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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25 views

Is the following rule a stopping time in regards to reverse filtration?

Let $X_1, \dotsc, X_n \sim F$, where $F$ is a distribution function with support in $[0,1]$. For $t \in [0,1]$, define the sigma-algebra: $$ \mathscr{F}_t = \sigma(1_{\{X_i \leq s\}}\;,\; 1 \geq s ...
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47 views

Prove that $\text{Var} \tau = \frac{1 − (p − q)^2}{(p-q)^3} $ where $\tau$-stopping time

Let $S_n = \xi_1 + \dots + \xi_n$ be asimetric random walk such that $P(\xi_i = 1) = p > \frac{1}{2}$ and $P(\xi_i = -1) = q $. Let $\sigma^2 =1-(p-q)^2$ and let $X_n=(S_n-n-(p-q)n)^2 - \sigma^2n $ ...
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64 views

Prove that $X_n = e^{(\theta S_n-n \ln \phi(\theta))}$ is martingale.

Let $\xi_1 \neq const.$ be a random variable with moment-generating function $\phi(\theta) = Ee^{(\theta \xi_1)}$. Let $S_n = \xi_1 + \dots \xi_n$. Prove that $X_n = e^{(\theta S_n-n \ln ...
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29 views

Weak convergence of probability measure on $\mathscr{C[0,\infty]}$

What is the reason for considering the spaces of probability measures on the space of all continuous functions and then considering weak convergence there ? Is it that we can then use Skorohod's ...
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92 views

Rotational invariance and distributions

Let $k\leqslant n$ denote two positive integers, $A$ an $n \times k$ matrix with $A'A = I_k$, and $X$ and $Y$ two independent random variables on $\mathbb R^n$, each rotationally invariant (that ...
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39 views

Moments of $|ax-by|$

Suppose that $X$ and $Y$ are independentr.v. uniform on $[0,1]$. What is the $E[|aX-bY|^p]$ for some constants $a,b,p>0$? What I did. \begin{align*} E[|aX-bY|^p]=\int_0^1\int_0^1|ax-by|^p dx ...
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43 views

Does the system of equations always have a nontrivial solution?

$f:[0,1]^2\to R_+$ is a continuous conditional density function. For $g,h\in C$ on $\{(x,y)\in [0,1]^2|x\geq y\}$, the system of equations is given by$$ \frac{\partial g}{\partial x}\leq ...
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68 views

Are there probability density functions with the following properties?

Given two distinct and continuous probability density functions on real numbers, $f_0$ and $f_1$ consider the following set of density functions: ...
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36 views

Convergence equivalent random sequences

Suppose we have a sequence of independent random vars $X_n$ and consider a sequence of truncated random variables $Y_n=X_n1_{X_n\le n}$ s.t. $E[Y_n]=0$. We know that $X_n$'s and $Y_n$'s are ...
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32 views

How to calculate expectation of Cantor distribution without using p = 0.5

The question goes like this: given F(x) is the distribution function of Cantor distribution, how to calculate $E(x) = \int xdF(x)$ without using the argument of the following: X is characterized by ...
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52 views

monotone convergence of $f_n$ plus weak convergence of $\mu_n$ implies convergence?

I would like to know if somebody is aware of some result that looks like the following. Let us consider the space $C_b(X)$ of continuous bounded function over a measurable space $X$. Suppose that: ...
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79 views

Kolmogorov backward equations for Birth-Death

I'm trying to solve the Kolmogorov backward equations for a Birth-Death Markov chain with three states. I have 2 equations: $$P_{00}'(t) = \lambda_0 (P_{10}(t)-P_{00}(t))$$ $$P_{10}'(t) = ...
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86 views

Chapman-Kolmogorov equations of time inhomogenous Markov chains

Let us assume that we are given a time inhomogenous Markov chain in continuous time (ICTMC) $(X(t))_{t \geq0}$ with a finite state space $\{1,\ldots,n\}$. Set $P(t)_{i,j} := \mathbb{P}(X(t) = j \mid ...
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99 views

Proof of Doob's decomposition theorem?

I feel confused about the proof of the theorem. First question: Why in step 2, $Z_{n+1}=E(X_{n+1}|F_n)-X_n+Z_n$, then we can get $Z_{n+1}$ is $F_n$ measurable? I can see $E(X_{n+1}|F_n)$ and $X_n$ ...
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130 views

Finite Expectation Value means random variables are bounded?

