0
votes
1answer
33 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
0
votes
2answers
19 views

probability related to uncountable set

Let $X$ be the collection of closed interval of the form $[a,1],$ where $a \in [0,1]$ and we fix a real number $t \in [0,1]$. Suppose an element $c$ is randomly drawn from $X$, what is the probability ...
0
votes
0answers
26 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 ...
1
vote
1answer
24 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 ...
3
votes
2answers
32 views

Substitution in integral

I am working on a probability theory excercise and encountered the following integral: $$ \iint_{(x,y)\in A}\frac{1}{2}(x+y)e^{-(x+y)}dA, $$ where $A = \{(x,y)\in\mathbb{R}^2\,:\,x+y\le z\,;\, ...
1
vote
0answers
21 views

The infinitessimal generator of Brownian Motion

Background: I know, and can almost prove, that the infinitessimal generator of BM is the Laplacian/2. For me (and I have never heard it done differently) we have Feller Processes, of which BM is one ...
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 ...
0
votes
2answers
23 views

Point between left and right limits of a CDF

This is from the Chapter 15 text of Gourieroux and Monfort's Statistics and Econometric Models II: Set Up: Suppose that there are 2 possible parameter values $\theta_0$ and $\theta_1$ from which ...
1
vote
1answer
88 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)| ...
1
vote
0answers
21 views

Growing of a score function

The argument that I'm dealing is very specific, I hope to make you understand the problem without going into detail. I have this score function: \begin{align} score = MargL^q + MargL^{\theta} ...
1
vote
0answers
15 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 ...
3
votes
0answers
56 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
23 views

choosing the function of a random variable with the lowest variance w.r.t.o the mean of that random variable?

Consider a real gaussian random variable with mean $\theta$ and unit variance. Let $y$ be an observation of the random variable. The objective is to estimate $\theta$ over all possible $y$. Let ...
1
vote
1answer
24 views

Why is this set an event?

As a part of a setup for another problem, my text remarks that it can be used without a proof that if $X_1, X_2, \ldots$ are random variables then $$C:=\{ \omega\in\Omega \ | \ \sum X_n(\omega) \ ...
0
votes
1answer
33 views

Continuity of probability measure

Sorry, I just wanted to know whether I understand this correct. Let $(x_n)$ be an increasing sequence such that $x_n \rightarrow a$, then we have for the probability measure on an arbitrary ...
0
votes
1answer
46 views

Independence random variables

I found two theorems in my notes and they seem to be somewhat complementary which made me doubt that both of them are true: a) Let $X,Y: \Omega \rightarrow \mathbb{R}$ be a measurable function and ...
2
votes
1answer
52 views

Below bound of the mesure of a finite intersection

Let $(X, \mathcal{M}, \mu)$ be a measure space, with $\mu(X)=1$. If $A_{1}, A_{2}, ..., A_{n} \in \mathcal{M}$, prove that $$\mu \left(\bigcap_{j=1}^{n} A_{j} \right) \geq \sum_{j=1}^{n} \mu{(A_{j})} ...
1
vote
1answer
34 views

How does one show that the series converges almost surely?

Let $X_1, X_2, \ldots:\Omega\to \mathbb R$ be random variables. Define $C:=\{ \omega \ | \ \sum X_n(\omega) \text{ converges} \}$. There is such $q\in(0,1)$ that for all $n\in \mathbb N: P\{ |X_n| ...
0
votes
1answer
43 views

Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
0
votes
1answer
64 views

Proof that a median minimizes 1-norm. [duplicate]

I was wondering whether there is an easy way to show the following: We have a data set $x_1,...,x_n$ and $m$ is a median if for at least half of the n data points we have that $x_i \le m$ and for ...
2
votes
1answer
99 views

Convergence in distribution ( Two equivalent definitions)

I read that for convergence in distribution it is equivalent to have that either the characteristic functions of the random variables convergence pointwise or we have that $F_{X_n} \rightarrow F_{X}$ ...
1
vote
1answer
58 views

Weak convergence implies uniform convergence of characteristic functions on bounded sets.

