0
votes
0answers
18 views

Mutual information staying constant under composition of channels

Consider the following scenario: one has 2 communication channels $C_1$ and $C_2$. Denote by $p(x)$ the input probability distribution. The mutual information between the input and the output of ...
3
votes
2answers
63 views

Is this generating family of a measurable space of point measures a pi-system?

I'm learning some probability and measure theory and working my way through the first few paragraphs of [1]. My question is perhaps too basic for Math Overflow, but I hope it is welcome here. Point ...
-1
votes
0answers
44 views

A question on Burdzy's proof of non-increase of Brownian motion [on hold]

I am reading Burdzys' paper "On non-increase of Brownian motion" and I don't understand why P(A0)=0 implies that a.s. Brownian motion has no point of increase. Can anyone help me please? Here is the ...
0
votes
0answers
21 views

Explicit construction of transition semigroup from generator for completely independent spin system (Feller process)

As an example of how to obtain the transition semigroup from the probability generator for a Feller process, I am looking at the easiest spin system, namely with all sites independent. Notationwise, I ...
1
vote
0answers
30 views

Jumps of jump diffusions

Let $W$ be a Brownian motion and $N$ a Poisson random measure defined on $\mathbb R_+ \times \mathbb R_0^n$ ($\mathbb R_0^n:=\mathbb R^n-\{0\}$) with compensator $\tilde N(dt,dz):= N(dt,dz) - dt ...
3
votes
1answer
141 views
+50

An efficient method to find the MLE of the combination of two point processes

I have a point process defined in two parts as follows. Consider first the main process which we call $A$ which is homogeneous Poisson process with conditional intensity $$\lambda(t) = \mu$$ For ...
2
votes
1answer
59 views

weak convergence of the solutions to stochastic heat equation

$W(t,x)=\sum_ic_ie_i(x)B^i_t$ is a Brownian motion in $L^2(R^d)$, where $\{e_i\}$ is the standard orthogonal basis and $\sum_ic_i^2<\infty$. $$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$ ...
1
vote
1answer
76 views

Intuition for the definition of a probability generator of a Feller process

I am working with the definition of a probability generator of a Feller process as stated in Liggett's book, "Continuous time Markov processes": Let $S$ be a compact state space and denote by $C(S)$ ...
3
votes
1answer
101 views

Approximating Markov chains by Brownian motion

I would like a result along the following lines to be true, but haven't been able to locate it in the literature; pointers would be welcome. Let $X_t$ be a finite-state, irreducible, aperiodic Markov ...
0
votes
1answer
86 views

Combine results with different veracity [closed]

I have 3 neural networks processing 3 different vectors of values. Each NN processes a sample of it's vector and gives binary result (y/n) that is correct with given probability. All 3 NNs give answer ...
1
vote
0answers
56 views

Branching Brownian Motion and the KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
0
votes
0answers
130 views

What is a.e. a.s [migrated]

I am reading a paper which uses almost everywhere almost surely (a.e.,a.s.) simultaneously, I am not quite sure what it means then. To be specific, they consider a stochastic process $\{X_t\}$ such ...
0
votes
1answer
109 views

On the superior of generalized Ornstein-Uhlenbeck process

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde: $dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $ $\partial_x X_t(0) = \partial_x X_t(1) = 0, $ $X_0 = 0, $ ...
1
vote
0answers
84 views

random walk with reflecting barriers [closed]

Consider a random walk on the line 1,...,d. You start at point 1. At each step you flip a coin: heads means go left, tails means go right. If you're at 1 and get a heads, just stay where you are (same ...
1
vote
0answers
183 views

Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
4
votes
3answers
457 views

How to explain “Feller process” to an undergraduate student?

I had to explain in informal terms what a Feller process was, to undergraduate students who understand Markov property, Poisson processes and such. It was easy to define Levy process as generalisation ...
5
votes
0answers
180 views

Strong Markov property for Poisson point process

The question is thoroughly contained in the title. I just say that I would only like to find a reference for this question. I have searched in some books, to no avail. Here is what I mean exactly. ...
5
votes
0answers
232 views

Do isonormal Gaussian processes have measurable sample paths?

