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 ...

learn more… | top users | synonyms (1)

2
votes
0answers
45 views

Development of measure and probability theory

I am interested in a reference (article, maybe a book chapter) on the development of mathematical probability theory - that is, mostly starting from the beginning of the 20th century. It is surprising ...
2
votes
0answers
26 views

Identify the possible weak limit

Suppose $X_1, X_2, \ldots$ are independent random variables with distribution: $$ \mathbb{P}(X_n = 0) = \frac{1}{n}, \, \mathbb{P}(X_n = 2n) = 1 - \frac{1}{n} $$ Let $Y_n = \frac{X_1 + X_2 + \ldots + ...
2
votes
0answers
36 views

Invariant Probability of Discrete Time MC from Continuous Time Markov Chain

Given rates α of an irreducible continuous-time MC on finite state space and told that π is the invariant probability measure of this chain, we define a discrete time MC as having transition ...
2
votes
0answers
22 views

Find the density for the random variable $Y=X_1+X_2$

Problem: Let $X_1, X_2$ be independent variables with uniform density on (-1,1). Find the density for the random variable $Y=X_1+X_2$. My attempt using convolution: $y_1=x_1+x_2, y_2=x_2 \Rightarrow ...
2
votes
0answers
22 views

Integrating a function of measures

I've been reading John Baez's series of posts on Information Geometry. I'm currently on part 6... Midway through the post he discusses Radon-Nikodym derivatives: The formula for information gain ...
2
votes
0answers
30 views

Proving an identity involving expectation

Let $S_t$ be a stochastic process satisfying $S_t = S_0 \exp \{ (r- \frac{ \sigma^2}{2})t + \sigma W_t \}$, where $S_0 >0$ and $W_t$ denotes a Brownian motion. Also, let $Z$ be a $N(0,1)$ random ...
2
votes
0answers
32 views

Challenging CDF of $\sup_t|B_t|$ ($B_t$ is a Brownian Bridge)

Question 1: Let $W_t$ be a Brownian motion. Then how could we prove that $$\Pr\left\{\sup_t|W_t|<b\right\}=1-\frac{4}{\pi}\sum_{j=1}^\infty \frac{(-1)^j}{2j+1} ...
2
votes
0answers
20 views

Finding a stochastic process that satisfies a few constraints

In a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, let $ \{ \mathcal{F}_t \} $ be a filtration generated by the Brownian motion $W$. Let $\mu$ and $\sigma$ be predictable processes such that ...
2
votes
0answers
34 views

How the second form of following equation is derived form first form (i.e. given first line, what are the steps involved in writing second line

How the second form of following equation is derived form first form (i.e. what are the steps involved in writing second line)
2
votes
0answers
22 views

(Billingsley, 2nd ed, 1968) D space, (12.33) inequality proof

Convergence of probability measures, Billingsley, 2nd ed, p132, Theorem 12.4 This is what I want to prove where $x \in D \equiv$ the set of cadlag functions defined on $[0,1]$ and $w_x^{''}(\delta) ...
2
votes
0answers
67 views

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
2
votes
0answers
61 views

Show that a measure is a probability measure

I have trouble with this question: We define an arc segment $B(\theta, \eta, r, R)=\{x \in \mathbb{R}^2\vert \omega(x)\in [\theta,\eta], \Vert x \Vert_2 \in [r,R] \}$ where $0 \leq \theta \leq \eta ...
2
votes
0answers
9 views

Monte Carlo Markov Chain Simulation Issues

The Markov Chain is uniformly distributed across all $50$x$50$ matrices of entries $0$ and $$1 with no neighboring $1's$. I am supposed to run a MC simulation to check the probability that the ...
2
votes
0answers
33 views

Show that $\sum_n P(X_n>M) <\infty$ implies $P(\sup_n X_n <\infty)=1$

Show that $\sum_n P(X_n>M) <\infty$, for some M, implies $P(\sup_n X_n <\infty)=1$. What I did. By Borel-Cantelli $\sum_n P(X_n>M) <\infty$ implies that $P(\limsup_n ...
2
votes
0answers
62 views

Convergence of Expectations (cont'd)

The question is related to this question. Suppose $\{X_n\}$ is a sequence of indep. random variables with zero expectation. Consider their sum $S_n$ which have the following properties: $S_n$ ...
2
votes
0answers
25 views

The independence of random variables

Here is my question: Consider a homogeneous ergodic Markov chain on a finite state space $X=\{1,\ldots,r\} $. Define the random variables $\tau_n \,,n\ge1$ as the consecutive times when the Markov ...
2
votes
0answers
25 views

Finding a general form of the density function when we have a four dimentional random variable.

