1
vote
1answer
17 views

Is an infinite product probability space compatible with an almost surely statement?

My question pertains to the following Theorem which can be found here. Theorem (Existence of product measures): Let $A$ be an arbitrary set and for each $\alpha \in A$ let ...
0
votes
0answers
15 views

Hermite polynomials with non standard variance (not equal to one)

It is known for probabilists that if $p(x)=\frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ then the derivatives of $p$ can be expressed in terms of the Hermite polynomials as follows: $$\frac{d^n}{dx^n} ...
-1
votes
1answer
26 views

How to derive this inequality

I learnt that for a standard normal random varialbe $Z$ and positive $x$, we have $$\mathbb P (Z > x) \geq \frac{x}{x^2+1} \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} = \frac{1}{x+\frac{1}{x}} ...
1
vote
1answer
54 views

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? [duplicate]

$X_n$ are r.v.s, is it true that $E[\sum_{n=1}^{\infty} X_n] = \sum_{n=1}^{\infty} E[X_n] $? My feeling is that this is not necessarily true. But cannot come up with an example. Can someone provide ...
-1
votes
0answers
26 views

Formula for conditional expectation and its computation w.r.t measures

Let $(X,M,\mu)$ be a probability space. Consider another $\sigma$-algebra $M'\subset M$ and define probability measure $\nu$ on $(X,M')$ by setting $\nu(E)=\mu(E)$ for $E\subset M'$. Show that there ...
0
votes
0answers
21 views

Can I use the triangle inequality to ensure a unique measure?

Let $\mathcal{C}$ be a finite set of objects, $\Delta\mathcal{C}$ the set of probability measures on $\mathcal{C}$, and $\mathscr{U}$ be a finite set of linear functions, $u: ...
1
vote
1answer
15 views

Expectation of Truncated Random Variables (2)

This is a follow-up question from here. Let $X_n$ be a sequence of $iid$ random variables with zero mean, finite variance $\sigma^2$ and partial sum $S_n:=\sum_{k=1}^n X_k$. Let $$0<\delta<0.5$$ ...
0
votes
0answers
23 views

Convergence in probability (in measure) of a weighted sum of random functions

Consider a mesh of points $\pi_{n} = (t_{1n},\ldots,t_{K_{n}{n}})$ with $0 < t_{1n} < \ldots < t_{K_{n}n} < 1$ and weights $(w_{1n},\ldots,w_{K_{n}n})$ such that ...
4
votes
1answer
46 views

Slowly varying function without limit at infinity

A function $f:\mathbb R \to \mathbb R$ is slowly varying at infinity if for any $t>0$ $$ \lim_{x\to +\infty}\frac{f(xt)}{f(x)}=1. $$ Is there a bounded function slowly varying at infinity whose ...
2
votes
1answer
81 views

About strong law of large numbers

I came across a problem: Let $X_1,X_2,...$ be independent random variables with uniform(0,1) distribution from probability space $(\Omega,\cal F,P)$. Prove that: ...
0
votes
1answer
28 views

Probability and Measure: Sigma-finite

What is a example that shows that $\mu$ $\sigma$ -finite does not imply $\mu \cdot T^{-1}$. I have a basic understanding of what $\sigma$-finite means but if $\mu$ is finite implies $\mu \cdot T^{-1}$ ...
3
votes
1answer
37 views

Convergence in equivalent probability measure

Let $(\Omega, \mathcal{F}, P)$ be a probability space and $A_n$ be a sequence of events such that $P(A_n)$ converge to 0. If $Q$ is an equivalent probability measure of $P$, does it mean that $Q(A_n)$ ...
1
vote
0answers
23 views

Can the Hamburger moment problem be solved for probability measures?

