Questions tagged [probability-theory]

For questions solely 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-distributions] for specific distribution functions, and consider [tag:stochastic-processes] when appropriate.

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Reference for a good multidimensional portmanteau theorem

I need references for a good n-dim portmanteau theorem that includes the following equivalent assertions: The sequence $(X_n)_n$ converges in distribution to a random vector $X$ of $\mathbb R^\nu$; $...
MikeTeX's user avatar
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Distribution of indefinite integral of Brownian motion multiplied by an exponential decay term w.r.t time

Consider the indefinite integral \begin{equation} I=\int_{0}^{\infty}f(t)e^{-t} W_t\mathop{dt}, \end{equation} such that $W_t$ is a standard Brownian motion. To ensure convergence, we'll suppose that ...
Jean Daviau's user avatar
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On the coupon collector problem

Originally from Proposition 8 of Tao's note: https://terrytao.wordpress.com/2015/10/23/275a-notes-3-the-weak-and-strong-law-of-large-numbers/comment-page-1/#comment-682885. Let ${N}$ be a natural ...
shark's user avatar
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Martingale example: a martingale on the partitioned unit interval

For an $L^1([0,1])$ function $f$, if we partition $[0,1]$ into intervals by setting $I_j = [x_j,x_{j+1}]$ with $0 =: x_0 < x_1 < \cdots < x_N := 1$ and let $\mathcal{F}$ be the sigma algebra ...
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How to expand this recurrence?

Here is a recurrence relation I have derived: $$E(Y_{r+1}^2) = E(Y_{r}^2) + 2E(Y_1)E(Y_r) + E(Y_1^2)$$ How can I get it in terms of $E(Y_1)$ and $E(Y_1^2)$? I've expanded the first few terms and I'm ...
user129393192's user avatar
3 votes
1 answer
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Poisson's distribution for probability

This question is more for learning purposes than anything however I came across this while trying to solving the following problem: The odds of winning the lottery are 1 to 50000 million. This week, ...
d0uble_a_b4ttery's user avatar
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Positive Correlation and the FKG Inequality

I am reading Chapter 2.2 of Percolation by Grimmett on the FKG Inequality and I am having a hard time visualizing the meaning of this inequality when I interpret it the statement in terms of integrals....
Shy's user avatar
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1 answer
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Do the stochastic integrals of two processes with same distribution also have same distribution?

I am trying to prove: Given two continuous functions $f,b$, and $X$ a stochastic process on $(\Omega,\mathcal{F},P)$, satisfying $X_t=\int_0^tb(X_s)ds+\int_0^tf(X_s)dW_s$, if $\hat X$ is a stochastic ...
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Control of Characteristic Function in Bounded Region?

I have a certain $n$-dimensional multivariate random variable $X$ I wish to study. I can compute its characteristic function, and can show that $$ \lVert \vec t\rVert_\infty \leq 1\implies \exp(-\frac{...
Mark Schultz-Wu's user avatar
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Construction of a coupling of a sequence of Bernoulli random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of Bernoulli distributed random variables defined on a probability space $(\Omega,\mathcal{F},P)$ and $(\mathcal{F}_n)_{n\in\mathbb{N}}$ be a filtration such ...
yannik0103's user avatar
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Random Walk Problem in Feller Probability Theory vol 2 Chapter XII page 425

Let $X_{i}$ be iid discrete random variable with distribution $P(X_{1}=-1)=q$ and $P(X_{1}=i)=f_{i}$ for all $i\geq 0$. Then consider the random walk $S_{n}=\sum_{i=1}^{n}X_{i}$ and $S_{0}=0$. Let $\...
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'Compactness' result for independent sigma algebras

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $A, B_1, B_2, \dots \in \mathcal{F}$. Let $\mathcal{F}_n := \sigma(B_1,\dots,B_n)$ and $\mathcal{F}_\infty := \sigma(B_i : i \in \...
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Additivity + $\sigma$-subadditivity implies countably additivity for set functions on semirings?

