# Tagged Questions

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

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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 ...
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### Question about the characteristics of semimartingales

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 $x\in D$....
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### Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$. Fix $N$ and consider now a discrete version ...
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### Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...
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### 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 (centroid)...
Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any $\... 1answer 51 views ### Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set? I have both a more general question (concerning stopping times), and then a more specific application (as described in the title). Let$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$be ... 0answers 85 views ### Is there a generalization of Polya urns to continuous outcome event? Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ... 0answers 59 views ### Subclass of semimartingales for which all characteristics can be estimated? I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great. An Ito semimartingale is a martingale for which the ... 0answers 72 views ### Da Prato's notion of Symmetric Operator For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ... 2answers 310 views ### Existence of strong solution to SDEs with non-Lipschitzian drift Consider the SDE: $$dX_t=b(X_t)dt+dW_t\quad X_0=x$$ If$b$is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution. I want to know if we assume$b$... 0answers 73 views ### Bounds on moving average process Let$X_1,X_2,\dotsc$be a sequence of i.i.d. random variables and define the average process$\{Y_t\}$as $$Y_t = \sum_{i=1}^p a_k X_{t-i}$$ with some constants$a_1,\cdots,a_p \in \mathbb{R}$. This ... 2answers 411 views ### Minimal expected absolute value of linear combinations of Gaussian random variables I am interested in the following question. Consider$n$independent standard normal random variables$g_i$. Cosider a linear combination$w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ... 1answer 162 views ### Does$E^{x,t}(f(X_T))$solve a PDE if$f$is not continuous? Many books [see below for references] explore the connections between partial differential equations and expectation values. Assume$X$is a diffusion with generator$A$, then they conclude, that ... 2answers 135 views ### Existence of an invariant measure on an infinite dimensional space via Lyapunov functional Set-up. Assume that we have a complete separable metric space$\mathcal{X}$that is not locally compact. Let$V: \mathcal{x} \to [0; +\infty]$be a functional such that$K_r :=\{x \in \mathcal {X} : V ...
Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have X_n = A_n \cdots A_1X_0.\$...