Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

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18 views

Equality of random variables measurable w.r.t. different sigma-algebras

I'm stuck trying to prove the following statement: Let $\tau $ be a non-negative random variable on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We'll consider the following ...
1
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0answers
12 views

powers of the transition matrix for a subshift of finite type

Is there a relation between powers of the transition matrix for a subshift of finite type and iterations of the (forward) subshift? It is very nice to understand the transition matrix via a graph and ...
2
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1answer
23 views

Square Integrable Martingales and the Unit Process

Let $X_t$ be a continuous square-integrable martingale. Then it is true (I think, please correct me if I am wrong) that: $$\forall t \in [0,\infty), \quad \int_0^t 1_{[0,t]}(s) dX_s = X_t - X_0$$ So ...
3
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1answer
33 views

Does it follow that $\mu$ is a measure? [duplicate]

Suppose $\mu_n$ is a sequence of measures on $(X, \mathcal{A})$ such that $\mu_n(X) = 1$ for all $n$ and $\mu_n(A)$ converges as $n \to \infty$ for each $A \in \mathcal{A}$. Cal the limit $\mu(A)$. ...
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0answers
36 views

Proving sigma-algebras equality

I'm not sure if my proof of the following statement is correct. Let $\tau : \Omega \to \mathbb{R}_+$ be a non-negative random variable, defined on a probability space $(\Omega, \mathcal{G}, \...
2
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1answer
69 views

Sequence of Erdos-Renyi random graphs convergent with probability 1

Definitions Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph ...
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0answers
12 views

How does commutative property play a big role in probability theory? Kindly, explain why do we need free-independence?

I got to know that as matrix operations are non-commutative, calculation of moments are not possible? To be able to do so, one need to ralax the independence to something called free-independence. I ...
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3answers
75 views

probability of having an ace on each hand when dealing 52 cards to 4 hands

The question from DeGroot's book: "probability and statistics" "Suppose that a deck of 52 cards containing four aces is shuffled thoroughly and the cards are then distributed among four players so ...
0
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0answers
14 views

What is Steiltjes transform (other Integral transform) and how does it helps in probability theory, specifically in random matrix theory?

I have started growing interest in random matrix theory. Trying to understand it from "Random Matrices" by Madan Lal Mehta and "An Introduction to Random Matrices" by Anderson and many sources on ...
1
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1answer
42 views

Help understanding convolutions for probability?

I have been trying to do some problems in probability that use convolutions but there has not been much of an explanation of what a convolution is or the purpose of using a convolution. For example ...
1
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0answers
38 views

Simplifying Multiple Integral for Compound Probability Density Function

Are there any ways to simplify this multiple integral? $$ \hat{f}\left(\left.y\right|\alpha\right)=\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\hat{f}\left(\left.y\right|\theta_{1}\right)\hat{...
2
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1answer
57 views

How to prove that if $E[X^2]$ is finite then $n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0$?

Let $X$ be a random variable with $E[X^2]<\infty$. I want to prove that $$ n\Pr[\lvert X\rvert>\varepsilon\sqrt n]\xrightarrow[n\to\infty]{}0 \text. $$ I tried to apply Chebyshev's inequality, ...
6
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1answer
71 views

Do we necessarily have that $\int g\,d\mu_n \to \int_0^1 g\,dx$?

Let $\mathcal{B}$ be the Borel $\sigma$-algebra on $[0, 1]$. Suppose $\mu_n$ are finite measures on $([0, 1], \mathcal{B})$ such that $\int f\,d\mu_n \to \int_0^1 f\,dx$ whenever $f$ is a real-valued ...
8
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2answers
50 views

$\{f_n\}$ is uniformly integrable if and only if $\sup_n \int |f_n|\,d\mu < \infty$ and $\{f_n\}$ is uniformly absolutely continuous?

Let $(X, \mathcal{A}, \mu)$ be a measure space. A family of measurable functions $\{f_n\}$ is uniformly integrable if given $\epsilon$ there exists $M$ such that$$\int_{\{x : |f_n(x)| > M\}} |f_n(x)...
1
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3answers
37 views

How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ such that $\int_A f_{n_j}(x)\,dx$ converges?

Let $\{f_n\}$ be a sequence of measurable real-valued functions on $[0, 1]$ that is uniformly bounded. How do I see that if $A$ is a Borel subset of $[0, 1]$, then there exists a subsequence $n_j$ ...
2
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1answer
29 views

Bernoulli Trials: Law of Large Numbers vs Gambler's Fallacy, the N paradox

I have asked this question before but I think it wasn't clear what I implied with my succinct question, so I will be a bit more verbose this time. Lets set the following example: Bernoulli trials, K=...
0
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1answer
14 views

Use Bayes rule to test whether patient has disease after several positive tests

I have solved one of those standard bayes rule application exercises a la: Given a prevalence value of a disease, the sensitivity and the specificity of a test, calculate the probability that the ...
2
votes
1answer
25 views

Integral of sequence converges? [closed]

Suppose $(X, \mathcal A, \mu)$ is a measure space, $f$ and each $f_n$ is integrable and nonnegative, $f_n \to f$ almost everywhere, and $\int_X f_n \to \int _X f$. Does it necessarily follow that for ...
0
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1answer
52 views

Infinite dimensional Borel-measurable function.