Consider a subject having time of the specific event $T_i$, which is a single sample from a distribution $F_i$ with density $f_i$ and support $[t_{\min},t_{\max}]$, for $i= 1,\ldots,n$. Let these ...
2
votes
0answers
60 views

Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, ...
2
votes
0answers
85 views

How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
2
votes
0answers
30 views

Sum of $\{X_n\}$ iid random variables contained in a compact interval implies each $X_i=0$ a.s.?

I am working through an exercise that starts with a sequence i.i.d. random variables where for $a\leq b$, $$\Pr\left(\lim\sup_n \sum_{i=1}^{n} X_i \in [a,b] \right) \neq 0.$$ Does this require $X_i ...
2
votes
0answers
38 views

Convergence in the space $C([a,b],M_1(\mathbb R))$.

Let $M_1(\mathbb R)$ be the space of probability measures on $\mathbb R$ with the weak(-*-)topology: $\mu_n \rightarrow \mu$ iff $\int f(x) \mu_n(dx) \rightarrow \int f(x) \mu(dx)$ for all $f \in ...
2
votes
0answers
21 views

Where have I used the assumption that $X\in L_2$?

Let $X\in L_2$ be a random variable and $g$ a positive real function. Let $I$ be an interval and $b>0$, and suppose that $\forall x\in I\ g(x)>b$. I have to show that: $$\operatorname ...
2
votes
0answers
35 views

how to related a weakly convergent random variable with its k-th moment

Let $\{X_n\}$ be independent random sequence with zero mean and unit variance. Suppose $$S_n:=\sum_{m=1}^n \frac{X_m}{\sqrt{n}} \Rightarrow X\sim \mathcal{N}(0,1)$$ holds. (Here "$\Rightarrow$" ...
2
votes
0answers
33 views

Analyzing a coin tossing game with cheating

Consider a game where you toss $N$ coins, and let $H$ denote the number of heads. Let's say you win the game if $|H - N/2| \geq K$, i.e. if the number of heads deviates east least $K$ from what you ...
2
votes
0answers
68 views

Measure-preserving maps on probability spaces

A professor posed me a problem a few days ago, and I have not been able to find an answer to it. Let $(X_1,\Sigma_1,\mu_1)$ and $(X_2,\Sigma_2,\mu_2)$ be probability spaces. Suppose the following ...
2
votes
0answers
37 views

to be 99% certain of making a profit? central limit theorem?

Let $X_i$ be the profit card $i$ makes when its sold. I let $S_n = X_1 + ... + X_n$ so total profit. I found the mean of $X$ to be $0.1$. and $E[X^2] = 25$ so variance $= 24.99$ Are these correct? ...
2
votes
0answers
28 views

Does $(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp (Y,W) \mid Z$ hold?

I know that $ X \perp (Y,W) \mid Z \implies (X \perp Y\mid Z) \ \& \ (X \perp W \mid Z)$ but does the converse hold? i.e. does: $$(X \perp Y\mid Z) \ \& \ (X \perp W \mid Z) \implies X \perp ...
2
votes
0answers
31 views

Is there a continuous version of the Borel-Cantelli lemma?

Given a sequence of events $A_n$ for $n\in \mathbb N$, the first Borel Cantelli lemma states that, if the sum over all probabilities $\sum_{i=1}^n P(A_n)$ is finite, then the probability of the limes ...
2
votes
0answers
101 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = ...
2
votes
0answers
27 views

Intersection of Probability measures

Let $X$ be a metric space with Borel $\sigma$-algebra $B(X)$ and suppose that $\mathbb{P}_1$ and $\mathbb{P_2}$ are two probability measures on $(X,B(X))$. Question: Suppose $G$ is an open set ...
2
votes
0answers
12 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
2
votes
0answers
27 views

Stratified Sampling: Total and mean estimate of population

Question: In a sample survey designed to estimate the total number of cattle, the universe of 2072 farms was stratified into 5 strata on the basis of the total acreage of farms. In the hth stratum (h ...
2
votes
0answers
67 views

Martingale Can't be Strictly Increasing

If the sample paths of a martingale are almost surely continuous and not constant on any interval, is it true that they are almost surely not increasing on any interval? Edit for clarity: Let ...
2
votes
0answers
46 views

Relation of stopped sigma-Algebra on cadlag sample space to arbitrary sample space

Let $X_t : \Omega \to E$ be a cadlag process with Polish state space $E$, $T$ a stopping time w.r.t. the canonical filtration $\mathcal{F}_t$ of $X$ and $X^T_t$ the stopped process. Then it should ...
2
votes
0answers
55 views