Let $\{\mu_n:n\in\mathbb N\}$ and $\mu$ be distributions on $\mathbb R$, and let $\{\phi_n:n\in\mathbb N\}$ and $\phi$ be their respective characteristic functions. We can easily show using a direct ...
0
votes
1answer
23 views

Which probability in this hypothesis test?

We have a hypothesis A (null hypothesis) such that $p\le 0.6$ and B such that $p>0.6$. Now we want to develop a deterministic test $\phi$ for 20 people that has a safety of 95%. Hence we would be ...
3
votes
1answer
71 views

Random variable exponentially distributed?

I just want to be sure about this: If I read the phrase ' a random variable is exponentially distributed'( which is often said in probability theory and then it is never explictely stated what $X$ ...
2
votes
3answers
87 views

Does it hold for any random variable that $E[X]$ exists iff $\sum_{n\geq 1}P(X\geq n)<\infty$?

If $X:\Omega \to \mathbb R$ is a random variable, show that $E[X]$ exists iff $\sum_{n\geq 1}P(X\geq n)<\infty$ I can prove it only under the additional assumption that $X\geq0$. So I was ...
0
votes
1answer
26 views

$X:=(X_1,X_2)$ is random variable iff $X_1$ and $X_2$ are

Let $X_1, X_2: \Omega \to \mathbb{R}$ and $X: \Omega \to \mathbb{R^2}$ such that $X(\omega):=(X_1(\omega),X_2(\omega))$. Where $\mathbb{R}$ and $\mathbb{R^2}$ are equipped with $\sigma$-algebras of ...
4
votes
1answer
69 views

Which types of convergence preserve this property?

Say, we have a sequence of random variables with $X_n \geq 0$ almost everywhere. Which of the following types of convergence: almost everywhere: $X_n \xrightarrow{a.e.} X$ in probability: $X_n ...
2
votes
2answers
91 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
0
votes
1answer
39 views

range of a function defined by expectation

Given $c>0$ and define a function $f:(0,\infty)\mapsto \mathbb{R}$ by \begin{equation} f(\sigma)=\frac1{\sqrt {2\pi}\sigma}\int_{-\infty}^\infty\max\{-c,\min\{x,c\}\}^2e^{-\frac{x^2}{2\sigma^2}}. ...
0
votes
1answer
32 views

almost sure convergence in distribution

Consider an i.i.d sequence $(X_i)_{i \geq 1}$ of scalar random variables distributed as $\pi$. It is not difficult to check that for almost every realization $(x_i)_{i \geq 1}$ of this sequence, the ...
0
votes
1answer
47 views

proving increasing function

Given $c>0$. Let $f:(0,\infty)\to (0,\infty)$ be a function defined by \begin{equation} f(x)=\frac1{\sqrt{2\pi}x}\int_{-c}^ct^2e^{-\frac{t^2}{2x^2}}dt. \end{equation} I'd like to prove that $f$ is ...
1
vote
1answer
22 views

A basic question on distribution function

The distance between two distribution functions is the infimum of those $\epsilon > 0$ such that $G(x-\epsilon) - \epsilon \leq F(x) \leq G(x+\epsilon) + \epsilon \forall x $. I want to show that ...
2
votes
3answers
51 views

Probability computation, tossing two dice

I have some ideas on how to solve the problem, but simulations do not support my analytical results :) Toss two dice and sum their value and write it down: Denote by $X_n$ the result at $n$-th toss. ...
3
votes
1answer
62 views

A problem on limit superior

Let $A_n$ be the square $[(x,y) : |x|\leq 1, |y|\leq 1]$ rotated through the angle $2\pi n\theta$. I need to find the geometric description of $lim sup_n A_n$ when $\theta$ irrational. I understand ...
0
votes
1answer
27 views

Probability qual problem about Polya's criterion (I guess)

Suppose $\mu$ is non negative, $\sigma$-finite measure on $(0,\infty)$ so that $$c:=\int_0^\infty x\mu (dx)\in(0,\infty)$$ Let $$\phi(u):=\exp\left(\int(e^{iux}-1)\mu (dx)\right)$$ Prove that there ...
1
vote
1answer
44 views