The hamburger moment problem states that given any real sequence $\{a_n\}$, there exists a positive Borel measure $\mu$ such that $$ a_n =\int_{\mathbb{R}} x^{n}\,d\mu. $$ In other words, the ...
4
votes
1answer
31 views

An application of law of large numbers

How can one apply a law of large numbers to a Poisson Process in order to deduce the analytic fact that $$\lim_{t\rightarrow\infty} e^{-t}\sum_{n=0}^\infty ...
1
vote
1answer
41 views

Normal Distribution and Iterated Logarithm

Let $X_n$ be independent $N(0, \sigma^2)$-distributed random variables with partial sum $S_n := \sum_{k=1}^n X_k$, $n \geq 1$. Then I read the following results. $$ \sum_{k = 1}^n \mathbb P (S_n > ...
3
votes
1answer
45 views

Construct independent $X_n$ such that $\sum_{n=1}^\infty Var(X_n)=\infty$

How to construct an independent random variable $(X_n)_{n\in\mathbb{N}}$ such that $\sum_{n=1}^\infty X_n$ converges and $\mathrm{Var}(X_n)$ is uniformly bounded by some constant C, but ...
1
vote
1answer
37 views

Convergence of the expectation of a non-continuous function

Suppose that $F_{n}$ converges to $F$ weakly, where $F$ is a continuous distribution function. Also, suppose that $g$ is a bounded, continuous function and $\{x_{n}\}$ is a real-valued sequence of ...
2
votes
2answers
34 views

Definition of multivariate random variable

Let $(\Omega,\mathcal E,P)$ be a probability space and $X_1,\dots,X_n$ random variables on $(\Omega,\mathcal E, P)$. Then I can define the vector $X=(X_1,\dots,X_n)\colon \Omega \to \mathbb R ^n$ and ...
1
vote
2answers
22 views

Integrating a function wrt different measures [duplicate]

Suppose that $(\Omega, \mathcal E, P)$ is a probability space and $X\colon \Omega \to \mathbb R$ is a RV defined on $\Omega$. Denote as $\mu\colon \mathcal B \to [0,1]$ the probability measure on ...
1
vote
0answers
43 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
1
vote
1answer
26 views

Every measure of natural numbers and the power of natural numbers as their sigma algebra looks like this…

Let X= $ \mathbb{N} $ ans S= P($ \mathbb{N} $) . Prove that every measure $\mu $ in $(X,\mathcal S)$ can be obtained by an unique non-negative extended sequence of real numbers $(a_{n})$ as follows ...
0
votes
0answers
45 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
5
votes
1answer
227 views

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
0
votes
1answer
37 views

Ask for a good reference for the calculus involving singular continuous measure

I am not an expert on measure theory. I am sorry if this question is too simple for some experts here. Suppose the measure $\mu$ is singular continuous on $\mathbb{R}$, such as the cantor measure. ...
1
vote
0answers
37 views

Continuity preserved unter expectation? Dominated convergence?

Let $Z$ be a random variable with $0<Z<\infty, 0<\mathbb{E}[Z]<\infty$ and $Z$ be atom-less, i.e. $\mathbb{P}(Z=z)=0$. Further, let $g:\mathbb{R}^+\to\mathbb{R}^+$ be continuous and ...
3
votes
0answers
91 views

Convergence in probability and convergence of Cesaro means

Consider the random variable $X_n$, not necessarily iid. If $X_n\rightarrow 0$ almost surely, then the Cesaro means $\frac 1n\sum_{k=1}^nX_n$ converge almost surely to 0. This cannot be weakened to ...
1
vote
2answers
75 views

Additive but not $\sigma$-additive function

Give an example of a measure space $(\Omega, \mathit{F})$ and a function $\mu$ on $\mathit{F}$ that is additive but not $\sigma$-additive, i.e. $\mu(\cup A_i)= \sum\mu(A_i)$ for a finite collection of ...
3
votes
1answer
37 views

Sufficient conditions for a sum over a countable set to be well-defined

Suppose $W$ is a countable set and $f:W\to\mathbb{R}$ is a real-valued function. I would like to know the sufficient conditions so that the concept $$ \sum_{w\in W}f(w)\tag{$*$} $$ is well-defined. ...
0
votes
0answers
69 views