I am not quite sure how this statement about infinite dimensional borel-measurable functions is true, but the author says it is an easy observation: Let $D([0,\infty))$ denote the space of all ...
2
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0answers
43 views

How to physically model/construct a biased coin?

A perfectly unbiased coin is one that has the same probability for heads and tails (i.e., 50%/50%). A perfectly biased coin is one that has (as the name suggests) different probabilities for head ...
0
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0answers
24 views

When is a stochastic integral a martingale?

In what follows, let the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ as well as the chosen filtration $(\mathcal{F}_t)_{t \ge 0}$ be known, and let $f$ denote an arbitrary locally bounded ...
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2answers
37 views

Why is a continuous Lévy process twice integrable?

In his textbook "Wahrscheinlichkeitstheorie" (de Gruyter, 2008), Jochen Wengenroth shows (p. 144) that if $(X_t)_{t\in[0,\infty)}$ is a continuous, real-valued Lévy process with $X_t\in \mathcal{L}_2$ ...
2
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0answers
19 views

How to compute integral of exponential martingale with respect to Poisson process?

Let $N=\{N_t:t\in\mathbb R_+\}$ be a homogeneous Poisson process with intensity $\alpha$ and $M_t=N_t-\alpha t$ the compensated process. I'd like to show that $N$ is not a natural process, i.e. that ...
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1answer
69 views

Asymptotics of $\lim\inf X_n$, missing part of exam question [closed]

So I was working through an old exam and encountered the hilarious situation that part of the statement of the exercise was illegible. I was wondering if anyone could figure it out for me, so that I ...
3
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0answers
32 views

Are martingales progressively measurable? (Application to square integrable martingales)

This is an incredibly dumb question, but I'm not sure if I know the correct answer, and it doesn't seem to be stated anywhere on the internet, so here goes: Are martingales progressively ...
4
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1answer
45 views

Bound for sum of normal distributions

I have encountered an exercise that was quite puzzling for me. Maybe someone can help me out here? So let $(X_n)_n $ be $N(-a,1)$ distributed, independent random variables where $a>0$. I need to ...
1
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0answers
52 views

Proof Attempt: Non-decreasing continuous CDF is standard uniformly distributed

Proof Attempt: For any random variable $X$ with non-decreasing continuous cdf $F(x)=\Pr(X≤x)$ (note that the inverse function does not necessarily exist due to flat regions in $F$), I wish to prove ...
1
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1answer
20 views

Inferring absolute continuity of summands from absolute continuity of sum

Suppose we have i.i.d. random variables $X_1, X_2$. If $\text{Law}(X_1 + X_2)$ is absolutely continuous with respect to the Lebesgue measure $\lambda$, can we infer that each $X_i$ is absolutely ...
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1answer
57 views

Confusion about the average distance traveled on a $1$D random walk

The average absolute distance on a one dimensional random walk is supposed to be $\sqrt{n}$. Where $n$ steps are taken from the origin or $n$ is the time. I don't have an intuitive understanding or ...
2
votes
1answer
44 views

Sample proportion and the Central Limit Theorem

Suppose that $ (\Omega,\Sigma,\mathsf{P}) $ is a probability space and that $ (X_{k})_{k \in \mathbb{N}} $ is a sequence of i.i.d. Bernoulli trials on $ (\Omega,\Sigma,\mathsf{P}) $, each with ...
1
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2answers
58 views

When a Markov chain converges to a steady state, what kind of convergence is it?

Let $A$ be a transition matrix, the steady state distribution $x$ satisfies the distribution $Ax = x$. One can prove that under certain circumstances, $$\lim_{n\rightarrow\infty}A^n q=x$$ where $q$ is ...
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0answers
39 views

Forming a triangle probabilistically [duplicate]

What is the probability that if you break a stick at $2$ points the three sides form a triangle? Is there a technique that avoids calculus?
3
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0answers
27 views

Representation of point process

Let $E$ be a polish space and $N(E)$ be the space of finite integer value measures. It is known that for every $\mu \in N(E)$ exists $x_1, \dots, x_n \in E$ such that $$\mu = \sum_{i=1}^n \delta_{x_i}$...
0
votes
1answer
24 views

Convolution mixture of a probability generating function (population genetics)

I'm trying to work through an old population genetics paper (see here). The following model assumes an infinite number of nucleotide sites and no recombination between different sequences (so you can ...
0
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1answer
46 views

Conditional expectation of a product of random variables

I have two independent continuous random variables $X$ and $Y$ with pdf's : $f(x)$ and $f(y)$ cdf's : $F(x)$ and $F(y)$ a constant $a$ I am trying to express using the given pdf/cdf functions the ...
1
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1answer
23 views