Let $X$ be a set, $\mathcal{S}\subseteq \mathcal{P}(X)$ be a semiring on $X$, and $\mu: \mathcal{S}\rightarrow [0,\infty]$ with $\mu(\emptyset)=0$. Assume $\mu$ is finitely additive and countably ...
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Supremum of average of i.i.d. stochastic processes

Suppose that $(X_t)_{t\in [0, T]}$ is a stationary stochastic process with continuous sample paths such that $\mathbb E[X_t] = 0$ and $\mathbb E[|X_t|^2] < \infty$, and assume that $$ S := \mathbb ...
Roberto Rastapopoulos's user avatar
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Prove that the number that the root of a given tree connect is decreasing in exponential.

Consider a regular rooted tree, each vertex has $d$ offsprings. Now for every edge there's a probability $p$ choosing it and probability $1-p$ not choosing it, and the root of the tree can connect ...
Ho-Oh's user avatar
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1 answer
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Conditional bakery problem

This question is from QuantGuide(Bakery Boxes): A bakery manager uniformly at random selects an integer k between 1 and 4, inclusive. He then chooses from k distinct desert types at his shop to create ...
Md Kaif Faiyaz's user avatar
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1 answer
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How to mathematically compute the transition probabilities in the secretary problem

My model for the secretary problem is as in (1) in this Mathstackexchange question. More precisely, I look at the unit interval $I=[0,1]$, and $\Omega = I^n \smallsetminus D$ where $D \subset I^n$ is ...
Chertopkhanov on Malek Adel's user avatar
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How to find $P\left(\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} > X_2 + X_1\right)$ where $X_1, X_2, Y_1, Y_2$ are uniformly distributed from [0,5]

How to find $P((X_2 - X_1)^2 + (Y_2 - Y_1)^2 > X_2 + X_1)$ where $X_1, X_2, Y_1, Y_2 \sim U[0,5]$ and iid. Rearranging the probability, we have: $$P\left(\sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2} > ...
Xerium's user avatar
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2 answers
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How to generalise inner product to measures without densities

Let $(E, \mathcal{E}, \lambda)$ be a metric finite measure space, and let $\mu, \nu$ be finite measures with densities $f,g$ with respect to $\lambda$. Then, I am interested in considering the ...
legionwhale's user avatar
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Distribution theory for random projections

Suppose $v$ is a fixed vector in $\mathbb R^n$, and let $u\in S^{n-1}$ (unit sphere in $n$ dimensions; $S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$) be uniformly generated. What is the distribution of $\...
Landon Carter's user avatar
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Rearranging sums (by multiplying out)

I know that to reverse the summation requires the double sum to either be absolute convergent or non-negative (but in the second case, it may diverge to $\infty$) I was wondering for something like ...
user129393192's user avatar
1 vote
1 answer
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Expectation for non-negative r.v

suppose I have $$\sum_{n=0}^{0} P(X=k)$$ Would this evaluate to $P(X=0)$ or $0$? In general, from my understanding: $$\sum_{n=0}^{k} P(X=k) = (k + 1)P(X=k)$$. Is this correct? I ask in this scenario, ...
user129393192's user avatar
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Kullback-Leibler Divergence between a random variable and the product of its entries

Problem Statement I'm currently working with a result about Kullback-Leibler divergence. Let $X$ be an discrete random variable taking values in $\mathcal{X} := \{0,1\}^p$, with $X = (X_1, X_2,...,X_p)...
Ollie's user avatar
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Which is the domain of the probability density function of $X^2$?

Let $X$ be a random variable with probability density function $$f_X(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}} \text{ for each } x \in \mathbb{R}.$$ We want to find the probability density function ...
Cyclotomic Manolo's user avatar
2 votes
2 answers
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Asymptotic convergence of sampling distribution of the sample variance

Let's consider a set $\{X_i\}_{i=1}^N$ of $N$ i.i.d. random variables drawn from the distribution $P_X(x) = \mathcal{N}(\mu, \sigma^2)$. Define the variable $$\hat{\sigma}^2 = \frac{1}{N} \sum_i (X_i -...
user1172131's user avatar
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Decomposing stationary point process $Y$ as sum of two point processes $X,Z$. Can $X$ be chosen stationary as well?

