# Tagged Questions

**0**

votes

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73 views

### Poisson approximation of random sub-graphs

I add the edges of $G(n)$ the complete graph on $n$ vertices one by one, at random and without replacement, and denote by $G(n,m)$ the resulting Erdos Renyi random graph process. At step $m$ in the ...

**2**

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**0**answers

37 views

### Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular,
Have the numerical implications, if any, of the results in ...

**0**

votes

**0**answers

31 views

### Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit:
$$
dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC
$$
where $W$ is a standard Brownian motion ...

**1**

vote

**1**answer

74 views

### Diffusion processes with different diffusion coefficients and absolute continuity

I would first of all like to say that I am an analyst, and so I am familiar with probabilistic methods only on a basic level.
My initial situation is the following. Consider two SDEs:
\begin{align}
...

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votes

**1**answer

47 views

### Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.

**2**

votes

**1**answer

59 views

### Proof for power-law tail of Poisson-Dirichlet distribution (Pitman-Yor process & Zipf's law)

I'm trying to understand the motivation of using Pitman-Yor (PY) processes in language modeling, in particular Teh's hierarchical LM based on PY processes. A motivation frequently stated in research ...

**7**

votes

**3**answers

234 views

### A learning roadmap to the Schramm-Loewner evolution (SLE) for the complex analyst

I would like some good references to learn about the Schramm-Loewner evolution (SLE), for a complex analyst with no background in probability.
A quick google search gave a lot of references on SLE ...

**2**

votes

**0**answers

33 views

### Random matrices whose limit gives exact Wigner surmise

Let $M$ come from an ensemble of $N\times N$ matrices. The Wigner surmise is density function $p^W_0(s)=\frac{\pi}{2}se^{-\pi s^2/4}$. From a random matrix point of view, we can write ...

**6**

votes

**2**answers

105 views

### Ising model on lattices with (vertical side length) $\neq$ (horizontal side length)

Consider the Ising model with nearest neighbours interactions on a rectangular lattice $L\times M$.
If $L=M$ ($2$-dimensional square lattice), it is known (e.g., by Peierls' argument or Onsager's ...

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37 views

### “Bad” lower functions for a Bessel process?

Let $(X_t, t \ge 0)$ be a Bessel($\delta$) process, for some dimension $\delta > 2$, starting, say, from $1$.
Let $f: \mathbb{R}_+ \to \mathbb{R}_+$ be an upper semicontinuous function; assume ...

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vote

**1**answer

62 views

### The uniform integrability of exponential of Poisson process

Let $\left\{N_t,\mathcal{F}_t\right\}_{t\ge0}$ be a Poisson process with intensity $\lambda>0$. Define
$$X_t=\exp{\left[N_t-\lambda t(e-1)\right]}$$
I can show that $\{X_t,\mathcal{F}_t\}_{t\ge0}$ ...

**3**

votes

**1**answer

107 views

### Conditional Form of Rosenthal's Inequality

Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following:
If $\{S_i, \mathcal{F}_i, 1\leq i \leq n\}$ is a martingale and ...

**2**

votes

**2**answers

186 views

### Ito diffusion with highly oscillatory diffusion coefficient

Consider the stochastic differential equation on $\mathbb R$
$$
dx_t = f(x_t) dt + g(\omega t)\, dW_t
$$
with $W_t$ a standard Brownian motion, $f:\mathbb R \to \mathbb R$ a smooth function, and ...

**3**

votes

**1**answer

174 views

### Unusual augmentation of a filtration

consider a probablity space $(\Omega,\mathcal{F}, \mathcal{P})$ and a filtration $(\mathcal{F}^0_t)$. In general $(\mathcal{F}^0_t)$ doesn't satisfy the usual conditions (it is not both complete at ...

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vote

**0**answers

68 views

### Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...

**3**

votes

**1**answer

86 views

### Regularity of finite variation kernels in the (intersection) of the semimartingale spaces $H^p$

Suppose you have a continuous semimartingale $S_t=M_t + A_t$ where $A_t$ is the continuous finite variation part which has the form $A_t = \int_0^t b_s \, \mathrm{d} s$, where $\int_0^{\infty} |b_s| ...

**3**

votes

**1**answer

120 views

### Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals?

If $(M_t)_{t \geq 0}$ is a continuous local martingale, one can define the iterated integrals $I_0=1$, $I_1(t)=M_t$ and for $n \geq 2$ $$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$ By noting ...

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votes

**1**answer

196 views

### Birkhoff Ergodic Theorem and Ergodic Decomposition Theorem for Continuous-Time Markov Processes

I have a couple of questions regarding ergodicity for Markov processes in continuous time. (In particular, the first question seems like it should be particularly basic, and yet I haven't managed to ...

**3**

votes

**1**answer

146 views

### Stability of convergence in distribution under randomization

Suppose you have a sequence of non-negative stochastic processes $(X^n)_{t \in \mathbb{R}}$, $n \geq 1$, with continuous paths and continuous in $t$ such that
$$\int_{-\infty}^{\infty} X^n_t \, ...

**3**

votes

**1**answer

143 views

### Can't figure out “standard application” of the Garsia-Rodemich-Rumsey Lemma

I'm currently reading the paper http://arxiv.org/abs/0908.2473 and can't figure out what they call a "standard application" of the Garsia-Rodemich-Rumsey lemma (see p.8). Summed up, they have a ...

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**0**answers

51 views

### Almost sure transversality of smooth random maps

I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO.
Let me first formulate the ...

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**0**answers

69 views

### Reference request: Stochastic integration and martingale theory on the whole real line

I'm looking for a thorough treatment of stochastic integration and/or martingale theory on the whole real line, i.e. a way to construct a Brownian motion $(B_s)_{s \in \mathbb{R}}$ (if a two-sided BM ...

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votes

**1**answer

85 views

### Measurable functions lifted onto a space of point measures are measurable

I've been reading [1] and attempting to prove statements given without proof. In the paper the authors construct a measurable space of measures over a base space, and as an aside show an elegant way ...

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46 views

### Reference for “Newtonian capacity estimates probability that A is hit by a Brownian motion”

I am looking for the following statement
"In fact, the Newtonian (logarithmic) capacity gives an estimate, up to a constant factor, the probability that A is hit by a Brownian motion started, say, ...

**2**

votes

**1**answer

79 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

**2**answers

87 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 ...

**0**

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24 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 ...

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47 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

**1**answer

163 views

### 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

**1**answer

117 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

**1**answer

92 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

**1**answer

120 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

**1**answer

95 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

**1**answer

101 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

**1**answer

170 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, $
...

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vote

**0**answers

112 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

**0**answers

227 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

**3**answers

480 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 ...

**6**

votes

**1**answer

270 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. ...

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245 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 ...

**6**

votes

**2**answers

519 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

**2**answers

198 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

**1**answer

162 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 ...

**5**

votes

**1**answer

267 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

**1**answer

269 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

**0**answers

51 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

**3**answers

496 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

**1**answer

115 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

**1**answer

85 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

**1**answer

153 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) ...