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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Completeness of space of processes

Under what metirc, the space of continuous, non-decreasing, bounded processes becomes a complete mctric space? Thank you.
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49 views

Strong law of large numbers for square-integrable and uncorrelated random variables with bounded variance

Let $(\Omega,\mathcal{A},P)$ be a probability space and $(X_n)_{n\in\mathbb{N}}$ be a sequence of square-integrable and uncorrelated (maybe we actually need independence) random variables $\Omega\to ...
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3answers
35 views

Why is $ -\sum_{i \in \text I} p_i \log_2(p_i)$ maximized for all $p_i$ equal? Is it true if $|\text I | = \infty$?

Reading a text it is stated without proof that $$ -\sum_{i \in \text I} p_i \log_2(p_i)$$ where $\sum_{i \in \text I} p_i = 1$ is maximized if $p_i$ is a constant. In the case of my theorem, the ...
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0answers
9 views

Introductory study of Survival Analysis and Decision Theory

I'm pursuing a compact Masters degree in Mathematics, a 4 year program at BITS Pilani, India. Except for a couple of introductory courses on statistical and probabilistic analysis, and operations ...
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1answer
10 views

Comparing marginal on product space with other measure

My previous post Unifying the treatment of discrete and continuous random variable, got successfully answered and allowed me to get further in my results. However I am facing a question that I can't ...
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1answer
23 views

A question about independence of sigma algebras (generated by random variables)

Let $X_1, X_2, \ldots$ i.i.d random variables. Is it possible that $$\{X_{n+1} \in B\} \in \sigma({X_1, \ldots, X_n})$$ for some $B$? Why yes/not? I want to show that $\sigma(X_{n+1})$ and ...
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2answers
47 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\sim Po(\lambda)$ $$\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, N(0,1)$$ as $\lambda \rightarrow \infty$. Obs. One is asked not ...
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1answer
12 views

Distribution of a Gaussian Random variable vector [on hold]

I read a slide on the internet which show that: If the random vector $w\sim N(0,I )$ then how can I prove: $x= A^{1/2}w+\bar{x}$ has the distribution $N(\bar{x},A)$ Here A is the covariance matrix ...
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1answer
21 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
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1answer
36 views

Reverse diffusion of a gas

I have a question about the Ehrenfest model of gas diffusion. Suppose I have two urns, each one with $N = 1,000,000$ balls in it. To start with, the first urn has only black balls, and the second ...
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1answer
38 views

Verifying that a certain process is not a Brownian motion

Let $B$ be a standard Brownian motion in $1$ dimension. Define \begin{equation} \tau = \inf \bigg\{ t \geq 0 : B_t = \max_{0 \leq s \leq 1} B_s \bigg\}. \end{equation} We want to show that $(B_{t+ ...
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1answer
36 views

Determining if some random variable is a stopping time

I am stuck on this issue: Let $(B_t)$ be a Brownian motion. We know that since $\{0\}$ is a closed set in $\mathbb{R}$ and that $(B_t)$ is a continuous adapted process, $$ \tau:= \inf \{ t\geq 0 : ...
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1answer
28 views

Position of Brownian motion at exit time from the upper half plane

I am currently reading some books on SLE and struggling on some problems regarding Brownian motion. For a Brownian motion in $\mathbb{R}^2$ starting from $(x,y)$, I don't know how to find the ...
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1answer
23 views

Hitting time process of Brownian motion [on hold]

I am stuck with this problem: Let $(B_t)$ be a standard Brownian motion in $\mathbb{R}$. For $t \geq 0$, let $$ H_t = \inf \{ s \geq 0 : B_s = t \}, \quad S_t = \inf \{ s \geq 0 : B_s > t \}. $$ ...
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1answer
10 views

Convergence in probability given that covariance matrix goes to $0$

Suppose I have a sequence of random vectors $\{X_n\}$ each of dimension $2\times 1$. Suppose also that I know $$ ...
4
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1answer
23 views

Stopped process of Brownian motion

I am baffled about the following problem: Let $(B_t)$ be a standard Brownian motion. Let $$ \tau:= \inf\{ t \geq 0 :B_t = x \} \wedge \inf\{ t \geq 0 :B_t = -y \}$$ be a stopping time, where $x,y ...
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1answer
31 views

