# Tagged Questions

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

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### Validating a probability density distribution forecast model for a Markov process

Let's say we have a Markov process $X_t$, and we come up with a forecast model that takes some information from outside world and says: "value $X_{t+1}$ has probability density distribution $P_t(x)$". ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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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### 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 (...
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### 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 (...
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### 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 ...
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### 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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### 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. ...
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### 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 ...
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### 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 $E(S^{2}_{n}| \tau \geq n)$ known?...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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 $T_n = \min\{t \in \mathbb{N} \... 2answers 247 views ### Is$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 $$Y_{t} =... 1answer 108 views ### probability in galton watson processes [closed] I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger. I don't understand what Z_(n,i) stand for. And in the proof of Theorem ... 2answers 186 views ### Principal bundles and Subriemannian Geometry In sub-Riemannian geometry, one considers manifolds P equipped with a subbundle \mathcal{H} of TP, the horizontal distribution. One then has a Riemannian metric only on this distribution \... 0answers 54 views ### Strong Markov Property of the joint process (B_t,L_t)_{t\ge 0} Let B=(B_t)_{t\ge 0} be a Brownian motion and L=(L_t)_{t\ge 0} be its local time in zero. Given two strictly increasing functions \phi_1, \phi_2: \mathbb R_+\to\mathbb R such that \phi_1(0)=\... 1answer 388 views ### Central limit theorem for biased random walk Define random variables X_n by X_0 = 0 and \begin{equation*}X_n - X_{n-1} = \begin{cases} 1 & \text{with probability } g(X_{n-1}) \\ 0 & \text{with probability } 1-g(X_{n-1}) \end{cases} \... 1answer 387 views ### Stochastic interpretation of heat kernel on fiber bundle I'm looking for a stochastic interpretation of the heat equation for vector valued function. The classical set up is the following : If (M,g) is a riemannian manifold then we could consider the ... 0answers 177 views ### Joint law of a standard Brownian motion and its local time at a nonzero level Let B_t be the standard Brownian motion and L_t^a be the local time at level a. It is known that the joint-density of (L_t^0,B_t) is$$ P\left(B_t\in d y, L_t^0\in d v\right) = \frac{|y|+v}{\... 0answers 54 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 ... 0answers 126 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 ... 1answer 156 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 ... 1answer 190 views ### Stochastic differential equation associated with an optimal control problem We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process$X_t$is controlled up until it is stopped at a stopping time$\...
Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in D$....