Equality of sigma-algebras

Let $X$ be a set, $\mathcal{C} \subseteq X$ and $\mathcal{A}$ a $\sigma$-algebra on $X$. Is it true or false that $\sigma(\mathcal{C}) \cap \mathcal{A} = \sigma(\mathcal{C} \cap \mathcal{A})$? Of ...
2
votes
0answers
29 views

Non-separability of the Cadlag functions equipped with the topology of uniform convergence on compacts

Consider the space $D:=D([0,\infty),\mathbb{R}^N)$ of component-wise right continuous functions with left limits. Endow $D$ with the metric ...
2
votes
0answers
17 views

Proving that an inductively defined function is a Markov chain

Let $X_0$ be a random variable with values in a countable set $I$. Let $Y_1,Y_2,\ldots$ be a sequence of independent random variables, uniformly distributed on $[0,1]$. Suppose we are given a function ...
2
votes
0answers
23 views

Asymptotic uniform integrability and moments of Student's $t$

I am working on an exercise where I am trying to show that the moments of a $t$ distribution converge in probability to the moments of a standard normal. I'm in need of help with what I think is the ...
2
votes
0answers
45 views

Almost sure limit of $\log(X_1 + X_2 + … + X_n) - \log(n)$

Let $X_n$ be an i.i.d. sequence of positive random variables with expectation 2 and variance 1. What is the almost sure limit of $$\log(X_1 + X_2 + ... + X_n) - \log(n)$$ as $n \to \infty$ Would it ...
2
votes
0answers
55 views

When can one represent the conditional expectation $E[X|Y]$ as $g(Y)$ with continuous $g$?

Given two random variables $X$ and $Y$ we know that $E[X|Y] = g(Y)$ where $g$ is a Borel function. Is it a good question to ask under which condition there exists a function $g$ which will be ...
2
votes
0answers
47 views

Independence of two processes

Suppose $X_t$ is the solution of the SDE $$dX=a(X)dt+b_1(X)dW_1+b_2(X)dW_2$$ $Y_t$ is the solution of the following SDE $$dY=p(Y)dt+q_1(Y)dW_1+q_2(Y)dW_2$$ Here, $W_1$ and $W_2$ are independent ...
2
votes
0answers
16 views

Sampling with an “oversampling” factor, in K-Means||

I'm trying to understand K-Means||, a scalable version of K-Means++, which itself is an "improved" version of the clustering algorithm K-Means. Please find here the link to K-Means|| paper ...
2
votes
0answers
46 views

Mean time spent in a state in a CT Markov Chain

I consider a continuous-time homogenous Markov chain: with discrete state $X$ taking values in $\mathcal{F}=\{1,\cdots,N\}$ with the transition rates satisfying: \begin{equation} \begin{cases} ...
2
votes
0answers
80 views

Is my proof correct? are the arguments right?

my assumptions: (i) $\lim_{t \to \infty}F_{t}(x)=F(x) \ \forall\ x\ \in\ C(F)$(set of continuity points of F) with $F_{t}(x)$ family of distribution functions and $F$ distribution function (ii) ...
2
votes
0answers
13 views

Pushfoward of measures Lipschitz continuous in total variation

Let $X,Y$ and $Z$ be metric spaces and $f:X\times Y\to Z$ be a measurable map. Suppose that we are given a probability measure $\mu$ on $Y$, and define a stochastic kernel $$ ...
2
votes
0answers
32 views

Stationarity of Hawkes processes with (partially) negative kernels

Consider a point process $N$. For the linear Hawkes process, the conditional intensity is given by $\lambda(t) = \nu + \int h(t-s) N(ds)$, with constant $\nu > 0$ and kernel $h(s)$. In almost every ...
2
votes
0answers
28 views

The variational formulation of entropy

For $f:\mathbb Z_2^n \to [0, \infty)$, the entropy of $f$ is defined as $$ {\rm Ent}(f) = \mathbb E[f(X) \log f(X)] - \mathbb E f(X) \log(\mathbb E f(X)), $$ where $X$ is a random element of $\mathbb ...
2
votes
0answers
60 views

How is conditional probability being used here?

Because of conditional probability: $P(A\mid B)=P(A,B)/P(B)$, $$P(C(t)\in dt\mid x(T^+_{i-1}),x(T^-_{i}))=\dfrac{P(C(t)\in dt,x(T^-_{i})\in dx\mid x(T^+_{i-1}))}{P(x(T^-_{i})\in dx\mid ...
2
votes
0answers
37 views

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian. Without loss of generality we may assume that $X$ and $Y$ are ...
2
votes
0answers
47 views

Mutual Information for Gaussian Process (and also Fano's Inequality)

According to this presentation: Bounding Gaussian Process Information Gain we have a closed-form expression for the information gain as follows: $$ I\left(\vec{y} \mid f\right) = \frac{1}{2} \log\det ...