1
vote
2answers
308 views

Brownian motion of every point in the plane

Suppose every point in the plane undergoes brownian motion for a time t. What is the probability n particles ended up at 0? For n finite, countable or uncountable? What proportion of the plane does ...
0
votes
0answers
23 views

Variance Gamma Distribution and Process

I have read that a variance gamma process $X_t=\theta G_t+\sigma W_{G_t}$ is such that $X_1\sim Variance Gamma(\theta,\sigma,\nu)$ but the variance gamma distribution has 4 parameters: $\mu$, ...
2
votes
2answers
169 views

How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim ...
2
votes
1answer
67 views

What are the generalized Gaussian laws that are infinitely divisible?

We consider the probability density, often called a generalized Gaussian density, $$f_{p,\lambda}(t) = \kappa_{p,\lambda} \exp (- \lambda |t|^p),$$ with parameter $0<p<\infty$ and $\lambda > ...
2
votes
1answer
30 views

probabilistic distribution of given data

let us consider following model $$y(t)=A_1 \sin(\omega_1 t+\phi_1) + A_2 \sin(\omega_2 t+\phi_2) + A_3 \sin(\omega_3 t+\phi_3)+ \ldots +A_p \sin(\omega_p t+\phi_p)+z(t)$$ we have three parameter ...
3
votes
1answer
150 views

Quantiles moments and Convergence

QUESTION: Let $F$ be an absolutely continuous distribution function with density $f$, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
2
votes
1answer
128 views

A calculation involving a uniform random variable quantile

THE PROBLEM: Let $U$ be a uniform distribution and $U_{n}$ be its nth empirical distribution. Suppose $t\in (0,1)$ and $n\in \mathbb{N}$ are constants. What's the explicit expression to ...
0
votes
1answer
71 views

Running supremmum of a Levy process

Let X be a cadlag Lévy process with $X_0=0$ and let $p$ be a real number in $[1,\infty)$. Then, the following are equivalent. 1): $X$ is $L^p$-integrable. 2): $X^*_t= \mathop{\sup}_{0\leq s\leq t} ...
7
votes
3answers
323 views

Maximum of the expectation of maximum of Gaussian variables

Suppose $X=(X_1,\ldots,X_n)$ is a Gaussian vector with each entry $X_i$ marginally distributed as $\mathcal{N}(0,1)$. Want to find out the possible maximum of $$\mathbb{E}\max_{1\le i\le n}|X_i|$$ and ...
2
votes
0answers
76 views

Learning resources for Probability Distributions/Models [closed]

I've a good background in basic probability. I need to learn and get a good grip on the probability distributions and stochastic processes, counting processes, and other related topics. I am already ...
2
votes
1answer
143 views

Characterizations of the GOE/GUE family of distributions

This question is somewhat related to this one. Loosely speaking, when should I expect a GOE/GUE distribution? The angle of my approach to this is not through statements such as "there is a natural ...
0
votes
0answers
68 views

Eigen value distribution of autocorrelated Wishart matrix

Suppose the matrix W is constructed as $W=XX^T$ where $X_i(t) = \phi_i X_i(t-1) + a_i(t)$, and $a_i(t)$ ~ $N(0,1)$. I am interested in knowing the eigen value distribution of W. My google search on ...
1
vote
2answers
112 views

Distribution similar to PPP

According to the definition of Poisson Point Process, I can't define a certain number of nodes which are distributed in an area as PPP. Is there a distribution (a certain number of nodes distributed ...
0
votes
1answer
239 views

Markov Chain: state reduction

Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following: Firstly we have a Markov chain $\{Y_k\}$ with finite ...
3
votes
1answer
269 views

Probability that no three events happen in a pre-defined window

Consider a Poisson process with arrival rate $\lambda$ arrivals per unit time. Given a window of time $W$ and a total of $k$ events, what is the upper bound of the probability that no three events ...
2
votes
0answers
152 views

Have you seen this one parameter family of distributions before?

