Questions tagged [probability-theory]

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

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How to see that the Wick product has $0$ expectation.

In the book "Gaussian Hilbert Spaces" (Svante Janson) the author introduces the Wick product of a finite sequence of $n$ random variables living in a Gaussian Hilbert space $G$ as the orthonormal ...
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Continuous time strong approximation (in the sense of Komlós--Major--Tusnády)

The Komlós--Major--Tusnády result asserts that, given a sequence of i.i.d. random variables $X_1,\dotsc,X_n$, one may extend the probability space and find normal i.i.d. variables such that the ...
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Convolution of two Cardioid and vonMises PDFs

for the past few days, I've been "on and off" with Mardia's and Jupp's "Directional Statistics" to learn something new about approximating circular distributions. In particular, I've been looking at ...
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Question about random variables convergence

Further going through old lecture notes I've stumbled upon this... Let's say we are dealing with a sequence of random variables $\{X_n\}_{n=1}^\infty$ such that $\sqrt{n}(X_n-1)\to N(0,2)$ in ...
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Prove the existence of a finite additive continuation for measure

Prove that for any algebra $A$, every finitely additive measure defined on A has some finitely additive extension to the $σ$-algebra generated by $A$.
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A line up question where the position and the number itself should not equal in an arrangement

There are $n$ numbers, from $1$ to $n$. Suppose that the numbers are arranged randomly, and asks there is no arrangement where the position and the number itself are equal. How many arrangements ...
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Are “independent events” in probability really independent? [closed]

This is a hard and deep question. I understand very well the concept of independence. But, let us take two events: Event A (I throw a dice) and event B (some star explodes in an near galaxy). Are ...
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Stationary distribution of a Markov chain on the nonnegative integers

Let $\lambda_k,\mu_k\in\mathbb R_{\ge0}$ $(k\ge1)$ be nonnegative real numbers such that $\sum_{k=1}^\infty k\lambda_k<\infty,$ let $S=\mathbb Z_{\ge0}$ be the nonnegative integers, let $T=\mathbb ...
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Limit of expected value of supremum and infimum

Let $(\Theta,d)$ be a metric space, $\mathcal{X}\subset\mathbb{R}^n$ and $f:\Theta\times \mathcal{X}\to\mathbb{R}$. Define $X$ as a random variable in $\mathcal{X}$. Suppose that the mapping $\theta\...
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Does joint distribution affect Radon-Nikodym derivative?

Given two real-valued random variables $X, Y$ with distributions $\mu_X, \mu_Y$. Suppose $\mu_X<\!<\mu_Y$, then the Radon-Nikodym derivative $\frac{d\mu_Y}{d\mu_X}(\cdot)$ exists $\mu_X$-a.e. on ...
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If $\lim_m \mathbb{P}(\{\sup_{j\geq 1}{|S_{m+j}(t)-S_{m}(t)|}\leq \epsilon\})=1$ then $\mathbb{P}(\{S_n(t)\text{ is a convergent sequence}\}=1$

Let $(X_n)$ be a sequence of random variables in Probability space. Take $S_k=\sum_{n=1}^{k}{X_k}$. We suppose that : for all $\epsilon >0$ $$\lim_m \mathbb{P}(\{\sup_{j\geq 1}{|S_{m+j}(t)-S_{m}(...
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I don't know how to solve this question, could someone help me?? [closed]

I don't know how to solve this question, could someone help me?? A telemetry system transmits 1 million bits. Each bit 0 or 1 is independent with equal probability. Estimate the probability of ...
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How to construct a joint pdf, other than the independent coupling, given two marginal pdf's?

I am trying to find joint densities for which the marginal densities are $f_U(u) = 2\exp(-2u), u\geq 0$ and $f_V (v) = \exp (-v), v \geq 0 $. Of course I can take the joint pdf $f(u,v) = f_U(u) \cdot ...
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General form of the open mapping theorem

Let $X,X_1,X_2,...$ be real valued random variables on the same probability space $(\Omega, \mathcal{F},\mathbb{P})$. Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function.We know that ...
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If $W$ is a Brownian motion, then how can I justify a statement about $\Delta W_{t_{i+1}^{n}}^{2}-\Delta t_{i+1}^{n}$

Let $W$ be the standard Brownian motion with $$ \Delta W_{t_{i+1}^{n}} := W_{t_{i+1}^{n}}-W_{t_{i}^{n}}\; \; \;\operatorname{and}\; \; \; \Delta t_{i+1}^{n}:=t_{i+1}^{n}-t_{i}^{n}.$$ In lecture notes,...
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Show that $\log{M_n}/\log{n}\to1$ a.s. where $M_n = \max\{X_k \mid 1 \leq k \leq n\}$ and the $X_n$ are iid with $\mathbb{P}(X_n \geq i) = 1/i$.