Rates of convergence of an OLS estimator

I have a linear regression model $$ y_t=x_t\beta+e_t,\quad t=1,\ldots,N. $$ Here $x_t$ is non-random and given by $(1,\delta_t t)$ where $\delta_t$ is 1 for odd $t$ and $0$ otherwise. Moreover, ...
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0answers
20 views

Distrubution of the maximum of a sequence of random variables. [on hold]

Let $\{Xn\}_n$ a sequence of independent random variables with the same distribution. Suppose their mean is 0 and their variance is 1. Consider $$\max_{0\le k \le n} X_k(t)-tX_k(1)$$ How can I ...
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0answers
15 views

Question about upcrossings of Brownian motion

I am very stuck on this problem: Given a Brownian motion $(B_s)$, write $S_t = \sup_{0 \leq s \leq t} B_s$, for each $t>0$. All stopping times and martingales are considered w.r.t the filtration ...
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1answer
13 views

Multinomial distribution: bounding sum of coordinates' deviations from mean

Okay, given a multinomial random vector $$X=\text{Multinom}(n,\;p_{1},\;\dots,\;p_{k}),$$ so that $$X=(X_{1},\;\dots,\;X_{k})\;\;\;\text{with} \;\;\;\sum_{i=1}^{k}X_{i}=n,$$ I'm looking for a bound ...
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1answer
286 views

Probability - rolling a fair die 10 times, what is the probability you would match a separate set of 10 numbers?

Having some trouble with this problem... Say someone is rolling a fair die 10 times, and using that roll as an attempt to guess what number (1-6) someone else has written down on a piece of paper for ...
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0answers
23 views

What is the explicit obstruction to the failure of pointwise convergence in the stochastic integral?

Let $B(\omega,t)$ be a Brownian motion defined on some appropriately filtered probability space $(\Omega,\mathcal{F}_{t},\mathbb{P})$, and let $f(\omega,t)$ be a stochastic process defined on $\Omega$ ...
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0answers
14 views

Bounding $\int_{\infty}^{\infty}|g(s)v^3k(v)|dv$ where $k$ is a second-order kernel

Suppose $k$ is a nonnegative, bounded real-valued function that satisfies $$ \int_{-\infty}^\infty k(v)dv=1,\quad k(v)=k(-v),\quad \int_{-\infty}^\infty ...
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0answers
17 views

Conditional expectation involving Brownian Bridge

I have no ideas on this problem: Let $(B_t, 0 \leq t \leq 1)$ be a standard Brownian motion in $1$ dimension. Let $Z^y_t = yt+ (B_t -tB_1)$. We call $\{Z^y_t\}_{0 \leq t \leq 1}$ a Brownian Bridge ...
2
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1answer
40 views

Conditional expectation of an uniformly distributed random variable

Suppose $U_1, \ldots, U_n$ are i.i.d. random variables with $U_1$ distributed uniformly on the interval $(-1, 1)$. Compute $\mathbb{E}(U_1 + \ldots + U_n |\max(U_1, \ldots, U_n) = t)$ for $t \in (-1, ...
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1answer
28 views

Does $a^2P(|X |\ge a )\le EX^2 $ hold when $a<0 $?

That is, does Chebyshev's inequality hold for when $a $ is negative? I have seen some authors to require that $a $ be positive, but when Reading the proof by Rick Durrett, I cannot see that this is ...
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0answers
29 views

Three state probability where one state has yet to factor results. [on hold]

I'm currently trying to explain something to someone else using probability to make it simpler to understand. As I have it now there have been 5 examples that have happened. In one case there are 3 ...
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1answer
36 views

Prove that $\int k(w)o(h^2w^2)dw=o(h^2)$ for $\int k(w)dw=1$

Suppose that $k$ is nonnegative real-valued function satisfying $$ \int k(w)dw=1,\quad\int wk(w)dw=0,\quad\int w^2k(w)dw=\kappa_2<\infty.\tag{$\star$} $$ (The limits of the integrals are all ...
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2answers
328 views

Poisson Distribution for Consecutive Figures

I am trying to find the probability for a Poisson distribution. The mean is two cars sold per day. The question is: "What is the probability that at least one car is sold for each of three ...
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2answers
22 views

probability of a brownian motion being equal to the running maximum

Let $B$ be a standard Brownian motion on $\mathbb{R}$. I would like to show that $$ \mathbb{P} \bigg\{ B_1 = \max_{t \in [0,1]} B_t \bigg\} =0 .$$ I argue that since $\max_{t \in [0,1]} B_t $ has the ...
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2answers
34 views

Probability in dice, Feller exercise

I am stuck with exercise 2 of Chapter 4 Feller vol 1 "an introduction to probability theory and its application". Here I report the exercise text: Five dice are thrown. Find the probability that at ...
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3answers
46 views

Where does this conditional probability law come from?

