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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57 views

Algebra involved in computations with (or extensions of) a probability measure on a lattice

Suppose we have a probability measure $\gamma$ defined on the $d$-dimensional lattice $\mathbb{N}^d$. Let us say $d = 5$ for simplicity. I will also write $\gamma_{1:5}$ for this measure and write say ...
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323 views

Doob inequality for continuous martingales

In our class we have proven Doob's inequality for discrete martingale, i.e. Let $(M_n)_{n \in \mathbb{N}}$ a martingale, then $$ \| \max_{0\le k\le n} M_k\|_p \le C_p \|X_n\|_p$$ for $p\in (0,\infty)...
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102 views

Application of Anderson's theorem to probability

From Wikipedia, Anderson's theorem is stated as: Let $K$ be a convex body in n-dimensional Euclidean space $\mathbb{R}^n$ that is symmetric with respect to reflection in the origin, i.e. $K = −...
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194 views

Derivative of measure-valued function

I have a measure $\mu^x$ which is the law of a random variable and depends on $x$. The specific situation I am thinking of is $\mu^x$ is the law of $X_t$, the solution of an SDE with $X_0=x$. If I ...
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106 views

Measurability of a simple Set-Valued Map

Let $\mu(\cdot)$ be a probability measure on $\mathbb{R}^m$, so that $\int_{\mathbb{R}^m} \mu(dw) = 1$. Consider a function $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ such that $\...
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380 views

Slutsky's theorem for random matrices

This image is from Applied Multivariate Analysis. In this image plim means convergence in probability. I could not find the reference about the statement for ...
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128 views

Cover times and hitting times of random walks, once again.

This is a followup to my question Cover times and hitting times of random walks. Consider a random walk on an undirected graph with $n$ vertices which, at each step, moves to a uniformly random ...
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240 views

Computing the stationary distribution of a markov chain

I have a markov chain with transition matrix below, $$\begin{bmatrix} 1-q & q & & & \\ 1-q & 0 & q & & \\ & 1-q & ...
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494 views

Martingale convergence almost surely

Let $(X_i)$ be a sequence of r.v. and define $Y_i^n:=X_i\mathbf1\{|X_i|\le n\}$ and $Z_i:=T^i(Y_i^i)$, where $T^i$ is defined as $$T^i(X) :=\sum_{l=-\infty}^\infty l2^{-i}\mathbf1\{X\in(l2^{-i},(l+1)...
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160 views

expectation of rademacher chaos

Let $\sum_{i=1}^n\sum_{j=n+1}^m\epsilon_i\epsilon_jb_ib_j$ be Rademacher Chaos of degree two (here $b_k\in R$ and $\epsilon_k$ are Rademacher random variables) and such tat $\epsilon_m=-\sum_{k=1}^{m-...
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128 views

The expectation after Bayesian inference of a Normal r.v

I'm confusing myself with this question. Suppose there is a Normal r.v $X \sim \mathcal{N}(\mu, \sigma^2)$. We known the variance $\sigma^2$ however don't know the mean $\mu$, and choose to use ...
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268 views

Random walk with increasing step size

A random walk in the plane with step size $f(n)$ is cool if there is some constant $C$ such that it returns within a distance $C$ of the origin infinitely many times with probability $1$. At the $n$-...
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57 views

Green kernel of Hunt processes

Let $X$ be a regular Hunt process on $R^+$ with starting at $x$ and $T_y :=\inf\{t>0: X_t=y\}, y\in R^+$. $G_q(\cdot,\cdot), q >0$ denotes Green kernel of the process $X$. We have the following ...
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93 views

Alternative way of showing convergence of central moments

I woud like to show for a random variable $X$ $${1\over n} \sum_{i=1}^n (X_i-\bar{X}_n)^q\to E(X-EX)^q \quad (\text{convergence in probability)}$$ My approach was to define $Y_i:= (X_i-\bar{X}_n)$ ...
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689 views

Relationship between Abstract Algebra and Probability Theory

Is there a relationship between abstract algebra and probability theory? I ask this because of the following laws: Axiom of Countable Additivity: If $A_1, A_2, \dots \in \mathcal{B}$ (where $\...
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79 views

Probability inequalities

Let $X_{1},X_{2},\ldots$ be i.i.d. random variables and let $S_{n}=X_{1}+ \cdots +X_{n}$. Given that $1<a<a'$ and $0<\sigma<\lambda$, how do I show that if $\sup_{1\leq b \leq a'-a}...
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363 views

Convergence rate for the p.d.f. of a normalized mean to Gaussian (i..e Berry-Esseen for pdfs)

Berry-Esseen Theorem states that the rate of convergence of the probability distribution of normalized sample mean converges to Gaussian at rate $O(1/\sqrt{n})$ (given that certain conditions are met, ...
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185 views

Questions about regular conditional probabilities

In Section 9.9. Regular conditional probabilities and pdfs of David Williams' Probability and Martingales: By linearity and (cMON), we can show that for a fixed sequence $(F_n)$ of disjoint ...
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284 views

Show that the stopped sequence is a martingale

This is a series problem and I'm struggling with the last part. I assume that the last part has nothing to do with previous ones, so i won't put up the other parts. Question is : Let $\tau$ be a ...
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103 views

Likelihood Function of Random Process

Given the following data: $$ x(t) = A + \omega(t) $$ where $ \omega(t) $ is an AWGN with zero mean, what would be likelihood function $p(x(t);A)$? I know it could be proven to be: $$ p(x;A) = C ...
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93 views

Random walks - two questions

Let us suppose that a person are throwing a coin. He'll get one dollar if he win, and he'll pays one dollar if he loses. I understand that the winning will trend to zero in the case of unlimited ...
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169 views

Family of measure that admits a continuous density

This question is a generalization of an example provided in Absolute continuous family of measures. Consider a metric space $(X,\rho)$ with a Borel $\sigma$-algebra $\mathcal B(X)$. Consider a ...
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109 views

Probability metric spaces for which $r \leq R$ implies $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$

I'm looking for examples of spaces $X$ such that: $X$ is a probability space. $X$ is a metric space. If $x \in X$ and $0 < r \leq R$ then $\frac{Pr(B(x,r))}{Pr(B(x,R))} \geq \frac{r}{R}$. I ...
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78 views

Optional stopping theorem for Hilbert valued martingales

Suppose $X_n$ is a Hilbert-valued martingale. Does the optional sampling theorem apply in this case? Does anyone know where I can find a proof? Thank you.
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223 views

How to obtain tail bounds for a linear combination of dependent and bounded random variables?

