A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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24 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 $C(S)$ ...

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**0**answers

30 views

### How to compute the likelihood (and MLE) function of a simple point process

I have a large number of event times for a point process that I would like to model with a univariate point process with event arrival rate:
$$ \lambda(t) = \mu + \sum\limits_{t_i<t}{\alpha ...

**2**

votes

**1**answer

94 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 ...

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votes

**1**answer

80 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 ...

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45 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 ...

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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 ...

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votes

**1**answer

93 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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53 views

### Variance of continuous stochastic process

In the paper "Directed Information, Causal Estimation, and Communication in Continuous Time" the author show an example of continuous Gaussian Channel:
Let $\{B_t\}$ be a standard Brownian motion and ...

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**1**answer

62 views

### Numerical computation of Skorokhod integral

How can I numerically compute the Skorokhod integral of a non-adapted process? If it is adapted, that is easy since the integral is just an Ito integral.
I have found that computing the Malliavin ...

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vote

**0**answers

82 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 ...

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167 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 ...

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votes

**3**answers

455 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 ...

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**0**answers

174 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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votes

**0**answers

30 views

### Markov decision processes: action set revealed at point of decision

I have a problem which looks like a finite horizon Markov decision process (MDP), except the action space at each time is revealed at the decision making point. There is no way to know before hand the ...

**5**

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**0**answers

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**

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**1**answer

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. ...

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**1**answer

80 views

### Increasing stochastic process

I have the following, seemingly simple question:
Consider a stochastic process $(X_t)$ satisfying $X_t\le X_s$ a.s. for all $t\le s.$ My question is: Does there exist a modification $\tilde{X}$ of ...

**4**

votes

**1**answer

199 views

### “Average” Voronoi diagrams without probability?

A plane Poisson process with uniform intensity scatters "sites" about the plane. If I'm not mistaken, in a sense the "average" Voronoi diagram of that set of sites is a honeycomb. I know it's been ...

**2**

votes

**2**answers

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 ...

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**1**answer

131 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 ...

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18 views

### Examples of POMDPs where the actions impact the transitions of the underlying markov Chain

I am not sure if the following is a legitimate question for this board.
I am looking for examples of Partially observed Markov decision processes (preferably infinite horizon, Discrete time, Discrete ...

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votes

**1**answer

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 ...

**2**

votes

**1**answer

149 views

### Equicontinuity and $L^2$ convergence imply uniform convergence

I'm currently working through an old Paper of Garsia, Rodemich and Rumsey (A Real Variable Lemma) and theres one thing i don't get. Suppose $(f_n)_{n\in\mathbb{N}}$ is a sequence of continuous real ...

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**1**answer

63 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 > ...

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votes

**1**answer

29 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 ...

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**0**answers

42 views

### question related to Tanaka Formulae

Supposse $X=(X_t)$ is a cadlag martingale taking values in $\mathbb{R}$. If $f:\mathbb{R}\to\mathbb{R}$ is a convex function, then we have Tanaka Formulae. Now let $g: ...

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vote

**0**answers

81 views

### Eigenvalues of matrix products [closed]

Hi my problem is with row stochastic matrices. Its known that if we keep multiplying this row stochastic matrices, we will get a rank one row stochastic matrix. Rank one means that all the rows has ...

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**0**answers

36 views

### question about the optimal decomposition of supermartingale

Given a filtered probability space $(\Omega, \mathbb{F}, \{\mathcal{F}_t\}_{0\le t\le 1}, \mathbb{P})$, let $X$ be a cadlag martingale and $V$ be cadlag supermartingale. Suppose $V$ has the following ...

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**1**answer

232 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-) $S^X_n=\sum_{i=1}^n X_i$, ...

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45 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 ...

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votes

**3**answers

440 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 ...

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votes

**1**answer

99 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 ...

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**0**answers

105 views

### Reference for a General Theory of Sequences?

Since decades, mathematicians are studying function spaces, discovering new structures more and more adapted for a general theory of functional analysis.
In that works, sequence spaces are generally ...

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**1**answer

80 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

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

**0**answers

102 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

**1**answer

225 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 ...

**7**

votes

**1**answer

314 views

### About the convergence rate for an approximation to the heat kernel

Let $G(t,x)$ be the heat kernel
$$
G(t,x)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}, \quad t>0, \:x\in\mathbb{R}.
$$
Here is one approximation to $G(t,x)$:
$$
G_\epsilon(t,x)=e^{-t/\epsilon} ...

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**0**answers

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

**2**answers

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

**1**answer

107 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 = ...

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**0**answers

91 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 ...

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**3**answers

69 views

### a special filtration satisfying $0$-$1$ law

Let $\xi$ be a uniformly random variable on $[0,1]$ defined on some probability space $(\Omega,\mathcal{F})$. Define the process $\xi_t:=\min(\xi,t)$ for $0\le t\le 1$. And let ...

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votes

**1**answer

123 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
...

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**1**answer

112 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
...

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vote

**1**answer

139 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
...

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votes

**1**answer

70 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} ...

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**0**answers

63 views

### a question about Dambis, Dubins-Schwarz Theorem

Let $M=(M_t)_{0\le t\le 1}$ be a continous $\mathbb{F}=\{\mathcal{F}_t\}_{0\le t\le 1}$-martingale s.t. $M_0=0$. Now my question is whether there exists a Brownin motion $B$ s.t.
...

**0**

votes

**0**answers

59 views

### Fredholm Integral Involving Stochastic Process

I wish to solve an integral equation of the form $$g(X) = c\int_0^1 K(X,t)f(t) \ dt $$ where $f\in L^1([0,1])$ and $g$ is some function on finite sequences of random variables. So, $X$ is a stochastic ...

**0**

votes

**0**answers

24 views

### How to get expectation of function of an optimal stopping time

Let $P_t$ be the posterior probability, $p_0$ be the prior probability. The evolving process of $P_t$ is: $ dP_t=\frac{P_t}{P_t+(1-P_t)(1-\lambda^kdt)}-P_t$.
The optimal stopping time problem is ...