Continuity of integral w.r.t. Lebesgue measure

Let $f:\Bbb R^n\times \Bbb R^m\to \Bbb R$ be a bounded measurable function, and $\mu$ be a probability measure on $\Bbb R^m$ which is absolutely continuous w.r.t. Lebesgue measure $\lambda$, e.g. $m ...
1
vote
1answer
31 views

Almost everywhere convergence of a function of rv

Let f(x) a continuous function on $\mathbb {\bar{R}^+}$(extended positive real line, $x\in(0,\infty]$). Take $y\in \mathbb R^+$. We can say that $\lim_{y\to 0}\frac{x}{y}=\infty $ almost everywhere ...
1
vote
0answers
62 views

Sequence of probabilities with monotone function

Let $\{X_k\}_{k=1}^{\infty}$ be a sequence of i.i.d. random variables with finite support $S = \{ 1, 2, ..., N\}$. Let $P$ be the corresponding probability measure. For all $k \geq 1$, define $A_k := ...
1
vote
1answer
36 views

A basic question on base-$r$ non-terminating representation of a number

For $i=0,\dots,r-1$, let $A_r(i_1,\dots,i_k)$ consist of the numbers in the unit interval in whose base-$r$ expansions the digits $i_1,\dots,i_k$ nowhere appear consecutively in that order. I need to ...
0
votes
1answer
23 views

Inversion formula as a way to check if a function is a characteristic function

Using the notion of characteristic functions from probability theory, suppose that a function from $\mathbb{R}$ to $\mathbb{C}$ is given. Lukacs says, in his book on ch. f., that the usual inversion ...
1
vote
1answer
109 views

Expected value, I do not get this “wikipedia triviality”

on wikipedia are two definitions of the expected value for some random variable $E(X)=\int t f_X(t) dt$ and $E(X)=\int X dP$. I do not see how they are equivalent. In cases, that $X$ would be ...
1
vote
2answers
115 views

Too stupid to understand random variable questions?

I have two excercises: 1.) Let $X_1,X_2,X_3$ be independent uniformly distributed random variables on $[0,1]$. What is the density function of $X_1+X_2+X_3$? 2.) Let $X_1,...,X_4$ be independet ...
0
votes
1answer
50 views

Inequalities of the quantile function [closed]

I'm trying to rigorously prove the following inequalities involving the quantile function $Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\}$ where $F$ is the distribution function: 1) $F(x) < a \iff ...
0
votes
1answer
24 views

Meaning of my calculation card game

I have made a calculation and now I do not understand what I did there. It is about the following question: Imagine you have n cards of which there are 2 aces, what is the expectation value to get ...
0
votes
1answer
35 views

If $S_1, S_2$, and $S_1 \cup S_2$ are $\sigma$-fields, then either $S_1 \subset S_2$ or $S_2 \subset S_1$

I wanted to show that If $S_1, S_2$, and $S_1 \cup S_2$ are $\sigma$-fields, then either $S_1 \subset S_2$ or $S_2 \subset S_1$. Here is what I've tried: Assume the opposite: $S_1 \not\subset ...
2
votes
2answers
56 views

I want to show if $\int |f|d\mu < \infty$ then almost everywhere $|f|< \infty$.

show that if $\int |f|d\mu < \infty$ then almost everywhere $|f|< \infty$. thanks for help.
1
vote
1answer
41 views

An application of LDCT

Consider the sequence of functions: $$F_n (t)=\int_{-\infty}^t \underbrace{ \frac{\Gamma \left[ \left( n+1 \right)/2 \right] }{\sqrt{\pi n} \Gamma \left(n/2 \right)} \frac{1}{\left(1+y^2 /n ...
3
votes
1answer
205 views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
2
votes
1answer
43 views

a question on measurability

Let $X$ be a compact metric space and $(\Omega, F, P)$ a nonatomic probability space. Suppose that $(\Omega, F', P')$ is the completion of $(\Omega, F, P)$. If $f: \Omega \rightarrow X$ is a ...