How to adjust estimation of probability according to new information

Suppose $\{a_1,a_2,\dots,a_n\}$ is a permutation of $\{1,2,\dots,n\}$. The probability of $a_i=j$ is estimated to be $p_{ij}$. The probability matrix might look like this $$ P=\left( \begin{matrix} ...
1
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0answers
7 views

$X-x_0=O_p(n^{-1/2})$ implies $g(X)-g(x_0)=O_p(n^{-1/2})$

Suppose that $X$ is a random vector and $x_0$ is a fixed vector such that $$ X-x_0=O_p(n^{-1/2}).\tag{$*$} $$ Let $Y=g(X)$ where $g$ has a continuous gradient that is nonzero at $x_0$. Let $y_0=g(x_0)$...
0
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0answers
47 views

Proving a lower bound on the value of a probability density at the solution to an equation

Assumptions and notation Let $f$ be a twice-differentiable log-concave density function on $[0,1]$, and let $F$ be the corresponding distribution function. Define $x^D$ by: $$\frac{1-2F(x^D)}{f(x^D)}=...
5
votes
6answers
582 views

Find the probability of getting two sixes in $5$ throws of a die.

In an experiment, a fair die is rolled until two sixes are obtained in succession. What is the probability that the experiment will end in the fifth trial? My work: The probability of not getting ...
1
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3answers
82 views

Representing pairwise-independent but not independent occurrences with venn diagram

For $A,B$ and $ C $ partially pairwise independent occurrences (i.e. $I(A;B)=0$, $I(A;C)=0$ ), it is not true to say that $I(A;B,C)=0$, since $I(A;B,C)=I(A;B)+I(A;B|C)$ [<-this is not correct, see ...
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0answers
33 views

Multidimensional change of variables for pdf integration

I have a very simple question, but I could not find the answer, so I have to ask this here: Given is a multidimensional pdf $f(x_1, ..., x_n)$. $x_1, ..., x_n$ are Carthesian coordinates. We want to ...
0
votes
1answer
15 views

Find the Expectation of $2\sum\limits_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$ for $y_i$ i.i.d. Cauchy$(\alpha,\beta)$

Find the Expectation of $$2\sum_{i=1}^n \frac {(y_i-\alpha)^2-\beta ^2}{(\beta^2+(y_i-\alpha)^2)^2}$$ given $y_1,y_2...$ iid ~Cauchy$(\alpha,\beta)$ with pdf $(-\infty < y<\infty , \beta>0)$: ...
2
votes
1answer
38 views

Problem involving Central Limit Theorem

The following problems is from Durrett 3.4.9: Suppose $X_i$ are independent and $S_n = X_1 + ... + X_n$. Assume that $$ \begin{split} P(X_m = m) &= P(X_m = -m) = m^{-2}/2, \text{ and }\\ P(...
0
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0answers
36 views

Random walk in a random environment

Consider the random walk in a random environment $\{X_n\}$, that is $P(X_{n+1}=z+1|X_n=z)= \alpha_{z}$ and $P(X_{n+1}=z-1|X_n=z)=1- \alpha_{z}$, where $\{\alpha_{z}\}$ is i.i.d random variables, with $...
1
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1answer
25 views

Representation of Matrix Calculations - Column Mean Subtract From Each Row

Suppose I have a matrix that looks like this: $$X = \begin{bmatrix}1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0\end{bmatrix}$$ The ...
-1
votes
1answer
20 views

the relation between the sigma-algebras of two isomorphic spaces [closed]

It crosses my mind the following question : if X and Y are two isomorphic spaces what can we say about the Borel sigma-algebras associated to each of them, otherwise what is the relation between $\...
1
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1answer
38 views

$X_i = \mathcal{N}(0, \sigma_i^2)$, with $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$

Let $\{X_k\}$ be independent random variables such that $X_i = \mathcal{N}(0, \sigma_i^2)$, with $\{\sigma_i^2\}$ such that $\sigma_1^2 \geq \sigma_2^2 \geq \dots \geq 0$ and $\sum_i \sigma^2_i = 1$. ...
1
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0answers
14 views

Rate of convergence for martingales, “merging of opinions” results

Let $(\Omega, \mathcal{F})$ be a measurable space, and let $P$ and $Q$ be probability measures on this space. Let $(\mathcal{F}_{n})_{n \in \mathbb{N}}$ be a filtration on $\Omega$. Assuming that the ...
3
votes
1answer
89 views

Probability that a Lévy-process is unbounded, zero-one law?.

For a Lévy-process, I need to prove that the probability that the trajectories are bounded on $[0,\infty)$ is either 0 or 1. Can you please help me? (The author says that this is a consequence of ...
0
votes
1answer
52 views

What at the chances of getting 20 heads on a row if tossed 100 million times? [duplicate]

I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such ...