Consider the following setup. We are given two $[0,1]$-marked point processes on $\mathbb R$ or $[0,\infty)$, denoted $X,Y$. Denote that the ground processes (i.e. the point processes ignoring marks) ...
Václav Mordvinov's user avatar
3 votes
1 answer
52 views

Estimating the expected supremum of the absolute value of a Gaussian process

I'm currently reading a famous paper of Talagrand and fail to "easily see" that for a Gaussian process $(X_t)_{t\in T}$ and any $t_0 \in T$ one has $$ E \sup_T | X_t | \leq E |X_{t_0}| + 2E \...
n_flanders's user avatar
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Finite-Dimensional Elementary Cylinder vs. Finite-Dimensional Cylinder

I'm currently reading Theory of Probability and Random Processes by Koralov and Sinai. In Chapter 2, the authors state the definition of a finite-dimensional elementary cylinder and a finite-...
A. Sun's user avatar
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Dimension of a measure space?

Let's consider $\Omega = [0,1]^2$ and functions $f, g \colon \Omega \to [0,1]$ defined via $f(x, y) = x$ and $g(x,y) = y.$ Clearly, $f$ and $g$ are independent random variables that have uniform ...
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Adjoint operator and random variable

Let $(\Omega,\mathcal{A},\mathbb{P})$ be a probability space and $\mathbb{D}$ a dense subset of $L^2(\Omega,\mathcal{A},\mathbb{P})$. I consider a linear map $D$ from $\mathbb{D}\subset L^2(\Omega)$ ...
coboy's user avatar
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4 votes
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Why are Gaussian measures seen as the standard measure for infinite dimensional spaces?

I'm learning about infinite dimensional probability, and most resources I've consulted so far motivate things by saying there is no infinite dimensional Lebesgue/translation invariant measure that ...
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3 votes
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Best strategy for choosing valued items. (variant of secretary problem) [closed]

Suppose we have to fill our bag with items and each item has a value $V_i \sim \text{Uniform}[a,b]$ where each random variable is iid. There are $N$ items in total but only a fraction $x$ of all items ...
Eduard's user avatar
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5 answers
86 views

If you roll a dice four times, what is the probability to get two consecutive threes?

I was reviewing brainteasers and this one is stumping me for some reason. Let X be a non-3. you have 3 cases. 33XX, XX33, and X33X where we have the successes. I suppose you also have 333X and X333, ...
jd_h2003's user avatar
  • 101
2 votes
1 answer
55 views

Integrability of random variable

I am interested in proving that $$ \lim_{x \rightarrow \infty} E(|X| 1_{|X| > x}) = 0 \Longrightarrow E(|X|) < \infty \tag{1} $$ where $X$ is a real-valued random variable. In this blogpost the ...
harisf's user avatar
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Which spherical symmetric graphs are there?

I am interested which "relevant" infinite graphs G=(V,E) (or in my use cases called networks) are out there that are spherical symmetric, i.e. for a fixed origin 0 and each pair $v,w \in V$ ...
GG314's user avatar
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1 answer
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Continuous random variable, positive part and probability of $P(X=0)$

We all know that for a continuous random variable the probability of the variable taking a specific value is zero. However, I have come across the following example. Consider a standard normal random ...
FM89's user avatar
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0 answers
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Bivariate random variable and transformation

Let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be non-negative absolutely continuous random vector and if $\phi(X_j)=Y_j$, $j=1,2$, are one-one transformation then $$H[Y;\phi(t_1),\phi(t_2)]=H(X;t_1,t_2)-E[\log ...
Unknown's user avatar
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1 answer
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Prove or disprove there exists a Borel subset $F$ of $[0,1]$, such that $\mu(F)=a$ and $\mathcal{L}(F)=\frac{1}{2}$ [closed]

Given $\mu$ a Borel probability measure on $[0,1]$, and $\mathcal{L}$ the Lebesgue measure restricted on $[0,1]$. $$a:=\sup \{ \mu(E) : \mathcal{L}(E) = \frac{1}{2}, ~E \text{ is a borel subset of [0,...
well's user avatar
  • 174
2 votes
2 answers
91 views