For a random variable $\xi $, does $|E(\xi)| < \infty $ means that there exists $M>0$, such that $|\xi | < M $ almost surely? A counter example I can think of is $\xi(x) = \frac{1}{\sqrt{x}} ...
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29 views

The inverse of the integration over a ball with radius $\epsilon$

First of all sorry for the nondescript title, but this seemed like the most suitable one. Now let $d\geq2, D\subset \mathbb{R}$ a domain and $G:D\times D\rightarrow[0,\infty]$ continuous. Define ...
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31 views

Sum of Lomax random variables

Suppose $X_1,X_2,\cdots X_n$ are $n$ i.i.d Lomax random variables with pdf $f(x)=\frac{m}{(1+x)^{m+1}},x\geq 0,m\in \mathbb N$. I need to determine the pdf (or cdf) of the sum $S_n=\sum_{i=1}^{n}X_i$. ...
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21 views

Looking for resources: Generalizations of martingales to $\mathbb R^2$

In most introductory courses, a martingale $Y$ is defined as a stochastic process $$Y: T \times \Omega \to S$$ ,which satisfies certain conditions. ($\Omega$ is a probability space and a filtration ...
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177 views

Slutsky's Theorem

In Slutsky's Theorem's proof as outlined in the link, we can get the general results that $g(X_n,Y_n)\rightarrow_d g(X,c)$ whenever $g$ is continuous. However, in the Continuous Mapping Theorem, it ...
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14 views

What is Law($Z$) under $\mu$ for a random variable $Z$, and distribution $\mu$?

Is it simply the probability measure $A\in\sigma(\mathbb{R})\mapsto\mu(Z^{-1}A)$? (Or correspondingly whatever the range of $Z$ might be.)
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61 views

Rate of Convergence in CLT for IID random vectors with dependent entries

Okay, so suppose I've got a sequence of IID random vectors $X_{n}$, where the entries of each $X_{n}=(X_{n,1},\dots, X_{n,i})$ are dependent. The motivating example here is the multinomial ...
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57 views

Relative Entropy and variation formula for $C_c$

Let $R(\mu \mid \nu ) = \int_{\mathbb R} \log \frac{d\mu}{d\nu} d\nu$ for $\mu, \nu$ probability measures over $\mathbb R$. By the varational representation formula of Donsker and Varadhan we know ...
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9 views

Time changes conditions to be adapted

Given a process $X_t$ and another process $T_t$ which is increasing, what conditions should we require such that the process $X_t$ is adapted to the time change $T_t$, that is such that $X_t$ is ...
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63 views

Measurability properties of processes that arise as limits of sequences of measurable processes

I try to reduce my problem to a more general statement from which I want to know whether this is true in general. I have a sequence of continuous-time stochastic processes $X_t^{(n)}, t \geq 0$ with ...
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35 views

Soft: Interesting examples of applications of the Itô–Nisio theorem

I'm writing up a presentation on the Itô–Nisio theorem and I'm looking for a simple (nontrivial) example showcasing it. I was thinking of a 2-dimensional simple random walk, but that seems ...
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70 views

Cameron Martin Theorem

I am struggling with two versions of the Cameron Martin Theorem. 1) We define the measure spaces $(\Omega,\mathcal{F},P)$ and $(C[0,1],\mathcal{C},\mathbb{L}_0)$, where $\mathcal{C}:=\sigma(f\mapsto ...
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101 views

Law of iterated logarithm proof

I am trying to master this proof of iterated logarithm. However, I get stuck at the last part. Here is a link In the last two line at fourth page. We calculate the probability that: $$ ...
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54 views

Weak convergence, measurability, uniform integrability

I've been facing the following problems: a) Let $ X_n \rightarrow X $ and $f $ be a measurable, bounded function. Prove that $ \mathbb{E}f(X_n) \rightarrow \mathbb{E}f(X) $ (we also assume that the ...
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40 views

Hitting time of two dimensional continuous martingale

Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...
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53 views

Application of Law of Iterated Expectations

Could you please explain something from a text I am reading? We're given that $E(\epsilon_i)=0$ and $E(\epsilon_i x_i)=0$ for $i\geq 1$. Write $g_i=\epsilon_i x_i$ and we further know that $$ ...
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21 views

Predictive analysis based on history

Let me first say that I am a CS person and my knowledge about statistics is quite basic. I am trying to see what predictive analysis to use for a problem I am trying to solve. I will try to make my ...
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54 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
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18 views

counterxample: Events with bounded minimum probability not having a subsequence with 0 probability

So, I'm trying to come up with a counterexample (Or proof if that would be easier that thi thing is false, but it seems like counterexample is the normal way to show something is false...) Anyhow, ...
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30 views

Probability question on 5 components operation after a given time

My attempt: $\lambda=1/2.5=0.4$ Since $P(T\geq t)=({1-e^{-\lambda t}})$ $P(T\geq3)=({1-e^{-0.4(3)}})^5$ However my book says: $P(T\geq3)=({e^{-0.4(3)}})^5$ Why is this? How did the book do ...
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33 views