I am looking for tail bounds (preferably exponential) for a linear combination of dependent and bounded random variables. consider $$K_{ij}=\sum_{r=1}^N\sum_{c=1}^N W_{ir}C_{rc}W_{jc}$$ where $i \...
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2k views

Expected value of max/min of random variables

I am trying to solve the following problem. Let there be $n$ urns and let us have $k$ balls. Assume we put every ball into one of the urns with uniform probability. Denote by $X_i$ the random ...
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159 views

Are Cox point process and Poisson random measure the same concept?

Reading Wikipedia articles for Cox point process and Poisson random measure, I was wondering if they are actually the same concept? If they are not, I wonder how to understand the concept for Cox ...
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985 views

Maximum of Correlated Gaussian Random Variables

Let $x_{1},x_{2},\ldots, x_{n}$ be zero mean Gaussian random variables with covariance matrix $\Sigma=(\sigma_{ij}^{2})_{1\leq i,j\leq n}$. In other words, $$ \mathrm{cov}(x_i,x_j)=\sigma_{ij}^{2} $$...
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103 views

Measurability of a point process or random measure at a measurable subset

Suppose $\xi$ is a point process on $(S, B(S))$, where $S$ be locally compact second countable Hausdorff space equipped with its Borel σ-algebra $B(S)$. I was wondering if $\xi(A), \forall ...
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532 views

Definition of vague convergence of distibution functions

I am reading Kai Lai Chung's A Course in Probability Theory. I understood the concept of vague convergence of a sequence of probability measures in the book. But it seems to me the book uses the ...
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25 views

What is probility to miss at least one test?

The probability that a teacher will give an unannounced test during any class is $\large \frac 15$. If a student is absent twice, then probability that he misses at least on test is $a) \ \...
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24 views

Weak-* compactness of probability measures on compact, non-Hausdorff space

Let $X$ be a compact topological space. How would I prove that the space of probability measures $\Delta(X)$ is weak-* compact? Without assuming $X$ being Hausdorff I can't apply to the usual Reisz ...
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20 views

If $X_n \stackrel{p, quickly}{\to} X$, then $X_n \to X$.

Probability with Martingales: Without using hint, can I just do something like this: http://math.stackexchange.com/a/1538503/140308 ? With using hint: By continuity of probability, I think ...
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57 views

Ito's Formula applied to a weird equation…

EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything I was just wondering if someone could explain how to solve this problem. I ...
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11 views

Central Limit Theorem Heuristics

Surrounding the central limit theorem there exist several heuristics which say when a normal distribution is a reasonable approximation to the mean $\frac{X_1 + \cdots + X_N}{N}$ of $N$ independent (...
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14 views

“Return probability” to origin of a variant of the random walk.

Let $\{\epsilon_t\}_{t\ge0}$ be an iid sequence of random variables and let $\lambda>1$. I am interested in the following process: Let $X_0 = 0$ and $$ X_{t+1} = \lambda(X_t+\epsilon_t). $$ This ...
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13 views

Nash Equilibrium

Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 1, 2, . . . , 100). Player B guesses the number that player A picked, and receives that amount in ...
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27 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
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40 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
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35 views

Lindeberg condition's counterexample (central limit theorem)

My aim is to find an example where the CLT is true but not the following (equivalent to Lindeberg's) condition: Find a sequence of independent $(X_k)\sim\mathcal{N}\left(0,\sigma^2_k\right)$, so ...
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0answers
20 views

Shifting the mean of a composite function of deterministic and random variables

For a project I am involved in relating to communication, I have the following model: $L = f(r).X$ where $X$ is a lognormal random variable with zero mean in the logarithmic scale and standard ...
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20 views

Convergence in law and sequence of reals

How to show that if I have $Y_n$ a sequence of random variables and $a_n$ a sequence of reals such that (i) $Y_n\to Y$ in law (ii) $a_n\to a$ i have $a_nY_n\to aY$ in law? I know that if I have (i)...
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42 views

One question regarding independence of $\pi$ systems.

Let G and H be sub-sigma-algebras of F and I and J are $\pi$ systems such that $\sigma(I)=G$ and $\sigma(J)=H$ Can anyone explain the following quote? Suppose I and J are independent, for fixed ...
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11 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
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23 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
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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 ...
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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 ...
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60 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 ...
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17 views

Probability of at least X same choices by N people choosing out of a set of Y options

I am currently facing the following problem: There is a set of integers from 1 to Y, and N people choosing a random number of this set. What is then the probability that there is at least one number ...
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40 views

Conditional expectation and partition theorem

Let $X,Y$ be two random variables and consider $E[g(Y)\mid X]$. Suppose $\cup_{i=1}^N A_i= \mathbb{R}$, then do we have some thing like $$E[g(Y)\mid X]=\sum_i E[g(Y)\mid 1_{Y\in A_i}, X]P[\{Y\in A_i\}...