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 ...

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multivariate convergence in law

Suppose $X_n \overset{\mathscr{L}}{\longrightarrow} X$, and $Y$ is another random variable which may be depending on $X_n$. Then it seems not true that we have the following joint convergence in law ...
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25 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
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69 views

If $X$ is such that $c P(X\geq c) \leq E(Y; X\geq c)$ for every $c$ then so is $X\wedge n$

I came across the following as I was reading the book Probability with Martingales by Williams (pg 143): Assumption: $X$ and $Y$ are non-negative random variables such that $$c P(X\geq c) \leq ...
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2answers
46 views

Resampling operation

I am reading from an arXiv.org paper the following math text: "Let $x\in \{−1, 1\}^I$ be random and uniform, and let $y$ be obtained from $x$ by resampling each coordinate with probability ...
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61 views

Show that $\frac1n\log X_n$ converges almost surely

Let $X_0$ follow $\mathrm{Uniform}(0,1)$. Define $X_{n+1}$ iteratively as $X_{n+1}$ follows $\mathrm{Uniform}(0,X_n)$, $n\geq0$. Show that $\dfrac{\log X_n}{n}$ converges almost surely and find the ...
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25 views

If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$

Let $\{X_k\}_{k=1}^n$ be iid random variables that are symmetric around $0$ i.e. $X=-X$ in distribution. Define $S_n=\sum_{i=1}^nX_i$. Then show $P(|S_n|\geq\max_{1\leq i\leq ...
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1answer
37 views

Show that $S_n$ converges almost surely to $\infty$

Suppose $X_n$ are independent random variables with $P(X_n=-n^2)=\dfrac{1}{n^2}$, $P(X_n=-n^3)=\dfrac{1}{n^3}$ and $P(X_n=2)=1-\dfrac{1}{n^2}-\dfrac{1}{n^3}$ for all $n\geq2$. Let ...
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2answers
66 views

piecewise weak convergence in $C[0,1]$

Let $P$ and the sequence $P_n$ be probability measures on $C[0,1]$ with the uniform metric. Fix $0<u<1$ and let $\Pi_1$ and $\Pi_2$ be the projections from $C[0,1]$ onto $C[0,u]$ and $C[u,1]$, ...
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1answer
36 views

Measurability of a version of a random variable

If $X$ is a ($\mathcal{F}$-measurable) random variable defined on a probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and $Y$ is a version of $X$ in the sense that $\mathbf{P}(X \ne Y) = 0$ and ...
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96 views

Zero conditional entropy

This question is related to this math.se question but I need a bit more guidance. For two discrete random variables $X,Y$ we define their conditional entropy to be $$H(X|Y) = -\sum_{y \in Y} Pr[Y = ...
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2answers
74 views

expected values of identically distributed random variables

Let $X$ and $Y$ be identically distributed random variables on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. Then, if I let $F_X$ and $F_Y$ denote the distribution functions of $X$ and ...
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47 views

Every random variable $X$ can be written as $X=\lambda Z_1+(1-\lambda)Z_2$, for $Z_1$ discrete and $Z_2$ continuous random variables.

Show that every random variable $X$ can be written as $$X=\lambda Z_1+(1-\lambda)Z_2$$ for a discrete random variable $Z_1$, a continuous random variable $Z_2$, and a real value $\lambda$. This ...
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1answer
76 views

Taylor expansion of characteristic function in probability theory

In probability theory, what is the Taylor expansion of characteristic function? I know this is a basic question but I couldn't find a full answer.
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33 views

Does the following result require the random variables to be independent?

I am sitting with the book Labelled Markov Processes by Prakash Panangaden, and on page 79 he defines what it means for a set of random variables on a probability space to be independent, and after ...
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2answers
63 views

Which one of the following versions of Bayes' theorem is correct?

I've seen two versions of Bayes' theorem: I've seen this very long version from a frequentist probability class: $$ P(B|A)=\frac{P(A|B)P(B)}{P(A|B)P(B) + P(A|B^c)P(B^c)} $$ where $B^c$ is the event ...
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48 views

A question on $\pi-\lambda$ system

I was reading Rick Durett's Probability Theory and Examples. In the proof of the $\pi-\lambda$ theorem i.e If $\mathcal{P}$ is a $\pi-$system and $\mathcal{L}$ is a $\lambda-$system that contains ...
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1answer
26 views

How is this an application of the independence property of events?

I'm currently working my way through Klenke's book on probability theory and do not understand a step in his proof of the Borel-Cantelli lemma (Theorem 2.7): The assertion is that for an independent ...
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1answer
14 views

Stochastic domination preserved by dilution?