Let $H$ be a real separable Hilbert space. Let $W=\{W(h):h\in H\}$ be a real-valued stochastic process defined on a complete probability space $(\Omega,\mathcal{F},P)$. Assume that $W$ is a centered ...
5
votes
1answer
327 views

References for a physicist migrating to stochastic processes

I've studied "Markov Chains" - Norris and "Measure, Integral and Probability" - Capinski, Kopp. Now, I'm looking for a couple of books (or other references) that help me bridging these two topics. ...
2
votes
2answers
163 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 ...
0
votes
1answer
152 views

maximum of certain Gaussian processes

Let $\mathbf{a}_k\in\mathbb{C}^n$ for $k=1,2,\ldots,m$ be i.i.d. standard complex normal random vectors with distribution $c\mathcal{N}(0,\mathbf{I})$. I am interested in a tight upper bound on the ...
3
votes
1answer
184 views

Properties of the algebraic self-difference set of Brownian motion zeros

As I was trying to exhibit new interesting(?) path transformations of Brownian motion, I became interested in the (random) set of times $t$ such that $B(t)=B(t+1)=0$, where $B(t)$ denotes a standard ...
1
vote
1answer
248 views

Comparing the expected stopping times of two stochastically ordered random processes (Added:(14.05.2014))

Information: a-) $X$ and $Y$ are two continuous random variables on $\mathbb{R}$ having continuous distribution functions $F$ and $G$ with $G(y)\geq F(y)$ for all $y$. b-) $S^X_n=\sum_{i=1}^n X_i$, ...
0
votes
0answers
46 views

Is Feller process time-homogeneous?

The first question is just the title. The second question is that can a Feller process which is not a Levy process has the same infinitesimal generators as Levy process? I am confused in the ...
8
votes
3answers
441 views

A question on Cramer's theorem

Almost everybody is familiar with Cramer's theorem: a sum $X+Y$ of of independent random variables is normal if and only if both $X$ and $Y$ are normal. Are there any other classes of distributions ...
2
votes
1answer
100 views

Linear or quadratic combinations of i.i.d. random variables [closed]

I already posted this question here http://math.stackexchange.com/questions/769920/law-of-large-numbers-for-linear-quadratic-combinations-of-i-i-d-random-variab but I received no answers. Let ...
4
votes
1answer
82 views

On the moments of Lévy processes

For a Brownian motion $B_t$, the evolution of the moments with $t$ obeys the simple rule: $$\mathbb{E}[|B_t|^p] = \kappa_p |t|^{p/2},$$ with $\kappa_p<\infty$. The proof only requires to remark ...
4
votes
1answer
144 views

Concurrency related problems in $n$ independent, parallel $M/M/1$ queues

Queueing Model: Consider $n$ independent, parallel $M/M/1$ queues with identical arrival rate $\lambda$ and service rate $\mu$. For each $M/M/1$ queue, we use the FCFS (First Come First Served) ...
1
vote
0answers
107 views

Fundamental theorem of calculus for iterated stochastic integrals

I'm trying to find the rate (or a bound for it) with which an iterated integral of the type $$\int_{-h}^0 \int_{-h}^{t} A_s d B_s A_t d B_t$$ converges to zero (in probability/distribution) for $h ...
6
votes
1answer
230 views

Properties of the time integral of Wiener process

Let $W_t$ be a Wiener process and consider the time integral $$ X_T:= \int_0^T W_t dt $$ It is often mentionend in literature that $X_T$ is a Gaussian with mean 0 and variance $T^3/6$. I am ...
1
vote
0answers
44 views

A counterpart of Karhunen theorem

According to the Karhunen theorem, if the correlation function of a process $X(t)$ can be represented as $$ R(t,s)= \int_{\Lambda} f(t, \lambda) \overline{f(s, \lambda)}d\nu(\lambda) $$ then the ...
1
vote
2answers
107 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
1
vote
1answer
108 views

M/M/1 Queue with probability of new customer leaving [closed]

I'm looking at a M/M/1 queue system and trying to show that $\{M_t\}_{t\geq}0$, the number of clients in the system, is a birth-death process. In the simplest of cases this is true if $\lambda_i = ...
1
vote
0answers
92 views

number of times Brownian motion hits boundaries

Any experts here please direct me to some appropriate keywords that I can search for. Consider a Brownian motion constrained to an upper and lower boundaries. Let's say I want to know that how many ...
3
votes
1answer
138 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
114 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 ...
1
vote
1answer
140 views

question about uniform continuity under Skorokhod Metric

Let $D=D([0,1], \mathbb{R})$ be the space of cadlag functions $x$ with $x(0)=0$ and $x$ is continuous on $1$. If we endow $D$ with Skorokhod Metric, see: http://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g ...
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} ...
3
votes
0answers
210 views

Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE

Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem $$ \partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x) $$ ...
1
vote
0answers
73 views

question about Doob-Meyer decomposition

Given a filtered probability space and let $X$ be a cadlag local martingale defined on this space. Let $V$ be a cadlag supermartingale and assume we know the following decomposition: ...
4
votes
2answers
98 views

Smoothness of $g(t,x)=\mathbb{E}[f(X_T)|\mathcal{F}_t]$

Assume a process with Itô dynamics of the generic form $$dX_t=\mu(t,X_t)dt+\sigma(t,X_t)dW_t$$ and let $f:\mathbb{R}\to\mathbb{R}$ be borel-measurable. Is the following function smooth ? ...
1
vote
0answers
94 views

Arctic Circle Theorems and the Wave Equation

I've seen the following remark in a number of papers but don't know what to make of it. In this paper by Cohn, Elkies and Propp, it is mentioned that the normalized average Height function ...
1
vote
1answer
165 views

Strong solutions on SDE (stochastic differential equations) with discontinuous drift and diffusion coefficients

I want to get some advice from you about the existence (and the uniqueness if possible) of a strong solution on my SDE. In fact, due to the structure of the problem that I consider, both the drift ...
7
votes
3answers
311 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
75 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 ...
1
vote
0answers
68 views

asymptotic estimate of random walk involved hitting time and return time

Consider a reversible random walk on (say) $\mathbb{Z}$, are there any estimate for the following probability $\mathbb{P}(\tau_n=m<\tau_0^+)$ where $\tau_n$ is the first hitting time at site n and ...
2
votes
0answers
73 views

Existence of a conditional distribution

Let $X$ and $Y$ be standard Borel spaces and let $J$ be an analytic subset of $X\times \mathcal P(Y)$ where $\mathcal P(\Omega)$ is a set of probability measures on a Borel space $\Omega$ endowed ...
3
votes
0answers
85 views

Nonlinear Markov process

Consider the following nonlinear $\mathbb{R}$-valued stochastic recursive sequence: $ X_{n+1} = F(X_n) + W_{n+1}, \quad (W_n)_{n\ge1} \stackrel{ \scriptsize \mathrm{i.i.d.} }{ \sim } \phi. $ How can ...
11
votes
1answer
346 views

Fictitious density of paths of diffusion processes outside the Cameron--Martin space

Let $X_t$ be an $n$-dimensional diffusion process satisfying the following Itō SDE over $[0,1]$: $$dX_t = f(X_t)\,dt + dW_t,$$ where $W_t$ is an $n$-dimensional Wiener process and $f$ is of class ...
1
vote
0answers
85 views

On the solution of a stochastic partial differential equation

Consider a simple SPDE as follows: $\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$, $u(t,0)=u(t,1)=0$, $u(0,x)=v(x)$, where $V$ is a bounded, smooth ...