Let $(X_n)_{n \geq 1}$ be a family of independent, identically distributed integer valued random variables with $\mathbb{P}(X_n \geq i) = 1/i$ for each $i \geq 1$. For each $n$, define $M_n = \max\{...
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If $X:\Omega\times[0,\infty)\times E\to E$ is a stochatic flow, is $\kappa_t(x,B):=\operatorname P\left[X^x_t\in B\right]$ a semigroup?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(E,\mathcal E)$ be a measurable space and $X:\Omega\times[0,\infty)\times E\to E$ be a stochastic flow, i.e. $X$ is $(\mathcal A\...
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Some doubts about proof of Strong Law of Large Numbers

I quote Jacod-Protter. Theorem: Let $\left(X_n\right)_{n\geq1}$ be independent and identically distributed and defined on the same space. Let$$\mu=\mathbb{E}\{X_j\}$$ $$\sigma^2=\sigma_{X_j}^2<\...
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Computation of the quadratic variation of Wiener Process.

My confusion arose from a commonly mentioned exercise: Show that the quadratic variation of Wiener Process is $\langle W\rangle_{T}=T$. Note that the quadratic variation here is the non-...
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How can I compute $\int_{\mathbb R}f(x,y)\mathbb P(X\in dx, |X|\in dy)$?

Let $X\sim \mathcal N(0,1)$. I want to find the join distribution of $(X,|X|)$ is given by $$F(x,y)=\mathbb P(X\leq x,|X|\leq y)=\mathbb P(X\leq x,-y\leq X\leq y)=\mathbb P(-y\leq X\leq \min(x,y))$$ $...
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Equality of expectations if identically distributed

Let $X,Y,Z$ be independent random variables on some probability space $(\Omega, P)$ with values in $\mathbb R$ such that $X \sim Y$, i.e. $X$ and $Y$ have the same distribution. Let $f \colon \mathbb ...
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How do concepts such as limits work in probability theory, as opposed to calculus?

When I am flipping a fair coin and say that as the number of trials approaches $\infty$ the number of heads approaches $50\%$, what do I really mean? Intuitively, I would associate it with the ...
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25 views

$f(X_n)\to f(x_0)\Rightarrow X_n\to x_0$

Let $f:\mathcal{X}\subset\mathbb{R}^n\to\mathbb{R}$ be a continuous function with unique maximum $x_0\in\mathcal{X}$, i.e., $$ x_0 = \text{argmax}_{x\in\mathcal{X}}f(x)\quad\text{and}\quad f(x_0)>f(...
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Proving Etemadi's inequality

Consider $\{X_i\}^n$ independent random variables in some Banach space, for all $t\geq0$ we have: $$P\left(\max_k \left\|\sum^k_iX_i\right\| > t\right) \leq 3 \max_kP\left(\left\|\sum^k_iX_i\right\...
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Relationship Between Entropy of a Measure and Hausdorff Dimension of Its Support

In this paper by Chhabra and Jensen, the authors make the claim (based on a theorem by Eggleston which is proven in this paper) that "for a special class of measures $\mu$ that arise from ...
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Using Chapman-Kolmogorov Property to prove v=Qv

How would you use the Chapman-Kolmogorov property ($Q_{t+s}=Q_tQ_s$) to prove that v (a column vector distribution over the sample space) is a stationary distribution of Markov Chain $X_t$ with ...
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Stationary Distribution of Ehrenfest Markov Chain

The example in my book for an Ehrenfest Markov chain is: A system of of two urns, A & B where there are 2n balls total in both urns. We are assuming that there are $i$ balls in urn A and $2n - i$ ...
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1answer
22 views

Expectation of Inverse Normal CDF

Suppose a r.v. $\mu$ is distributed Normal $N(\theta,\sigma^2)$. Is there any way to derive the expectation $\mathbb{E}(\frac{\mu}{\Phi(\mu)})$ where $\Phi$ is the CDF of a standard Normal random ...
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Poisson Arrival Process and Uniform Distribution

I'm brushing up on some basic probability and have this question: If we have a Poisson arrival process with arrivals $A_{1}, A_{2}, \dots$, and we know that there is one and only one arrival in a ...
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Conditions for Weak and Strong law of Large Numbers

I am a second-year undergraduate student and was reading up the topics of the law of large numbers from Casella and Berger's Statistical Inference. They state the laws as follows: The link to the ...
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1answer
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$f, g$ are probability density functions of an normal distribution N(0,1), prove h is $N(0,\sqrt 2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. When $f$ and $g$ are $exp(\lambda)$ it's solved by $\int_{0}^{...
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Why (multi marginal) optimal transport?

I recently learned about optimal transport (OT) and its generalization to comparing multiple distributions jointly, called multi-marginal optimal transport (MMOT) In a nutshell, the OT does $ \...
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Counterexample for continuous time sub-martingale convergence

We know that if $X_t$ is a right continuous sub-martingale with $\sup_t \mathbb{E}[X_t^+] < \infty$ then $\lim_{t \rightarrow \infty} X_t$ exists almost surely, but I haven't been able to find a ...
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confusion regarding pdf?

