsilon }(x)$), $\mu $ a Borel probability measure, and $A$ a Borel set with positive probability. Do we have that $\lim_{\varepsilon ...
1
vote
1answer
38 views

Symmetrisation of function

Consider the probability space $\Omega = \{-1, 0, 1\}$ with the $\sigma$-algebra of all possible events and a probability measure $P$. Consider also the smaller $\sigma$-algebra $$F = \{\emptyset, ...
0
votes
0answers
29 views

The mean of $\mu_{P}(\theta)=\frac{1}{Z}P(x|\theta)$

Consider a parametrized probability measure $P(x|\theta)$, that is for each $\theta\in[a,b]$ it is a valid probability measure on $x$. Denote $f(\theta)$ its mean and $\Sigma(\theta)$ its variance. ...
1
vote
3answers
76 views

A basic question on integration [closed]

$x^{k}{\rm e}^{-x^{2}}$ decreases to zero "exponentially" when $x \to \pm \infty$, $\int_{\mathbb R}{\rm f}\left(x\right)\,{\rm d}x < \infty$. Which theorem is being used here ?
2
votes
0answers
79 views

Kolmogorov extension-type result

I would like to prove the following, using the standard Kolmogorov extension theorem (e.g. http://en.wikipedia.org/wiki/Kolmogorov_extension_theorem): Let $(\Omega, \mathscr{F}, P)$ denote our ...
3
votes
3answers
86 views

Two random variables from the same probability density function: how can they be different?

The definition of $X$ as a random variable according to Wiki is as follows: $Let (\Omega, \mathcal{F}, P)$ be a probability space and $(E, > \mathcal{E})$ a measurable space. Then an $(E, ...