Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

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Solution to Billinglsey (1995) problem 20.22

Let $Y_1\leq Y_2\leq ...$ be random variables s.t. $\mathrm{plim} Y_n = Y$. Show that $Y_n \to Y$ with probability 1. Some hints? My strategy would be to prove that $\sum P(\lvert Y_n -Y \rvert > ...
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Can infinite random sequences be asymptotically compressed?

A number $0.5<p<1$ is chosen at random and given to two people A and B whom are allowed to communicate before beeing separated. A is then given a sequence S of N random bits where each bit has ...
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92 views

Prove random vector with covariance matrix $\Sigma$ has non-degenerate distribution iff $\Sigma$ is positive definite

Let $X$ denote a $d$-dimensional random vector with covariance matrix $\Sigma$ satisfying $|\Sigma| < \infty$. Prove $X$ has non-degenerate distribution iff $\Sigma$ is positive definite. ...
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How can subscripts converge to $\infty$ almost surely?

Here is the question: Give an example of random variables $X_n \in \{0,1\}$, $X_n \rightarrow 0$ in probability, $N(n) \uparrow \infty$ almost surely and $X_{N(n)} \rightarrow 1$ almost surely. Here ...
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130 views

Showing almost sure divergence

Doing some exercises as preparation for an upcoming exam, but im sort of stuck at this exercise: \ Assume that $X_1,X_2,...$ is an i.i.d sequence, such that $X_1 \sim \mathcal{N} (\xi , \sigma^2)$, ...
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34 views

Framing a problem in the context of $\Omega,\mathcal{F},P$

There is a problem in my probability book about lightbulbs going out and getting replaced. It states that $X_1, X_2,\dots$ i.i.d with $0<X_i<\infty$. "Let $T_n = X_1 + \cdots + X_n$ and think ...
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61 views

Prob: Moment Generating Functions

I found the moment generating function of W=4Y-2 which is e^(-2t) * e^(3*((e^4t) -1)). Is this moment generating function poissonly distributed? And Why?
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224 views

Probability Theory: Moment generating functions

If $Y$ has a binomial distribution with $n$ trials and probability of success $p,$ show that the moment-generating function for $Y$ is $m(t) = (pe^t + q)^n,$ where $q = 1 − p.$ I got to $$m(t) = ...
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37 views

A problem on distribution functions

I have a quick question here. From the definition of a distribution function (DF), $\text{A real-valued, nondecreasing, right continuous function} \; F \; \text{defined on} ...
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49 views

probability problem(3 urns)

There are 3 urns. Urn 1 has 2 black and 3 white ball. Urn 2 has one black and 2 white balls. Urn 3 has 2 black and 1 white. A person who is blindfolded, picks a ball from urn 1, puts it into urn 2, ...
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64 views

transition kernel

I've got some trouble with transition kernels. We look at Markov process with statspace $(S,\mathcal{S})$ and initial distribution $\mu^0$. We have a transition kernel $P:S\times ...
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86 views

Relative sizes of Skorokhod and product topologies on space of sample paths

Let $E$ denote a compact metric space. Let $T$ denote the non-negative reals. Let $E^T$ denote the class of all functions from $T$ to $E$. Let $C$ denote the subset of $E^T$ consisting of càdlàg ...
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54 views

Prove Multinomial Coefficient (Probability Theory)

Prove that the multinomial coefficient given by: $$ \binom{n}{n_1}\binom{n-n_1}{n_2}\binom{n-n_1-n_2}{n_3}\cdots\binom{n-n_1-n_2-\dots-n_{k-1}}{n_k} $$ equals the following expression $$ ...
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Field and Algebra

What is the difference between "algebra" and "field"? In term of definition in Abstract algebra. (In probability theory, sigma-algebra is a synonym of sigma-field, does this imply algebra is the same ...
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132 views

Differentiation under (measure theoretical) integral sign

I am looking for a citable reference for the result on differentiation under the integral sign for integration against a measure. The result states that if $R \subset \mathbb R$, $(X,\mathcal F, ...
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85 views

Quantifying the “flatness” of functions which are the Fourier transforms of positive functions

I have a question which I admit is a little cumbersome for me to try to state succinctly, and which I fear may not have a simple answer, but I figured I'd give it a shot. In broad terms, I'm trying to ...
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80 views

Confusing cont. r.v. problem

Let $X_1$, $X_2$, $X_3$, $X_4$, $X_5$ be independent continuous random variables having a common distribution function $F$ and density function $f$, and set $$I = \mathbb{P}\{X_1 < X_2 < X_3 ...
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Upper bounds on the sum in a Martingale process

My question is related the hitting time of not a random walk, but a more general martingale process. Suppose we start with an arbitrary $x_0=x$ with $0\leq x\leq 1$. We compute $x_{t+1}$ from $x_t$ ...
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41 views

If $f^n$ is mixing then $f$ is mixing?