Finding test of critical region for sum/variance of normal distributions

Let $Y_1,....,Y_n$ denote independent, identically distributed random variables such that $Y_1$ has a normal distribution with mean $\theta$ and standard deviations $\theta$, where $\theta$ > 0. ...
0
votes
1answer
33 views

Category theoretic view of coupling measures/RVs

Here is a general definition of the word "coupling" that covers every use I've seen of it. (And this generality is necessary because sometimes one does not define a coupling on an exact product space, ...
3
votes
0answers
48 views

Simplifying an integral involving Gaussian PDF

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty ...
2
votes
1answer
71 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
0
votes
1answer
36 views

Putting a bound on some probability inequality

Assume that we have the following polynomial: $$ax^2 + bx =c$$ and a, b, c are i.i.d uniform random variables in [0, 1]. I'm trying to calculate the probability that the root is real, and that ...
0
votes
2answers
33 views

Total boundness of Lipschitz densities

In the article Almost Sure Testability of Classes of Densities by Devroye and Lugosi in 1999. They claim in Example 10 (page 9) that Lipschitz densities on [0,1] with Lipschitz constant bounded by ...
1
vote
1answer
33 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
1
vote
1answer
42 views

Reverse Fatou's lemma on probability space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets. Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq ...
2
votes
2answers
51 views

Can any measure be made into a bounded measure?

Is it possible to derive a bounded measure from any measure on a measure space? For example can the Lebesgue measure be made into a probability measure?
0
votes
1answer
37 views

differential equation with random coefficient

I am confused with a problem I encountered at hand, not on how to work on it but rather understanding the problem itself: Let $A(x;\omega)$ be a random field taking values in $[a,b]$ where $a,b < ...
1
vote
2answers
35 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
2
votes
1answer
42 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
4
votes
0answers
52 views

A formula similar to $\int_a^bf(x)dx=\mu\left[a,b \right]$ for $f^p$.

Let $\mu$ be an absolutely continuous measure with respect to the Lebesgue measure on $\mathbb{R}$ , and $f:\mathbb{R}\to \mathbb{R^+}$ its Radon-Nikodym derivative . We can write $\int_a^bf(x)dx$ in ...
0
votes
1answer
25 views

Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
2
votes
1answer
35 views

What is the proper definition of cylinder sets?

in class we defined the terminal $\sigma$-algebra for a sequence of random variables $(X_i)$ with $X_i:\Omega \rightarrow \mathbb{R}$ as $G_{\infty}:=\bigcap_i G_i$, with ...
6
votes
1answer
220 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
0
votes
0answers
18 views

Conditional expectation for discrete random variables

Is it correct that for two discrete random variables $X,Y$ we just have $$E(X|Y \in A) = \sum_{x \in ran(X)} xP(X=x|Y \in A)?$$ This should follow from $$E(X|Y \in A) = \sum_{y \in A}E(X|Y = {y}) ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
1
vote
0answers
22 views

Spectral Representation for a real valued process

So I just finished reading a section in a book which discusses how every stationary stochastic process $\xi(t)$ can be expressed as $\xi(t)=\int_{\mathbb{R}}e^{it\lambda}\,dZ(\lambda)$ where ...
1
vote
0answers
41 views

Central limit theorem does not converge to random variable

Recently, we investigated whether the expression in the central limit theorem converges to a random variable pointwise almost sure? The answer was negative due to $P ( \text{limsup} ...
4
votes
1answer
41 views

How to show convergence in distribution

Let $([0,1],B,\lambda)$ (B Borel Sigma-algebra) and $\lambda$ the Lebesgue measure. I want to show that this sequence converges in distribution. $$X_n(\omega)= \left(\begin{matrix} 1, & ...
0
votes
2answers
39 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...