# Tagged Questions

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

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44 views

### Find the joint density function?

Assume that $X_t$ is the OU process , i.e,
$dX_t=\kappa(\theta-X_t)dt +\sigma dW_t$ where $0\leq t\leq T$ and $X_0=x_0>0$.
Let $q(x)=\frac{\kappa}{\sigma}(\theta-x)x +\frac{\sigma}{2}$.
I want ...

**-2**

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**0**answers

20 views

### Branching process and process stochastic [on hold]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$
Note: This ...

**1**

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**0**answers

21 views

### Spectral densities of stationary Feller processes with no diffusion, constant positive drift and negative jumps

For a (real valued, finite variance, centered) stationary process $X_t$ on $\mathbb R$, the auto-correlation function $k(\tau) = \mathbb E(X_{t+\tau}-X_t)^2$ and its inverse Fourier transform $\rho$, ...

**0**

votes

**1**answer

45 views

### Sum of two parts of a continuous stochastic process

Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all ...

**0**

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**0**answers

20 views

### Writing eigen functions of one Stochastic Process in terms of the eigen functions of another

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...

**0**

votes

**1**answer

318 views

### Generating independent random variable from two correlated random variables

Suppose two random variables $X$ and $V$ are given. I am wondering what kind of condition we need to impose on joint distribution of $V$ and $X$ to make sure that there exists a random variable $Z$ ...

**0**

votes

**1**answer

74 views

### Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...

**2**

votes

**1**answer

174 views

### explicit characterization of the stochastic integrand

Let $V$ be a cadlag positive supermartingale with the following decomposition:
$$V_t=V_0+\int_0^tH_sdX_s-K_t$$
where $X$ is a cadlag local martingale and $K$ is an adapted increasing process with ...

**2**

votes

**0**answers

82 views

### Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

[I've decided to rewrite the question, to make the essential point clearer.]
Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset ...

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vote

**1**answer

119 views

### Arc Sine law for Random Walk conditioned to non-absorption or not?

Let $S_n$ be simple symmetric Random walk on the integers in $[-N,N]$ with states $N$ and $-N$ absorbing. Let $\tau$ be the time to absorption when $S_0 = 0$.
Is the $E(S^{2}_{n}| \tau \geq n)$ ...

**6**

votes

**7**answers

522 views

+50

### Semicircle law universality elsewhere

Wigner's semicircle distribution is:
$$f(x)=\frac{1}{2 \pi}\sqrt{4-x^2}, \ \ -2\leq x\leq 2.$$
Under reasonable conditions, the rescaled eigenvalue density of random symmetric matrices $M_n$ follows ...

**2**

votes

**0**answers

53 views

### Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may I ask for your help?
First the setting:
I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...

**0**

votes

**0**answers

70 views

### Why is this distribution exponential?

Take the interval $[0, 1]$.
Now sample 10000 points in this interval randomly according to the uniform distribution.
The fact is that the distribution of the distances between adjacent points on ...

**5**

votes

**1**answer

388 views

### Law of Iterated Logarithm for autoregressive process

Suppose that $\{X_i\}$ is an $\mathrm{AR}(r)$, defined by:
$X_{i}= h(i) + \varepsilon_i $,
$h(i)=\alpha_1 X_{i-1} + \dots + \alpha_{r} X_{i-r}$
where $\{\varepsilon_i\}$ are i.i.d. ${\cal ...

**4**

votes

**1**answer

254 views

### Quasi-stationary distribution for a death process

In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a ...

**1**

vote

**0**answers

46 views

### Convergence to equilibrium for time in-homogeneous diffusions

Consider the long time behavior for a time in-homogeneous diffusion such as
$$dX_t = dB_t - \nabla V(X_t)\,dt + b_t(X_t)dt,$$
where $V(x)$ is a smooth convex function and $b_t(x)$ is a time-dependent ...

**3**

votes

**1**answer

292 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 $\sum_ic_i^2<\infty$.
$$\partial_t u(t,x)=\Delta u(t,x)+u(t,x)\dot{W}(t,x)$$
...

