1
vote
1answer
18 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
33 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 ...
1
vote
0answers
67 views
+100

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
30 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
35 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 ...
1
vote
0answers
39 views

Partial sums are alternate upper and lower bounds for $\mathbb{P}(\cup A_i)$

Show that $$ \sum_{k=1}^m(-1)^{k+1} S_k \leq \mathbb{P}(\cup_{i=1}^n A_i) \leq \sum_{k=1}^{m'}(-1)^{k+1} S_k$$ where $m, m' \leq n$, $m $ is even and $m'$ is odd, and $S_k = ...
3
votes
0answers
68 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 ...
0
votes
2answers
53 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 ...
2
votes
1answer
27 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
13 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. ...
1
vote
0answers
36 views

Expected values of continuous and bounded functions are equal then random variables are equal, too.

I have seen several of reasoning based on the following fact: Real random variables $X, Y$ in $\mathbb{R}^n$ are equal almost surely if and only if $\mathbb{E}g(X)f(X) = \mathbb{E} g(X)f(Y)$ for ...
0
votes
1answer
31 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, ...
2
votes
0answers
42 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
70 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 ...
2
votes
0answers
40 views

Conditions on Poisson random variables to convergence in probability

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants. a)Find conditions ...
0
votes
1answer
34 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
41 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
35 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
31 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
38 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
51 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
23 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
30 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
201 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
39 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
39 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
38 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 ...
1
vote
0answers
55 views

Entropy and Gibbs measures (mathematical formalism) [closed]

Entropy and Gibbs measures are both phrases that are thrown around at probability seminars and other analysis events, and sadly this is not "only" bad because I am confused about it, but also because ...
1
vote
0answers
31 views

monotonicity of a complex function referring normal distribution

In my research I need to make clear the following point: Suppose that a random variable $\theta\sim N(\mu, \sigma^2)$. There are two imperfect signals about $\theta$: $X=\theta+\sigma_x\xi$ and ...
0
votes
1answer
38 views

Sub sigma algebra and probability spaces — definition

I am reading this book and I am a bit lost with the definitions because they are not provided and I can't seem to find it online: Let $L_2(\Omega,A,P)$ be a probability space such that $f \in L_2$ ...
1
vote
1answer
56 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
1
vote
0answers
42 views

Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
0
votes
0answers
33 views

Distribution function inequality for a transformed random variable ?!

I'm stuck with the following problem. Let $A:=\{g: g:(0,1)\to\mathbb{R}^+, non-decreasing\}$ and $U\sim U[0,1]$ be uniformly distributed on the interval $[0,1]$ on the probability space $(\Omega, ...
1
vote
1answer
62 views

Probability of event in normal distribution

Let $X$ be a random variable that is normally distributed and $X_1,\ldots,X_n$ be (independet) copies of $X$, then we can estimate this probability by using a simple Monte-Carlo estimator: $p := P (X ...
0
votes
0answers
30 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
0
votes
1answer
30 views

Is this a Markov chain property

For $A,B$ measurable sets and $(X_n)_n$ a Markov chain. Do any of the following properties hold? $$P(X_2 \in B | X_1=x_1,X_0 \in A) = P(X_2 \in B|X_1=x_1)$$ or $$P(X_2 \in B|X_1 \in A,X_0=x_0) = ...
0
votes
0answers
33 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
2
votes
1answer
66 views

Convergence almost sure pointless?

A very common type of convergence in probability theory is 'almost sure convergence'. I don't understand why this type is used at all. In principle, we should always be able to substitute it by a ...
2
votes
1answer
100 views

Property (ii) of increasing functions in Chung's “A Course in Probability Theory”

I am a bit confused by the line of reasoning on page 2 of Kai Lai Chung's "A Course in Probability Theory". In particular, he is considering a real-finite valued function $f$ which is defined and ...
0
votes
2answers
41 views

Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
0
votes
1answer
40 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
1
vote
1answer
32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
0
votes
1answer
73 views

Difference between Borel Sigma algebra and Cylindrical sigma algebra?

I see that there are two differen concepts for Sigma Algebras on cartesian products over the real numbers. The first one is the Borel Sigma Algebra created by the product topology. The other one is ...
0
votes
1answer
41 views

Multivariate normal distribution independet iff uncorrelated

I found a few threads about this but none of them answered my question. I am supposed to show that if you have random variables $X_1$,$X_2$ that are gaussian distributed and they fulfill that ...