Consider an at most countable set $S$ and the corresponding bit space $\{0, 1\}^S$ that is often considered in percolation, interacting particle systems, and other lattice models. Suppose that $\le$ ...
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61 views

Expectation in measure theory

I'm reading a book on measure-theoretic probability, and the author defines the expectation of a random variable $X$ on a probability space $(\Omega,\scr H,\mathbb{P})$ as $\int_\Omega Xd\mathbb{P}$, ...
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27 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
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15 views

$\mathbb{P}(d(X,Y)>\alpha)<\beta$ if $\mathbb{P}(X\in E)\leq \mathbb{P}(Y\in E^{\alpha})+\beta$ for all measurable E

Given two random variables X,Y with measures P,Q (i.e. $P(A)=\mathbb{P}(X\in A)$). Show that if $P(E) \le Q(E^\alpha) + \beta$ for all measurable $E\subset\mathbb{R}$ then ...
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33 views

Almost surely diverging sum implies almost surely diverging sum of conditional expectations?

Suppose $\sum_{n=1}^\infty X_n = \infty$ almost surely for nonnegative $X_n$. Let $\mathcal F_n = \sigma(\{X_0, X_1, \ldots, X_n \})$. Can we show that $\sum_{n=1}^\infty \mathbf{E} (X_n | \mathcal ...
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31 views

Conditional independence expansion

I have four random variables A,B,C and S. A,B and C are conditionally independent given S. So, I need to obtain P(A,B,C,S) By the chain rule: $$P(A,B,C,S)=P(S)P(A|S)P(B|A,S)P(C|A,B,S)$$ By the ...
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41 views

Expectation of quadratic variation

I got stuck in a step of a proof and need some help. The situation is the following: Let $M$ be a continuous local martingale (which satisfies $\mathbb{E}[\langle M\rangle(T)]<\infty$ - I don't ...
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1answer
25 views

Probability measure on the space of $n \times n$ symmetric matrices with non negative integer coefficients

I know that there exists a particular measure, called Haar measure, defined on random matrices, i.e. $n \times n$ orthogonal complex matrices. My question is the following: can we define a ...
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1answer
40 views

Independence of random variables

Let $\{X_n\}$ be a sequence of independent random variables on some probability space. Then, by definition(according to the book that I am reading), I know that $\{\sigma(X_1),\sigma(X_2),\dots, \}$ ...
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1answer
33 views

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent?

Let $X$ denote the number of tosses required to get the 5th head and $Y$ the number between the 6th and 7th heads. Are $X$ and $Y$ independent? Y will always depend on X . NO ? i know geometric ...
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1answer
23 views

X denotes government will increase payment. x~Bin(2,2/3) . if one increment =9%. expected increment =?

If Government increases payment then they increase it by 9% . now if whether government will increase payment follows binomial distribution with parameters n=2 and p=(2/3) , then what percentage of ...
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1answer
81 views

Almost sure convergence of nonrandom sample

This is a question about almost sure convergence. Consider the following set-up: There are $B$ banks. Each has size $S_{b}$, which follows a size distribution $f_{S}$ with mean E[S]. $f_{S}$ is ...
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1answer
116 views

Can we have a Brownian motion under two different probability space?

Is it possible to construct a stochastic process $B_t$ such that $B_t$ is a Brownian under $(\Omega, F, P)$ and $B_t$ is a Brownian under $(\Omega, F, \hat{P})$? If not, how to argue that $P=\hat{P}$? ...
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1answer
46 views

how that $P(G)=1$ iff $\sum_n \Bbb P(A \cap E_n )=\infty$ for all events $A$ having $\Bbb P(A)>0$.

Two probability problems: 1. Let $a>0$ and let $X_n$, $n \geq 1$, be iid r.v. that are uniform on $(0,a)$ and let $Y_n = \prod_{k=1}^{n} X_k$. Determine all values of $a$ for which $\lim_{n ...
2
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1answer
41 views

Adapted random variable

Let $\{Y_n\}_{n=1}^{\infty}$ be a sequence of random variables, and let $F_n = \sigma(Y_1,Y_2,\dots,Y_n)$ for each $n$. Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of random variables adapted with ...
4
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1answer
90 views

Uniformly integrability and convergence

Question 1: $X_n$'s are non-negative, uniformly integrable. Then $E\left[\dfrac{\max_{1\leq k \leq n} X_k}{n}\right]\rightarrow 0$. Question 2: If u.i. is dropped then above the may fail. My ...
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1answer
40 views