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vote

**1**answer

51 views

### When is the hitting time of an open set a stopping time?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_t)_{t \in [0,T]}$ a filtration. Consider an adapted, right-continuous process $X$ taking values in $\mathcal{X}$ and let $B$ be ...

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**0**answers

37 views

### characterization of the equivalence between two probability measures

Let $X=(X_1,...,X_n)$ be a canonical process defined on the Euclidean space $R^n$, i.e. $X(x)=x$ for all $x\in R^n$ and $\mathbb F=\{\mathcal{F}_k\}_{1\le k\le n}$ be its natural filtration, i.e. ...

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vote

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44 views

### Quadratic Variation of a Martingale in Hlibert Spaces

I'm looking at a Martingale (actually a Martingale difference sequence),
$$
M_n = \sum \delta M_n,
$$
and I'd like to prove something about convergence. If Martingale is Hilbert space valued ...

**1**

vote

**0**answers

29 views

### Stochastically coloring a graph in a local way

Suppose you are assigning values in $S$ (assume $|S|<\infty$) to nodes of a (directed) graph in a stochastic way. At the beginning, none of the node is assigned values. At the $i^{th}$ step, you ...

**0**

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**0**answers

48 views

### conditionning by a Gaussian field

I know that if $(X,Y)$ is a Gaussian vector, then $(X|Y=y)$ is a Gaussian vector which covariance matrix is explicit in function of the covariance matrix of $(X,Y)$, and does not depend on $y$.
What ...

**1**

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34 views

### Asymptotics of Variable Drift Ornstein–Uhlenbeck Process

The Ornstein–Uhlenbeck process is defined as the stochastic process that solves the following SDE:
$dx_t = \theta (\mu-x_t)\,dt + \sigma\, dW_t$
where $\theta>0$, $\mu$ and $\sigma>0$ are ...

**0**

votes

**0**answers

20 views

### Probability that a Lipschitz Gaussian field vanishes nearby

Let $X(t),t\in T$ be a Gaussian field on some bounded open set $T$ of $R^d$. Let $x\in T,\varepsilon>0$. Is there a finite number $K$ such that $P(\exists t\in B(x,\varepsilon):X(t)=0)\leq ...

**2**

votes

**1**answer

120 views

### Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...

**0**

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**0**answers

55 views

### Markov chains on a polyhedron

A modification of a question from Gerard Letac (1976): A m-sided q-adjacent-faced polyhedron has one of its faces "up." Each round, the polyhedron rolls so that any of the adjacent faces is now up. ...

**2**

votes

**1**answer

197 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 stochastic ...

**1**

vote

**0**answers

56 views

### permutations sampling by probability matrix [closed]

I am looking for effective and reliable algorithm which is able to generate random samples of permutations by square doubly stochastic probability matrix $P$ (n x n) distribution ($\sum_{i}p_{i,j} = ...

**2**

votes

**2**answers

215 views

### Time integral of a diffusion

Define $\bar\sigma^2_t=\frac{1}{t}\int_0^t\sigma^2(X_s)ds$ where $\sigma(x)\geq0$ is a measurable function and $X_t$ a diffusion process defined by
\begin{equation}
...

**3**

votes

**1**answer

265 views

### Convergence of random variables with hypergeometric distribution

This is a very interesting conjecture of large scale property of hypergeometric distribution.
Let $a>1$ be a integer constant, $N\in\mathbb{N_+}$, for any $x<N-1$, consider $N+(a-1)x$ balls in ...

**1**

vote

**1**answer

407 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 ...

**2**

votes

**1**answer

152 views

### Question about the stochastic integral of martingales

Let $M=(M_t)_{t\ge 0}$ be a continuous martingale defined on some filtered probability space taking values in $R$. Let $H=(H_t)_{t\ge 0}$ be some bounded progressively measurable process, i.e.
...

**8**

votes

**1**answer

1k views

### Gluing Markov processes

I am looking for a reference on the gluing together of strong Markov processes to get a new one.
Here is an example of what I have in mind. Let $B^1, B^2, \ldots $ be independent one-dimensional ...