I am reading a question regarding probability I have attached snaps of both question and answer I am confused regarding encircled equations of answer especially how we proceed from 2nd row/equation ...
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$f, g$ are probability density functions of an exponential distribution, prove h is $\gamma (\lambda ,2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. Then is asked: $f, g$ are probability density functions of an ...
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Quadratic variation of true martingale

We know that for a continuous local martingale $M$ the quadratic variation $\left<M \right>$ is such that $M^2- \left< M \right>$ is a continuous local martingale. Is it true that if $M$ ...
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$L^p$ Boundedness of a martingale

So I recently read a paper where the authors claim that if for some martingale $(M_t)_{t\geq 0}$ we have $$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$ ...
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Probability of $\epsilon$-ball in Wiener space

Let $f \in C(\mathbb{R}^n)$ and equip this space with the classical Wiener measure $\nu$. Let $\epsilon>0$ be given. How can we compute $$ \nu\left( \left\{g \in C(\mathbb{R}^n):\, d(g,f)\leq \...
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Does almost sure convergence imply almost sure convergence of conditional expectations?

So I'm looking for a counter example to show : $X_n\xrightarrow{a.s.} X \not\Rightarrow E(X_n|\mathcal{D})\xrightarrow{a.s.} E(X|\mathcal{D}).$ I thought maybe it would be helpful if $X_n \downarrow ...
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Describe all martingales that only take values ​in $\{−1, 0, 1\}$.

Describe all martingales that only take values ​​in $\{−1, 0, 1\}=:\Omega$. In the first instance i would try to find a filtration of $$P(\Omega)=\{\emptyset,\{0\},\{1\},\{-1\},\{1,-1\},\{1,0\},\{-1,...
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Stochastic processes - Why do we need filtration?

In the theory of stochastic process, besides the $\sigma$-algebra $\mathcal {F}$, we have an increasing sequence of $\sigma$-algebras $\{{\mathcal {F}}_{{t}}\}_{{t\geq 0}} $ called filtration. ...
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If $\gamma$ is a coupling of $\delta_x$ and $\delta_y$, can we show that $\int f\:{\rm d}\gamma=f(x,y)$?

Let $(E,\mathcal E)$ be a measurable space, $\pi_i$ denote the projection of $E^2$ onto the $i$th coordinate, $\delta_x$ denote the Dirac measure on $(E,\mathcal E)$ at $x$ for $x\in E$ and $\gamma$ ...
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1answer
22 views

Proving that $ \sup E[X_n] \geq E[X]$

Consider the sequence of random variables $\{X_n\}_{n\geq1}$ such that $X_n$ are non-negative and $X_n \rightarrow X$ almost surely, with $\sup E[X_n]<\infty$. Prove $E[X]\leq \sup E[X_n].$ My ...
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If $t\mapsto X_t$ is continuous almost everywhere and $(X_t)$ has independent increments, then $X_t - X_s$ follows a normal distribution?

The following statements can be found at Glasserman's Monte Carlo Methods in Financial Engineering. Given a stochastic process $(X_t)_{t\in [0,T]},$ if the mapping $t\mapsto X_t$ is continuous ...
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Non-uniqueness in the $L^1$ martingale representation

Let $\xi \in L^1(P,\mathfrak F_T)$ on some probability space with measure $P$, supporting a Brownian motion, we consider the augmented filtration $\mathfrak F$ associated to $W$, and a time $T>0$. ...
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27 views

$E[XY]=E[ZY]$ implies $X=Y$ almost surely? [closed]

Consider the hypothesis: $X,Z$ random variables in $L_1$ and $Y$ bounded random variable. Can I prove the following result with the above hypotheses? If $E[XY]=E[ZY]$ (for each bounded r. v., Y) $\...
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Any normal random variable is ''essentially'' surjective

Let $\Phi$ be the cumulative distribution function of the standrad normal distribution. Denote $X: (0,1) \rightarrow \mathbb R$ its inverse. Then $X$ is a standard normally distributed random variable ...
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Negative moments of variables

Let $f_i, i=1, \ldots, n$ be independent Steinhaus random variables, i.e. variables which are uniformly distributed on the complex unit circle. Let $a \in R^n$. Find $E\left(\sum_{i=1}^nf_i a_i\...
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23 views

chernoff bound application

say we sample a finite set a finite number of times (ie we have finitely many iid random variables $X_1,...,X_k$ taking values in a finite set). i read that the multiplicative chernoff bound can be ...
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29 views

A proof that $L^{\infty}$ is complete

I know this has been proved in other links, but I am wondering about the validity of the following proof: Suppose $X_n$ is a Cauchy sequence in $L^\infty$. Then there exists a subsequence $Y_k \...

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