# Tagged Questions

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] ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 $\mathcal{G}$. ...
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 ...
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 ...
35 views

89 views

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 ...
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 ...
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 ...
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 ...