Convergence of a distribution function

I have a problem that has me stumped. Frustratingly enough I can't really get the problem started. It goes as follows: Let $X_{1},X_{2},...$ be independent $C(0,1)$-distributed random variables. ...
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49 views

Integral of a standard brownian motion

I am working on the following problem which is on an introductory chapter of Brownian motion: Let B(t) be the standard Brownian motion. Define $X(t)=[1/\sqrt{t}]\int_{0 \to t}{f(B(s))}ds$ where $f$ ...
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12 views

Proving $Corr(\hat{e}_{ij}, \hat{e}_{jk}) = \frac{-1}{n_i-1}$ for $ j \neq k$

For the model of a single factor experiment: $y_{ij}= \mu + \alpha_i + e_{ij}$, $(1 \leq i \leq a, 1 \leq j \leq n_i)$, where a = the number of treatments, $n_i$ = the number of experimental units ...
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20 views

Principal Components vs Principal Directions

I'm trying to do statistical downscaling of some climate data and there is a module of principal component analysis by regression method required. I am confused with the different terms here. What is ...
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68 views

Random walks: number of crosses between $-\sqrt{x}$ and $\sqrt{x}$

Let $S_n = \sum_{k=1}^n X_i$ be a simple random walk, where $X_1, X_2, \dots$ are independent Bernoulli random variables, $\mathbb{P}(X_k = 1) = \mathbb{P}(X_k = -1) = \frac 1 2$. Let $T_1 = 1, ...
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32 views

Subgaussian bounds for $X$ imply subgaussian bounds for $X-E(X)$

A random variable $X$ is called sub-gaussian, if there exist positive constants $C,c$ such that for all positive $\lambda$ we have $$P(|X|\geq \lambda)\leq Ce^{-c\lambda^2}.$$Now I'm reading a text, ...
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73 views

Lindeberg condition of order $2k$

Prove that if $ (X_n),\: n \ge 1$ are independent r.v.s with $ E X_n = 0, E X_n^2 = \sigma_n^2$ which obey central limit theorem and $$\lim_{n\rightarrow\infty}E\bigg(\frac{S_n ...
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162 views

Property of uniformly tight random variables?

I'm stumped on the following question, which is problem 1.3.9 in the book Weak Convergence and Empirical Proceses by van der Vaart and Wellner. It is based on the following notion of asymptotic ...
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17 views

Measurability of the composition of sections and probability measures

Let $X,Y,Z$ be Polish spaces endowed with their respective Borel sigma algebras, and $\Delta(Y),\Delta(Z)$ be sets of Borel probability measures on $Y$ and $Z$, endowed with the weak*-topologies (and ...
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40 views

Why is the covariance a measure of how much two random variables change together?

In descriptive statistics, the empirical covariance of a point cloud $\left\{(x_i,y_i) : 1\le i\le n \right\}$ is defined as ...
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47 views

Show convergence in distribution of gamma distribution by the central limit theorem

So, I'm not sure how to solve this problem: $X \in \Gamma(a,b)$. Show that $$\frac{X-E[X]}{\sqrt{Var(X)}} \rightarrow^{d} N(0,1)$$ as $a \rightarrow \infty$ Use the central limit theorem. I've come ...
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147 views

Does clever noise exist?

This question is about a random noise, which is called clever if its distribution function satisfies a certain condition. Let $Y_0$ and $Y_1$ be two continuous random variables on the same ...
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157 views

Can we find a correlation between states of a Markov chain?

I have a fair bit of knowledge on Markov chains but I recently wondered if there is a way to find out a correlation between the states of a finite Markov chain. I could not find any material on this. ...
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22 views

Finding $a$ such that a product of iid $U(0,a)$ converges to $0$ a.s.

Let $a > 0$, let $X_n$, $n \geq 1$, be iid random variables that are uniform on $(0,a)$, and let $Y_n = \prod_{k=1}^n X_k$. Determine, with a proof, all values of $a$ for which $\lim_{n \rightarrow ...
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49 views

Removing balls from an urn in pairs: expected number of pairs in which both are red

I am sorry that I cannot make the title more clear. The following is from Sheldon M. Ross: Introduction to Probability Models (11th Edition). I am able to reach the desired answer but actually I don't ...
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27 views

How to show a limit law for record times $N(n) = \min\{j: j > N(n-1), X_j > X_{N(n-1)}\}$, $X_n$ iid.

I have the same question here which I asked a few days ago, but now I have really worked on it and I could not get the solution, so I need some more help. Let $X_1, X_2, \ldots, X_n$ be i.i.d. ...