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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0
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1answer
28 views

For an invertible measure preserving system, $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$

For an invertible measure preserving system, show that $\lim_NA_f^T(N)=\lim_N A_f^{T^{-1}}(N)$. Here we consider the measure preserving system $(X,\mathcal A,\mu,T)$ where $T$ is invertible and $\mu$...
1
vote
1answer
39 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
-2
votes
0answers
19 views

Probability calculation [closed]

There are n number of data. n data contain x ideal data and y raw data. Question: how to select randomly one ideal data. Please give me calculation with explanation. Thanks in Advance
3
votes
1answer
47 views

Law of large numbers for moving mean

Consider the following process: For $n = 1,\ldots$ $U_n \sim U[0, 1]$, that is, uniformly distributed on $[0, 1]$, $X_n = U_n 1_{U_n > q_n}$, where $q_n = \frac{1}{n-1} \sum_{i=1}^{n-1} X_i$, ...
5
votes
3answers
72 views

Can $Y$ and $\frac{X}{Y}$ be uncorrelated if neither $X$ or $Y$ is constant?

Suppose I have two variables $X$ and $Y$ with $Y>0$. Can the random variables $Y$ and $\frac{X}{Y}$ ever be uncorrelated, i.e., $$\mathbb{E}(X)=\mathbb{E}(Y)\mathbb{E}\left(\frac{X}{Y}\right).$$ ...
0
votes
1answer
20 views

Question regarding probability density function?

I was going through the concept of probability density functions and had a small confusion about the notation that a pdf can take values greater than one.I found this How can a probability density be ...
2
votes
1answer
40 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
4
votes
2answers
33 views

Distribution of throws of die rigged to never produce twice in a row the same result

A die is “fixed” so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probablity 1/5. If the first score is 6, what is the probability that the ...
3
votes
0answers
36 views

Comparing different definitions of tightness for measures

Let $X$ be a Hausdorff space, $\mathcal{B}(X)$ the Borel $\sigma$-algebra and $\mu : \mathcal{B}(X) \to [0, \infty]$ a measure. Consider the following properties: (1) $\forall A \in \mathcal{B}(X): \...
1
vote
1answer
23 views

Show that MLE estimator convergences in probability to actual parameter

For iid stochastic variables $X_1, ..., X_n$ whose distribution is defined by 2 parameters, I have found the MLE estimators. They are $\hat{\mu} = \sum x_i/n$, and $\hat{\lambda}$ given by $$ \frac{...
2
votes
1answer
20 views

Probability: Find Dispersion of X + Y

$X = \operatorname{Bi}(3,\frac14), Y=\operatorname{Bi}(4,\frac12), \operatorname{Cov}(X,Y) = -\frac34.$ Dispersion of $X+Y =?$ $D(X) = npq = \frac9{16}. D(Y) = npq = 1$ $D(X + Y) = D(X) + D(Y) = \...
0
votes
1answer
80 views

weak L1 convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
2
votes
1answer
35 views

An ancillary result from convergence in probability

I was reading a paper concerning probability theory. We have that $X_i$, $i = 1,2,...$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $$\frac{S_n}{n} \...
2
votes
5answers
87 views

Compare $\mathbb{E}[XY]\mathbb{E}[XY]$ with $\mathbb{E}[X]\mathbb{E}[XY^2]$

$\newcommand{\E}{\mathbb{E}}$So this was a question asked to me in an interview where $X$ and $Y$ are two random variables and I was asked to compare the $\E[XY]\E[XY]$ with $\E[X]\E[XY^2]$ . The ...
-1
votes
1answer
31 views

limit superior and law of large numbers [on hold]

I am wondering whether the following result is true: Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ ...
0
votes
1answer
27 views

Isometric embedding of $\ell^2$ into $L_1$.

Let $\{Y_n\}_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables on some probability space $(\Omega, \mathcal{F}, P)$ following a standard complex Gaussian distribution (that is, the ...
-1
votes
1answer
18 views

Converse to continuous mapping theorem [closed]

If $X_n\xrightarrow{d}X$, and $g$ is a.s. continuous, then $g(X_n)\xrightarrow{d} g(X)$. What if we know that $g(X_n)\xrightarrow{d}g(X)$, and $g$ is some continuous function. Can we claim that $X_n\...
1
vote
1answer
47 views

What is the uncertainty of a discrete sum given the uncertainty of an individual element?