Let $(X,\mathcal{A},\mu)$ be a probability space and $f:X\to X$ be a measurable map that preserves $\mu$. Fix $n\in \mathbb{Z}^+$. It's not hard to see that $f$ ergodic does not necessarily imply ...
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130 views

Which is a good textbook on stochastic processes which takes measure theoretic approach?

I was looking for an intermediate-advanced textbook on stochastic process. I have graduate level probability knowledge.
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102 views

Fractal dimension of Gaussian white noise is infinite?

I read in this paper that the fractal dimension of Gaussian white noise is infinite. The paper does not prove it nor give a reference to support it. I failed to find a reference from online searching. ...
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Heteroskadasticity and Linear Probability

Question Suppose $(Y,X,U)$ be a random vector such that $$ Y = X'\beta + U. $$ Suppose $Y$ takes values in $\{0,1\}$ and that $E[Y\mid X] = X'\beta$. Is it reasonable to assume that $Var[U\mid X]$ ...
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Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
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If $A∩B=A′∩B′$ for $A,A'$ and $B,B'$ independent, do we have $A=A′$ and $B=B′$?

Let $\mathcal{A}$ and $\mathcal{B}$ two independent sigma-algebra, and $A,A' \in \mathcal{A}$, $B,B' \in \mathcal{B}$ such that $A \cap B=A' \cap B' $, and $P(A)P(B)>0$. Do we have $A=A'$ and ...
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69 views

What causes the change in the expected value of the product of random variables?

The following question is part of a homework exercise on portfolio theory that I have to do. Suppose that $Y$ is a random variable representing the returns on an investment. Now, let $f$ be a ...
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Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
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233 views

Can you split up expectation over multiplication?

I was wondering if the property exists where if you have $\ E[(Y- \mu)^3]$ you can write it as $\ E[(Y- \mu)^2] E[(Y- \mu)] $ ?
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27 views

Local-Global properties of ``almost surely'' statements

Given some infinite collection of objects, $X$, if any $x \in X$ almost surely has property $\mathcal{P}$ what can we say about $X$? To clarify what I mean consider the following example. A random ...
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Sigma-finiteness of a point mass measure

Let $\mathcal{e}_{a}(x)$ denote point-mass at the point $a\in \mathbb{R}$. Let $a_{n}$ be a sequence of points in $\mathbb{R}$ and let $\nu = \sum_{n=1}^{\infty}\mathcal{e}_{a_{n}}$. What are ...
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275 views

Result and proof on the conditional expectation of the product of two random variables

My problem is the following: $X$ and $Y$ are two random variables and $\mathcal{F}$ is a $\sigma$-algrebra. Given that $X$ and $Y$ are independant, and that $X$ is independant of $\mathcal{F}$, can I ...
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68 views

A problem on conditional expectation

I am studying conditional expectation and found some problems. I tried to solve them to understand the subject better, but I'm stuck now. Let $X$ be a random variable with strictly positive density ...
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65 views

Show that a function of a Cauchy-distributed r.v. Is also Cauchy-distributed.

Let X be Cauchy with parameters $\alpha$ and 1. Let $Y=\frac{a}{X}$ with $a\neq 0$. Show Y is also a Cauchy r.v. And find its parameters. I'm supposed to get ...
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Exchangability of inner product and integral in bochner spaces

For the linear operator $e \in \mathcal{L}(V,V^{*})$, and sufficiently small $\delta s \in V := L^2(0,T,L^2(D))$ and $p \in V$ we have; $$ E[\langle e^{*}p, \delta s \rangle_{V}] = \langle E[e^{*} p ] ...
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Extensions of probability measures on fields

I am trying to solve exercise 3.3 in Billingsley's "Probability and measure" and I am not sure I correctly understood the text. I am not going to copy all the text, but I will ask directly some ...
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74 views

Let (X,Y) have a Dirichlet Distribution with paramters $(\alpha_1, \alpha_2, \alpha_3)$ Establish that X~Beta$(\alpha_1, \alpha_2 + \alpha_3)$

If the joint pdf of (X,Y) is $f(x,y)=\frac{\Gamma(\alpha_1 + \alpha_2 + \alpha_3)}{\Gamma(\alpha_1) \Gamma(\alpha_2) \Gamma(\alpha_3)} x^{\alpha_1 - 1} y^{\alpha_2 - 1} (1-x-y)^{\alpha_3 -1}$ ...
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87 views