**7**

votes

**1**answer

488 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. ...

**0**

votes

**1**answer

270 views

### Supremum in a Markov chain model

A Markov chain $X$ with finite state space $\{1,2,\cdots,N\}$ is defined on a probability space $(\Omega, P, \mathcal{F})$ equiped with filtration $\{\mathcal{F}_t\}$. And we assume that we can reach ...

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vote

**0**answers

34 views

### Oscillating Markovprocess Transition Probabilities

Suppose we have an irreducible positive-recurrent Markov process $\{X(t), t\geq0\}$ with generator $G$. Let $P(t)$ be its transition probability matrix and $\pi$ its stationary distribution. Then we ...

**7**

votes

**1**answer

243 views

### understanding of rough path

A rough path is defined as an ordered pair
$ (X, \mathbb X)$, where $X$ is a path mapping from $[0,T]$ to some Banach space $V$
and $\mathbb X:[0,T]^2 \mapsto V^2$ is another mapping for additional ...

**0**

votes

**0**answers

27 views

### On the induced norms of stochastic operator and its adjoint operator

The background: when studying the paper published in Automatica named '$H_{\infty}$ control and filtering of discrete-time stochastic systems with multiplicative noise' (volume 37, pp. 409-417), I ...

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**0**answers

23 views

### Truncated Robbins-Monro

I'm reading Han-Fu Chen's book "Stochastic Approximation and Its Applications", and in Chapter 1, he's got a statement of a theorem and proof on a truncated Robbins-Monro algorithm. In this version, ...

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vote

**0**answers

70 views

### Scaling of First-passage times for Random Walk on integer lattices

Consider simple symmetric random walk $S_{n} = (S_{n}^{(1)},\dots,
S_{n}^{(d)})$ on the d-dimensional integer lattice with starting point the origin.
Let $\tau_{N}$ be the first time $S_{n}$ exits ...

**4**

votes

**1**answer

150 views

### Upper bound of the waiting time of a sum process

Let $n \in \mathbb{N}$, $x_1, \ldots, x_n \in (0,1)$ fix but arbitrary, s.t. $\sum_{i=1}^n x_i = 1$. Let $X_i \sim \operatorname{Unif}(\{x_1, \ldots, x_n\})$ i.i.d., and $T_n = \min\{t \in \mathbb{N} ...

**2**

votes

**0**answers

62 views

### Cramer-Rao type bound for absolute estimation error

Let $\{X_1, X_2, \ldots, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown ...

**0**

votes

**1**answer

51 views

### How can two random variables are continuous infers that their jointly random variable is continuous [closed]

We assume that $\forall a,b$ suchthat $a^2+b^2>0$, $aX+bY$ is continuous random variable.
But we don't assume that $X$ and $Y$ are independent.
My question is the following:
Is it true that the ...

**4**

votes

**2**answers

226 views

### Is $B(t-1)$ an Ito process?

Let $I(\cdot)$ be an indicator, and $B_{t}$ be an 1-dim standard Brownian motion in a nice filtered probability space
$(\Omega, \mathcal{F}, P, \mathcal{F}_{t})$. We consider a random process
$$Y_{t} ...

**7**

votes

**1**answer

209 views

### Central limit theorem for biased random walk

Define random variables $X_n$ by $X_0 = 0$ and
\begin{equation*}X_n - X_{n-1} = \begin{cases}
1 & \text{with probability } g(X_{n-1}) \\
0 & \text{with probability } 1-g(X_{n-1})
\end{cases}
...

**0**

votes

**1**answer

102 views

### probability in galton watson processes [closed]

I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger.
I don't understand what Z_(n,i) stand for.
And in the proof of Theorem ...

**1**

vote

**2**answers

132 views

### Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution ...

**0**

votes

**0**answers

39 views

### Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that ...

**2**

votes

**1**answer

115 views

### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...

**1**

vote

**0**answers

117 views

### Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = ...