# Tagged Questions

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

**5**

votes

**2**answers

242 views

### Infimum of Gaussian process

Consider a Gaussian Process $g\sim GP(\mu,k)$ with mean zero $\mu\equiv0$ and continues covariance $k(t_1,t_2)=k(|t_1-t_2|)$ defined on the interval $A=[0,T]$. I'd like to make no assumptions about ...

**0**

votes

**0**answers

38 views

### Uniform convergence problem of the iterative function series

A process $\{\theta_{t}\}_{t=1}^{\infty}$ with finitely continuous state space $\mathcal{S}=[\underline{\theta},\bar{\theta}]$.The transition density is $\phi(\theta_{t},\theta_{t+1})$.I have known ...

**1**

vote

**0**answers

42 views

### Basic results for chi square processes

I could not find any introductory material with basic results regarding chi-square processes. Their definition from The Supremum of Chi-Square Processes
is as a sum of $d$ squares of independent ...

**3**

votes

**1**answer

61 views

### Number of samples needed as input to Bernoulli factory

Let $\{X_i\}$ denote an i.i.d. sequence of Bernoulli variables with parameter $p$. A Bernoulli factory is a procedure that generates events with probability $f(p)$ using the observations $\{X_i\}$, ...

**0**

votes

**0**answers

65 views

### Markov chain matching local time

Let $\left(X_{t}\right)_{t\geq0}$ be a Markov process taking values in
a finite state space $E$. Its local time at $y\in E$ started at
$x\in E$ is defined as
$$
...

**1**

vote

**0**answers

59 views

### Stationary distribution of two-dimensional Markov Process?

A two-dimensinal Markov process $\{\theta_{t},S_{t}\}_{t=1}^{\infty}$ where $\theta_{t} \in \Theta$ and $S_{t} \in S$.$\Theta$ is a continuous state space and $S$ is a discrete state space. Suppose I ...

**2**

votes

**0**answers

69 views

### integrability of Brownian motion stopped at some stopping time

Let $B=(B_t)_{t\ge 0}$ be a standard Brownian motion starting at zero and denote by $S=(S_t)_{t\ge 0}$ its running maximum, i.e. $S_t=\sup_{0\le s\le t}B_s$. Given a fixed number $p>1$, define the ...

**0**

votes

**0**answers

39 views

### Systems of stochastic differential equations with non-Lipschitz coefficients

I am looking for references to any literature which might consider the existence / behavior / regularity of solutions to systems of stochastic differential equations with non-Lipschitz coefficients.
...

**-2**

votes

**1**answer

146 views

### expected value of cosine wirh Gaussian phase

Is there a solution to the expected value/variance for a Gaussian with random phase:
$$\cos(\omega_0 t + \phi), \qquad \phi \sim \cal{N}(0,\sigma^2) $$
?
For $t=0$, the solution is for example ...

**4**

votes

**0**answers

50 views

### Existence of martingales given some constraint on laws

Let $X=(X)_{0\le t\le 1}$ be a continuous martingale starting at $0$, then denote by $\mu$ and $\nu$ the probability laws of $\int_0^1X_t \mathrm{d}t$ and $X_1$. Then it is easy to see that the couple ...

**2**

votes

**2**answers

300 views

### Can I use Birkhoff's Ergodic Theorem for Vector Valued Process?

I have a stationary process $\{u_n\}$ and I have a function $f:\mathbb{R}^L\to \mathbb{R}^+$. I want to evaluate the following limit $$\lim_{n\to \infty}\frac{1}{n}\sum_{k=1}^n g(f(\mathbf{u}_{k}))$$ ...

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votes

**0**answers

39 views

### Proof of Linear Stochastic Sate-Space Model is Gaussian Process

I would like to prove that the vector-linear stochastic state space model
$$
\dot{x}(t)=A(t)x(t)+B(t)u(t)+G(t)q(t) \\ y(t)=C(t)x(t)+D(t)u(t)+F(t)r(t)
$$
corresponds to a particular multi-output ...

**2**

votes

**0**answers

128 views

### The existence of stationary measures for certain Markov process

My question is that:For a discrete-time random process $\{x_{t}\}_{t=1}^{\infty}$ and $x_{t} \in \Omega$ where $\Omega$ is a general state space(If $\Omega$ is a discrete space, it is a discrete-time ...

**0**

votes

**1**answer

129 views

### Stationary distribution with exponential transition density function

A Markov chain with continuous state space has a transition exponential density function:
$$p(x_{t},x_{t+1})=\frac{1}{x_{t}}exp(-\frac{x_{t+1}}{x_{t}})$$
i.e. the realized value in period t is the ...

**0**

votes

**1**answer

114 views

### Distribution of bounded summation of i.i.d random variables

We have a set of positive random variables $\boldsymbol X=\{X_1, X_2,\ldots\}$, where $X_1, X_2,\ldots$, are independent and identically distributed (i.i.d.). The CDF $F(x)$ and PDF $f(x)$ for $X_i$ ...

**0**

votes

**0**answers

43 views

### Construction of a path of quadratic variation

This question has been posted to Stack Exchange earlier, and no answer is available yet.
Consider a path $x: [0,1] \to \mathbb R$. it's $p$-variation on an interval
is defined by
$$V_{p}(x, [a, b]) ...

**0**

votes

**0**answers

44 views

### Ergodicity property for continuous-time Harris positive Markov process

I have posted this question on there, but got no answer.
The following theorem is Theorem 13.3.3 of Meyn and Tweedie's Markov Chains and Stochastic Stability on page 328:
Theorem 13.3.3. If ...

**2**

votes

**0**answers

41 views

### The inter-request time distribution after aggregating some arrivals in the renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process"
The inter-arrival time of a renewal process, $t$, conforms to a general distribution, denoted by PDF ...

**1**

vote

**0**answers

78 views

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

For a (real valued, finite variance) stationary process $X_t$ on $\mathbb R$ with $\mathbb EX_t=m$, the auto-correlation function $k(\tau) = \mathbb E[(X_{t+\tau}-m)(X_t-m)]$ and its inverse Fourier ...

**0**

votes

**0**answers

36 views

### Writing eigenfunctions of one stochastic process in terms of the eigenfunctions of another

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

**3**

votes

**1**answer

114 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 ...

**2**

votes

**0**answers

67 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 $\mu_{i,j}$. ...

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votes

**0**answers

82 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 ...

**7**

votes

**7**answers

673 views

### 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 ...

**1**

vote

**0**answers

62 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 ...

**1**

vote

**1**answer

217 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 ...

**0**

votes

**0**answers

46 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. ...

**1**

vote

**0**answers

91 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

39 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 ...

**1**

vote

**0**answers

99 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

52 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 ...

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votes

**0**answers

37 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 ...

**0**

votes

**0**answers

75 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. ...

**1**

vote

**0**answers

48 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 ...

**1**

vote

**1**answer

179 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 $E(S^{2}_{n}| \tau \geq n)$ ...

**8**

votes

**1**answer

462 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 ...

**1**

vote

**1**answer

124 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 ...

**0**

votes

**1**answer

60 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

**1**answer

219 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 ...

**4**

votes

**1**answer

173 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} ...

**4**

votes

**2**answers

246 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} ...

**0**

votes

**1**answer

107 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

185 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 ...

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votes

**0**answers

54 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 ...

**7**

votes

**1**answer

378 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}
...

**2**

votes

**1**answer

385 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

172 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) = ...

**0**

votes

**0**answers

52 views

### Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...

**1**

vote

**0**answers

120 views

### Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...

**6**

votes

**1**answer

153 views

### Does a Gaussian process shrink under a contraction map

Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...