Independence of a sequence converging in probability implies independence of the limit

Let ${X_1,X_2,\dots}$ be a sequence of scalar random variables converging in probability to another random variable ${X}$. Suppose that there is a random variable ${Y}$ which is independent of ${X_i}$ ...
shark's user avatar
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2 votes
2 answers
62 views

variance of $n$ bernoulli trials and central limit theorem

I'm trying to figure out how to calculate the variance of $n$ Bernoulli trials. I need this information to use the central limit theorem to calculate some probabilities. here are my thoughts so far. I ...
qwerty's user avatar
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0 answers
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What does it mean to deviate macroscopically from the mean? And what does it mean to deviate in terms of fluctuations?

Let $\{X_{n}\}_{n\in \mathbb{N}}$ be a sequence of random variables. Suppose that $\{X_{n}\}_{n\in \mathbb{N}}$ fulfills certain regularity conditions described below. The random variables $X_{n}$ are ...
Elias Costa's user avatar
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3 votes
1 answer
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Am I computing conditional expectations correctly?

I am building a working model and would like to know whether my computations so far look correct. I am working with a generally bivariate distribution of two variables $\displaystyle X\sim [ 0,1]$, $\...
Weierstraß Ramirez's user avatar
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1 answer
32 views

Find regular conditional distribution

Question 1.1) Let $X_1 = 1_{A}$ for $A\in \mathbb{F}$ and $X_2$ be a R.V. Where $X_2$ takes value in an arbitrary Borel Space. Let $X_1,X_2$ take values in $(A_1, \mathbb{A}_1)$, $(A_2, \mathbb{A}_2)$ ...
Overkill123's user avatar
3 votes
1 answer
118 views

Hilbert space under a mean value inner product

I am looking for a (Hilbert) space of (real-valued) functions on $\mathbb{R}^n$ where the following map defines an inner product: $$ (f,g) \mapsto \lim_{T\to\infty} \frac{1}{\mathrm{Vol}[-T,T]^n}\int_{...
Bram's user avatar
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0 answers
25 views

Independence of Hilbert-space-valued random variable

Let $X,Y$ be two independent random variables taking values in a separable Hilbert space $U$. Prove that $X$ and $Y$ are independent if and only if for all $u,v\in U$, $(X,u)$ and $(Y,v)$ are ...
George's user avatar
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1 vote
1 answer
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Nonstandard Probability Axiom Construction

I am working on extending the concept of a probability space to the surreal numbers, the main reason which being that I think that having a set whose measure is nonzero while the measure of each ...
opfromthestart's user avatar
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0 answers
19 views

Help understanding periodic Markov chain

I'm interested in a Markov chain with, say, 5 kernels $P_1, \dots, P_5$ that cycle deterministically over the time steps of the chain. This is nominally a time-inhomogeneous chain. According to this ...
caitlin's user avatar
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1 answer
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Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events.If $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$

Question: Let $(A_k)_{k \in \mathbb{N}}$ a sequence of events. Prove that if $P(A_k)$ does not converge to $0$ then $\exists$ an event belonging to an infinity of $A_k$ My answer: 1-Let writte $A'_1=...
OffHakhol's user avatar
  • 712
1 vote
0 answers
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Rate of convergence of conditional expectation

Let $X_{0}\sim N(0,1)$, $X_{1}$ another RV independent of $X_{0}$. Let $X_{t}=(1-t)X_{0}+tX_{1}.$ We know that as $t\to1,$ $\mathbb{E}[X_{0}|X_{t}]\to\mathbb{E}[X_{0}|X_{1}]=E[X_{0}]=0.$ Can we ...
Santiago Aranguri's user avatar
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1 answer
70 views

Convergence in probability and $E(|X_n|) \to E(|X|)$ implies convergence in mean

It is well known that in probability theory that, if $X_n$ converges in probability to $X$, the following are equivalent. $X_n$ converges in mean to $X$ $E(|X_n|) \rightarrow E(|X|) < \infty$ $\{...
Randy Lai's user avatar

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