Existence of moments and slowly varying function at infinity

I have a somewhat advanced question involving the role of slowly varying functions and their relation to moments. I want to use them to derive certain results for domains of attraction. My problem is ...
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0answers
37 views

How to determine law of a random variable from its cumulative distribution

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, and let $X$ be a random variable. Suppose that we are given the distribution of $X$, denoted by $F_X$. i.e. $F_X(x) = P(X \leq x)$. ...
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1answer
64 views

Probability via Geometry, applications and examples

People, I am going to teach a lesson including something about Probability via Geometry and, if you have, I would like to know some references or materials (or even some good ideas) that can help me. ...
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1answer
39 views

Independence of a random variable and its conditional expectation

Let $(\Omega, \mathcal{F},\mathcal{P})$ be a probability space. Let $\mathcal{H} \subset \mathcal{F}$ be a sub $\sigma$-algebra, and let $X\in L^1(\Omega, \mathcal{F},\mathcal{P})$ be a random ...
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3answers
69 views

Is convergence in probability sometimes equivalent to almost sure convergence?

I was reading on sufficient and necessary conditions for the strong law of large numbers on this encyclopedia of math page, and I came across the following curious passage: The existence of such ...
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1answer
91 views

Integration with respect to a concrete measure

I got the problem of integrating with respect to a measure in concrete detail. Im just finding formal stuff elsewhere. The measure $Q(A)=\int_0^\infty P(f(r,X)\in A)dr$ is given and i need to show ...
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1answer
104 views

Coupon collector problem doubts

The Coupon Collector problem off Wikipedia: Suppose that there is an urn of $n$ different coupons, from which coupons are being collected, equally likely, with replacement. How many coupons do you ...
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71 views

Bounding Expected Value of a piecewise function

Let X and Y to be two independent random variables with known pdfs. Get a bound for the expected value of the following expression in terms of $E[X]$, $E[Y]$, VAR[X] and VAR[Y]: \begin{equation} ...
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29 views

The domain of the sum rule(probability: The logic of science)

Anyone read Probability Theory: The logic of Science. Please help, I've been really stuck for ages at how the sum rule has it domain derived and I don't have any teacher to ask. Question How is ...
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1answer
47 views

Inequality in proof of SLLN

This comes from theorem 5.1.2 of KL Chung's A Course in Probability Theory. Suppose ${X_n}$ are uncorrelated and their second moments have a common bound. Then For each $n \ge 1 $, $D_n:= ...
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2answers
65 views

Average time until one disk drive fails

I have two hard drives. The given Mean to Failure Time is $100$ hours. So if I have one hard drive, the average amount of time until it fails will be $100$ hours. ...
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41 views

Finding a polynomial approximation of a PDF

I would like to find a polynomial $P(x)=\sum_{d=1}^D P_dx^d$ of degree $D$, where its derivative is larger than or equal to a given pdf $f(x)$ in $[0,1-\epsilon]$, for any $\epsilon>0$. Note that ...
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2answers
65 views

Question about Kolmogorov extension theorem

I need some help understanding the relationship between the following two theorems Theorem 1: Let $\{\mu_n\}$ be a sequence of probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$, where ...
3
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1answer
35 views

$\left \| \left \| f \right \|_{L^{p}} \right \|_{L^{q}} \leq \left \| \left \| f \right \|_{L^{q}} \right \|_{L^{p}} $ for $0<p\leq q$

Let f be bounded on $X\times Y$ measure space with $\mathbb{P}\times\mathbb{Q}$ probability measure, show that for $0<p\leq q$: $\left \| \left \| f \right \|_{L^{p}(\mathbb{P})} \right ...
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1answer
56 views

$\limsup \frac{|S_n|}{n}=\infty$

$X_n$'s are i.i.d symmetric with $E|X_1|=\infty$. Then $\limsup \frac{|S_n|}{n}=\infty$. How do I show $\limsup \frac{S_n}{n}=\infty$ and $\liminf \frac{S_n}{n}=-\infty$? My attempt: Let $c=\limsup ...
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81 views

Proof - Limits of CDF

For a cdf, defined as $F(x)=P(X\le x)$, in order to prove $\lim\limits_{x\,\uparrow\,\infty}F(x)=1$, I've two concerns: (1) Some concern about a proof from a book, and (2)Validity of a proof that I've ...
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2answers
56 views

What's the probability that a particle jumps out of an interval?

Suppose I have a small particle and put it on the center of an interval on a 1-D axis. If the particle undergoes a motion that satisfies: It chooses its direction freely and randomly It ...