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $...
0
votes
0answers
22 views

No point masses in a conditioned distribution

Assume you have a probability space $(\Omega, \mathcal{F},\mathbb{P})$ and a filtration $(\mathcal{F}_t)_{t=0}^T$ where $\mathcal{F}_T=\mathcal{F}$. Suppose there are given two real valued random ...
0
votes
0answers
42 views

Jensen's inequality for two random variable

Prove: Let $X$ and $Y$ be two random variables in probability space $\left ( \Omega ,\mathcal{F},\mathbb{P} \right )$ , and $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a convex function, then $$f\left ( ...
1
vote
1answer
23 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
3
votes
1answer
92 views

Lindeberg CLT application

Let $X_1,X_2,\dots$ be a sequence of independent Random Variables with $$\mathbb{P}(X_k = -1) = \mathbb{P}(X_k = 1) = \frac{1 - 2^k}{2}$$ $$\mathbb{P}(X_k = 2^k) = \mathbb{P}(X_k = -2^k) = \frac{1}{2^{...
1
vote
0answers
63 views

Sum of random variables that are independent but not identical [closed]

For a real number $t$, let $X_t$ be the random variable that is uniformly distributed in the interval $[t/2, 3t/2]$. If $\{t_n\}$ is a sequence of positive real numbers, is there anything we can say ...
1
vote
0answers
18 views

What are the applications of quantum shuffle multiplication?

I am writing a research paper on quantum shuffle multiplication, and there is just one piece that I'm missing: I want to give one or several examples of its applications, and so far I have not been ...
0
votes
0answers
81 views

brownian bridge and supremum

I want to show that: $$ \sup_{u \geq 0} \frac{1}{u} \left( | B_u | - 1 \right) = \sup_{u \geq 0} \left( |B_u| - u \right) = \sup_{0 \leq u \leq 1 } b_u^2 $$ in distribution; with $ B_u = (1+u)b_{\frac{...
0
votes
3answers
82 views

Probability: Prove that events are independent

I'm self-learning probability and struggle on the following task: If $A$ and $B$ are independent events, prove that $A \cup B$ and $A \cap B$ are also independent. This is one of those cases where ...
2
votes
1answer
179 views

Find the almost sure limit of $X_n/n$, where each random variable $X_n$ has a Poisson distribution with parameter $n$

$X_{n}$ independent and $X_n \sim \mathcal{P}(n) $ meaning that $X_{n}$ has Poisson distributions with parameter $n$. What is the $\lim\limits_{n\to \infty} \frac{X_{n}}{n}$ almost surely ? I ...
-1
votes
0answers
57 views

Probability: How do I prove this inequality? [closed]

While studying probabilities and I have encountered this inequality. I'm trying to prove that for any random variable $X$ and any $\epsilon \gt 0$ this inequality is correct. $P(|X - EX| \ge \epsilon)...
0
votes
1answer
39 views

brownian bridge definition [closed]

I am trying to solve an exercise and I have trouble with the definition of brownian bridge. "Let (bu , 0 ≤ u ≤ 1) be the Brownian bridge derived by conditioning a one-dimensional Brownian motion (Bu ,...
1
vote
1answer
22 views

Martingale wrt natural filtration: $e^{ \sum_{i=1}^t X_i - t/2}$

Let all $X_i$ be standard normal and iid for $i \in [1,T]$, let $X_0 = 0$. Define for each $t \in [0,T]$ $S_t = e^{( \sum_{i=1}^t X_i) - t/2}$ Is this process a martingale wrt its natural filtration?...
1
vote
0answers
22 views

Sets not in a sigma-algebra

I have a question concerning some sets which are not in a given sigma-algebra. More precisely, I have two questions closely related: Let $\mathcal{A}(\mathbb{R}_{\ge 0}, \mathbb{R}^d), d \ge 1$, be ...
2
votes
1answer
33 views

Show that $tE(1/X;X>t)\to0$ when $t\to0+$

Let $X\geqslant 0$ be a random variable,then $$\lim_{t\rightarrow 0+}{ \,\,t\int_{\left [ X> t \right ]} \frac{1}{X} \, {\mathrm{d}\mathbb{P}} }=0$$ I have no idea of how to prove it.
0
votes
1answer
28 views

Understanding the solution of finding the number of red balls drawn before the first black ball is chosen

Question: An urn contains $n + m$ balls, of which n are red and m are black. They are withdrawn from the urn, one at a time and without replacement. Let $X$ be the number of red balls removed ...
3
votes
0answers
65 views

Are there any modern mathematicians whose research interest is in “Probability Theory”?