This is a one parameter family of distributions. Choose some parameter $\lambda > 0$ and define the measure $\nu_\lambda$ which is absolutly continuous with respect to the Lebsegue measure with the ...
5
votes
2answers
216 views

Liverani's CLT (a question)

Let $(\Omega,\mathcal{F},P)$ be a probability space where $\Omega$ is a complete separable metric space, let $T:\Omega\to \Omega$ ` be an ergodic transformation, let $\hat{T}:L^{2}_{_P}(\Omega)\to ...
0
votes
1answer
77 views

Multinomial — how many trials in order to see all the values with prob 1-\alpha

Let suppose that I have a box with $k$ different balls, each one with a different color. At each time I have to extract a ball and observe the color. Then I put the ball back in the box. How many ...
1
vote
2answers
512 views

minimum of different independent Poisson random variables

Let $X_1,\ldots,X_N$ be independent Poisson distributed random variables with unequal parameters $\lambda_1,\ldots,\lambda_N$. Is there any closed form expression or at least a good approximation for ...
1
vote
1answer
135 views

First moment of a function of a normally distributed random variable

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the ...
0
votes
1answer
173 views

Limit of the stochastic process at time 0

This is not a homework question so please be kind not to remove it right away. I am working on some research but have to justify the following argument: Assume $S_t$ is a continuous stochastic ...
0
votes
0answers
82 views

Proving that a property holds for random sequences with given marginal distribution by rearrangement

I am currently investigating the property of random sequences with a special marginal distribution function $F(x)$. Given any random sequence $X_1, X_2, \cdots, X_n$, supposing their joint ...
1
vote
1answer
125 views

Ergodicity and convergence time in Probabilistic Cellular Automata

Has the following conjecture been prooved, or has any step in the direction of its proof been done? "ANY Probabilistic Cellular Automata converge fast on the stationary probability distribution iff ...
3
votes
2answers
208 views

Ergodicity for a Probabilistic Cellular Automaton on a finite space

Let's consider a Probabilistic Cellular Automaton on a one dimensional lattice $S$. Each site of the lattice can have two states, $0$ and $1$. The transition probability acting on each site is: ...
0
votes
0answers
99 views

Stationarity of an Integral Process

Let $f$ be a continous deterministic function defined on $\left[0,c\right]$ and $(B_{t}^{H})_{t\geq 0}$ be a fBM with $H\in \left(0,1\right)$. We define a Process $\left(X_{t}\right)_{t\geq 0}$ with ...
1
vote
1answer
152 views

Tail of solutions of a stochastic differential equation

As we know, solution to $dX_t=\mu dt+\sigma dW_t$ is normal distributed and is light tailed; solution to $dX_t=\mu X_tdt+\sigma X_t dW_t$ is log-normal distributed and is heavy tailed. Is there any ...
3
votes
2answers
285 views

Continuity of hitting distributions

Hi everybody Let $U$ be the domain (as shown in the picture) and $\bar{U}$ its closure, further more set $\partial_r U$ to be the reflecting boundary and $\partial_a U$ the absorbing one. The process ...
1
vote
1answer
247 views

Limit of a rescaled random sum of i.i.d. random variables

Consider a sequence of i.i.d. random variables $(X_i)_{i \in \mathbb N}$ and let $S_n=X_1+\dots+X_n$ For every $\alpha \in ]0,+\infty[$, let $N(\alpha)$ be a discrete random variable on $\mathbb N$, ...
2
votes
1answer
200 views

Applications of this project

Hi Guys, Just wondering if you could suggest applications of distribution of the supremum of a fractional Brownian motion process with a drift ? Also if you could possibly recommend how to approach ...
0
votes
1answer
109 views

Does a definition for delta sequences in the multidimensional case exist?

Hi, does anybody know a good book on multidimensional delta sequences? Thanks in advance Imma
2
votes
3answers
503 views

exchangeable normal r.v.s

Usually "exchangeable normal random variables" means jointly normal random variables $X_1,\ldots,X_n$ (i.e. so distributed that every linear combination of them is normally distributed) that are ...
0
votes
1answer
1k views

kalman filter: understanding the mathematical part

i am currently reading the Probabilistic robotics book where the filters are discussed. Such filters as kalman filter or particle filters. Now I can understand one thing while reading about the ...