# Tagged Questions

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

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### Is there a natural random process that is rigorously known to produce Zipf's law?

Zipf's law is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ (...
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### Growing random trees on a lattice $\rightarrow$ Voronoi diagrams

Imagine growing trees from $k$ seeds on a square $n \times n$ region of $\mathbb{Z}^2$. At each step, a unit-length edge $e$ between two points of $\mathbb{Z}^2$ is added. The edge $e$ is chosen ...
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### Random Walk on $\mathbb{R}$ with Uniformly Distributed Steps and “Reflective” Boundary at Origin

A particle lies on the real number line at the origin. For each step taken, the particle moves from its current position a distance (and direction) chosen equi-probably from range $[-1,r]$. However, ...
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### Existance of certain almost invariant functions related to amenability and piece-wise transformations

We would like very much to know the answer to the following question: Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
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### Age of Stochasticity?

One user on MSE made an interesting question, which was unanswered so I suggested him to post it here but he refused for personal reasons and said I could ask it here. The question is this: Today ...
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### Bochner integral of stochastic process = path by path Lebesgue integral?

After some helpful comments, I realized that I had to repost this question in a more systematic way. On a complete probability space, let $\mathcal{H}_0$ denote the Hilbert space of square ...
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### A Markov process which is not a strong markov process?

Can anyone give an example of a Markov process which is not a strong Markov process? The Markov property and strong Markov property are typically introduced as distinct concepts (for example in ...
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### How much universality is there for contact processes?

A couple of weeks ago I had to pick my daughter up from her nursery because of suspected chicken pox. It turned out to be a false alarm, but while I was waiting at the doctor's surgery to establish ...
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### Correlated Brownian motion and Poisson process

Is there an (easy) way to construct, on the same filtered probability space,a Brownian motion $W$ and a Poisson process $N$, such that $W$ and $N$ are not independent ? I first asked this question ...
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### Brownian bridge interpreted as Brownian motion on the circle

Is it reasonable to view the Brownian bridge as a kind of Brownian motion indexed by points on the circle? The Brownian bridge has some strange connections with the Riemann zeta function (see Williams'...
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### Does Brownian motion immediately visit both sides of a Jordan curve?

Let $C$ be a Jordan curve in $\mathbb{R}^2$. By the Jordan curve theorem, $\mathbb{R}^2 \smallsetminus C$ is uniquely partitioned into two connected regions $A$ and $B$ (the interior and exterior). ...
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### How far can a particle travel from its origin if it exhibits self-avoiding Brownian motion in two-dimensions?

Let's say I have a point-like Brownian particle undergoing two-dimensional diffusion on an infinite plane with the caveat is that the particle can never return to a coordinate that it previously ...
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### Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ...
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Consider a Galton-Watson branching process, with offspring distribution $\mathbf{p}=(p_0, p_1, \dots, p_n, \dots)$. Let $O$ be the root of the branching process. Write $\eta=P(\text{process survives ... 2answers 904 views ### Why do we want maps to be measurable (in countably-additive setting) When I have to explain things that I am doing to people who did not do (or even did not learn) measure-theoretical probability, I think of getting a question in the title, and I am not sure I have ... 2answers 421 views ### Minimal expected absolute value of linear combinations of Gaussian random variables I am interested in the following question. Consider$n$independent standard normal random variables$g_i$. Cosider a linear combination$w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ... 2answers 1k views ### Markov chains: invariant measures and explosion The following seems like such an elementary question, but I didn't get anywhere with it. Suppose you are considering a Markov chain in continuous time which is transient and has an invariant measure (... 1answer 897 views ### Hardy spaces: analysis <---> martingales Let$H^p$be the Hardy space of analytic functions on the open unit disk$\mathbb{D}$:$f \in H^p$if$f$is analytic on$\mathbb{D}$and$\sup_{r < 1} \int_0^{2\pi} |f(re^{i\theta})|^p d\theta <...
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I often see in stochastic calculus books the terms 'adapted process' and 'progressively measurable process'. I know there is a small difference between them (every progressively measurable process is ...
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### Trapping a particle

A particle starts a brownian walk in the middle of a long tunnel in the plane, at one end of the tunnel is a region Y of given area A. Does the shape of region Y affect average time for the particle ...
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### Does the strong law of Large Number hold for an infinite dimensional Brownian motion?

For finite-dimensional Brownian motion $W_t$, it is well known that $$\lim_{t\to \infty}\frac{W_t}{t}=0,\text{ a.s. }\ \ \ \ \hspace{1cm} \langle 1\rangle$$ Now suppose we ...
A (standard, real-valued) Brownian motion $W = \{W(t): t \geq 0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t \... 1answer 354 views ### Extending state space to make a process Feller Let$X$be a locally compact Hausdorff space, and let$Y_t$be a continuous Markov process on$X$with transition function$P(t, x, \Gamma) := \mathbb{P}_x (Y_t \in \Gamma)$. Let$T_t$be the ... 1answer 367 views ### a question on 0-1 valued stochastic process Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ... 2answers 11k views ### Coin Pusher Game While doing laundry at my local laundromat, I saw a coin pusher game. Below is a picture, and here is a video depicting how it works (disregard non-coins). Essentially, one has a distribution of ... 1answer 1k views ### Karhunen–Loève approximation of Brownian motion and diffusions The Karhunen–Loève theorem says that Brownian motion on the interval [0,1] can be represented as follows:$W_t = \sum_{n=1}^\infty Z_n \frac{\sin((n-1/2)\pi t)}{(n-1/2)\pi},$where$Z_n \sim \mathcal{...
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 ...