I have seen professors in universities list "stochastic calculus", "stochastic analysis", "stochastic processes", "stochastic geometry" and "applied probability" as research interests, but are there ...
1
vote
1answer
31 views

Does $X_n \xrightarrow{L_1} X \implies X_n \xrightarrow{\text{qm}} X$?

Let $X_n$ and $X$ be a sequence of random variables. According to All of Statistics (pg. 81), we have that: $$ X_n \xrightarrow{\text{qm}} X \implies X_n \xrightarrow{L_1} X $$ But the book doesn't ...
0
votes
0answers
27 views

Time-changed Brownian Motion

Let $B_t$ be a standard Brownian motion and let $\tau_{-1}= \inf \{ t \geq 0: B_t(\omega) = -1\}$. By the Continuous Time Stopping Theorem we know that \begin{align} Z_t = B_{t \wedge \tau_{-1}} \...
5
votes
1answer
142 views

Strong Law of Large Numbers for a i.i.d. sequence whose integral does not exist

Prove: Let $X_1 ,X_2 , ... , X_n , ...$ be i.i.d. random variables with $\mathbb{E}[X_1^+]=\mathbb{E}[X_1^-]=+\infty$. If $S_n=\sum_{i=1}^{n}{X_i}$, then $$\limsup_{n\rightarrow\infty}{\frac{...
2
votes
1answer
27 views

Sharpen Doob's Maximal Inequality

Let $B_t$ be a Brownian motion, $B_T^* = \sup_{0\leq t \leq T} B_t$ and $\lambda > 0$. Applying Doob's maximal inequality gives: \begin{align} P(B_T^* \geq \lambda)\leq \frac{\mathbb{E}[B_T^p]}{\...
2
votes
1answer
42 views

Distribution of $\lceil X \rceil - X$ where $X$ has an exponential distribution

Suppose $X$ is a random variable with exponential distribution of parameter $\lambda > 0$. That is, $X$ has density $f(x) = \lambda e^{-\lambda x} \mathcal{1}_{\mathbb{[0,\infty [}}$. The question ...
-2
votes
0answers
22 views

What is the probability of 2 people being in the same class? [closed]

Person A is a girl. Person B is a boy. There must be 10 boys, 26 girls per class (36 people per class) There are 8 classes (288 people in total) What is the probability of 2 people (Person A, Person ...
0
votes
0answers
20 views

Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
2
votes
1answer
69 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
0
votes
0answers
30 views

In what sense does does linear dependence correspond to random variable dependence?

In linear algebra, there is a theorem that states that $\langle v, w \rangle = 0$ implies that $v$ and $w$ are linearly independent. Now let $V$ be a vector space of real-valued random variables on ...
6
votes
2answers
186 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
2
votes
1answer
50 views

Missing crucial step in the derivation of the Stirling's Formula via the Poisson Distribution and CLT

I believe that this is a particular neat example that we've done in class. Unfortunately there is one step I do not quite understand and my Professor had to skip due to the lack of time. I think this ...
1
vote
1answer
32 views

Show that $\mathbb{E}[h(x)^{2}-2h(x)y+y^{2}]=\mathbb{E}[h(x)^{2}-2h(x)\mathbb{E}[y|x]+\mathbb{E}[y^{2}|x]]$

Let $\mathcal{D}$ be a distribution over $Z=X\times Y$. I am trying to understand the following: Why $\mathbb{E}_{(x,y)\sim\mathcal{D}}[h(x)^{2}-2h(x)y+y^{2}]=\mathbb{E}_{x\sim\mathcal{D}_{x}}[h(x)^{...
1
vote
0answers
56 views

Surveys in probability? In the current literature sense

I mostly come from an economics background so when I want to find where the current state of knowledge is in specific fields I look for surveys. These are basically primers so that a researcher can be ...