I was trying to follow a computation done in my class notes, and was having difficulty seeing the inspiration for a part of the manipulation in a question regarding probability. I did some Googling, ...
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0answers
59 views

Central limit theorem extends to absolutely continuous measures

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $\mathbb{Q}$ be a probability measure on $(\Omega, \mathcal{F})$ that is absolutely continuous w.r.t. $\mathbb{P}$. Let the ...
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1answer
38 views

Measurability of adapted processes

Let $(\Omega, \mathscr{A}, P)$ be a probability space, $(E, \mathscr{E})$ a measurable space and $X_t : \Omega \to E$, $t \geq 0$ a measurable stochastic process, i.e. the map $X : [0, \infty) \times ...
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Intuitive explanation of variance and moment in Probability

While I understand the intuition behind expectation, I don't really understand the meaning of variance and moment. What is a good way to think of those two terms?
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2answers
46 views

A question about sum of n random variables

Let $X_1, \ldots, X_n$ be random variables. We know that $X_1, \ldots, X_n$ are $\sigma(X_1, \ldots, X_n)$ - measurable. But how about $X_1 + \cdots + X_n$? Is it $\sigma(X_1, \ldots, X_n)$ - ...
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1answer
99 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
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2answers
23 views

Prove that lim sup of a function belongs to a certain sigma algebra

I am so baffled with this problem: Let $B$ be a standard Brownian motion, $\{ \mathcal{F}_t \}$ be the filtration generated by the Brownian motion. I would like to show that for any $k>0$, ...
4
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1answer
45 views

probability circle determined by chord determined by two random points is enclosed in bigger circle

Two points $A$ and $B$ are chosen uniformly at random from the interior of a circle $X_1$. Let $X_2$ be the circle whose diameter is the segment $AB$. What is the probability that $X_2$ is contained ...
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1answer
36 views

Definition of $\sigma$-algebra. Axioms.

""Def. A family $\mathcal F$ of subsets of $\Omega$ is said to be a $\sigma$-algebra on $\Omega$ if: (A.1) $\Omega\in\mathcal F$ (A.2) $\ A\in\mathcal F\implies\ A^c\in\mathcal F$ (A.3) $\ ...
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true story about probability? [duplicate]

A women's organization was contemplating suing a famous American university when it learned that the percentage of women who received tenure in the university was smaller than the percentage of men. ...
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a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ and Kolmogorov-Riesz compactness theorem

Let $\lambda$ be the Lebesgue measure on $\mathbb{R}^{d}$ , $\mathcal{F}$ a set of all probability densities $f$ such that $\mathcal{F}$ is a totally bounded subset of $\mathbb{L}_1(\mathbb{R}^{d})$ ...
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1answer
38 views

What does it mean to say the smallest σ-algebra?

I am just starting out on measure theory. What does it mean to say the smallest σ-algebra?
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42 views

Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in ...
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2answers
37 views

Four dice, probability that difference of some outcomes is equal to others

I roll four dice which gives me outcomes $x_1, ..., x_4$. I want to determine the probability $$P\left((x_2-x_1) = (x_4-x_3)\right)$$ I have already calculated other probabilities in this setting and ...
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2answers
52 views

What is a non-decreasing sequence of sets?

What is a non-decreasing sequence of sets and how come it can have a limit? It appear in a probability theory book
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1answer
27 views

$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
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1answer
229 views

Probability of a zero product given one previous zero product

Consider two random vectors $v=(v_1,\dots, v_n)$ and $w=(w_1,\dots, w_{n+1})$. Each element of $v$ is independently $\pm1$ with prob $1/2$. Each element of $w$ is independently $1$ with probability ...
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1answer
27 views

Approximate normal distribution(this is different from what I asked earlier $\log(n)$ is replaced by $\sqrt{\log{n}}$)

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
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33 views

Partial Correlation Coefficient

I have the following questions on computing the correlation coefficient. Let us say we have two discrete random variables $X_1$ and $X_2$, where $X_1$ has $n_1$ outcomes and $X_2$ has $n_2$ ...