Inner product of random vectors

Suppose we have 2 vector random variables $X, Y\in \mathbb V$, $X: \Omega \rightarrow \mathbb V$ and $Y: \Omega \rightarrow \mathbb V$where $\mathbb V$ is a vector space with inner product $(X,Y)$. I ...
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57 views

How to prove the ratio of two random variables is also a random variable

If $X, Y$ are two random variables, how to prove $X/Y$ is also a random variable? I understand I have to prove $\{\omega \in \Omega;\frac{X}{Y}(\omega) \leq t \}\in F$ where $\Omega$ is the sample ...
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48 views

Evaluation of Standard Normal Integral

I have always wondered how we calculate the percentiles of the Standard Normal Distribution given that the CDF cannot be obtained in closed form: $$F(x)=\int_{-\infty}^{x} \frac{1}{\sqrt{2\pi}} ...
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114 views

Optimal combination of two estimates

I have a set of random variables, $X_1,\dots,X_N$. They are i.i.d. Gaussian with zero mean and $w$ variance. I observe $Y_1,\dots,Y_N$ where $Y_i=\sum_{j=1}^N a_{ij} X_j+N_i$ where all $a_{ij}$s are ...
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31 views

Box-Muller Type Problem

Let $U_{1}$, $U_{2}$ be two independent uniform random variables on (0,1). Let $\theta = 2\pi U_{1}$ and let $S=-\ln(U_{2})$. Show that S has an exponential distribution and $R=\sqrt{2S}$ has a ...
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Delta Method : Ambiguity in the proof by K. Knight

I am posting this message because I read the proof of the Delta Method (theorem 3.4) in K. KNIGHT, Mathematical Statistics, Chapman \& Hall CRC, 2000, p120-121. and there is something that ...
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Can the characteristic function of a multivariate normal distribution be extended from a neighborhood of the origin?

Let $x$ be a scalar random variable. There is a theorem that states that if $E[\exp(ixs)]= \exp\Big( i{s}\mu - \tfrac{1}{2} {\sigma^2s^2} \Big)$ for some neighborhood around the origin (i.e. ...
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Time after last jump and waiting time before the next jump of Poisson process

Consider $N =(N_t)_{t\geq0}$ a Poisson process of intensity $\lambda > 0$ and $(T_n)_{n\geq 1}$ its jump instants. Then consider for all $t \geq 0$, $Z_t = t- T_{N_t} \mathbb 1 _{\{ t \geq ...
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Conditional distribution of compounded Poisson process

Consider a Poison a process $N = (N_t )_{t\geq 0}$ of intensity $\lambda >0$ whose instants of jumps are $(T_n)_{n\geq0} $ $(T_0 =0)$ and a process $\tilde{N} =(\tilde N_t )_{t\geq 0}$ defined as ...
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90 views

Finding a Distribution When Introducing an Auxiliary Random Variable

Let (X,Y) be uniform on the unit ball; that is, $f_{(X,Y)}(x,y)=\begin{cases} \frac{1}{\pi}, &\text{if $x^{2}+y^{2}\leq 1$}\\ 0, &\text{otherwise.} \end{cases}$ Find the distribution of ...
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Chance of marrying a girl

My girlfriend's father has a magic - fair - coin, he agrees to let me marry his daughter if I play his game: I have to toss the coin couple times until I see the head comes up. Then if the number of ...
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21 views

Checking continuity result given a probability space and metric

Suppose we have a probability space ($\Omega$, $F$, $P$) is a probability space and $A_1,A_2 \in F$ and define the distance $d:F \times F \rightarrow \mathbb{R}$ by $d(A_1,A_2) = P((A_1-A_2) \cup ...
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73 views

Is there a surjection $f:[0,1]^{\omega} \to [0,1]$ such that the set of points that have non-unique primages is a null set in $[0,1]$?

Let $\Bbb H = [0,1]^{\omega}$, $\pi_i$ is the canonical projection that sends a sequence of elements in $[0,1]$ to its $i$th coordinate. The Borel structure $\mathcal{B}([0,1])$ and probability ...
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55 views

Approximation of Stochastic integral with Stieltjes integrals

Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is ...
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206 views

Probability of throwing balls into bins

You are throwing n balls into m bins randomly. What is the probability to be empty of the first $k$ bin? Given $k$ bins are empty. What is the probability to be empty of $(k+1)th$ bin? Forget the ...