All Questions
Tagged with ds.dynamical-systems pr.probability
120 questions
66
votes
4
answers
4k
views
Perron number distribution
A Perron number is a real algebraic integer $\lambda$ that is larger than the absolute value of any of its Galois conjugates. The Perron-Frobenius theorem says that any
non-negative integer matrix $M$ ...
- 25.1k
25
votes
6
answers
6k
views
Proof of Krylov-Bogoliubov theorem
Where can I find a proof (in English) of the Krylov-Bogoliubov theorem, which states if $X$ is a compact metric space and $T\colon X \to X$ is continuous, then there is a $T$-invariant Borel ...
- 897
21
votes
3
answers
1k
views
Central Limit Theorem(s) for irrational rotation
Let $\alpha$ be irrational and $T: S^1 \rightarrow S^1$ be the rotation by $\alpha$. I'm interested in what type of Central Limit Theorem (if any) can hold for sums $Y_n = \frac{1}{\sqrt{n}}\sum_{k=1}^...
- 3,323
20
votes
1
answer
2k
views
Roadmap to Ergodic Theory
I have recently been interested in going deeper into ergodic theory, beyond an introductory level of knowledge. Background wise, my training has mostly been in stochastic analysis, and I have a ...
- 6,155
18
votes
3
answers
1k
views
Not-lonely runners
The lonely runner conjecture
has several formulations.
They all involve a number $n$ runners running on a circular track,
each with a different speeds, and the conjecture is that each runner is ...
- 151k
18
votes
0
answers
667
views
The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
- 151k
15
votes
1
answer
1k
views
In how many steps a random walk visits all the elements of a finite group, with a probability 1/2?
This question is a variation of the return to the origin problem.
Let $G$ be the finite group $\mathbb{Z}/n \times \mathbb{Z}/n$ and let the random transformation $T: G \to G$ such that $T(a,b) = (...
15
votes
2
answers
571
views
Spearing rolling hula hoops
Or: Stabbing rolling disks.
Imagine there are $n$ unit-diameter disks rolling between $x=0$ and $x=d$,
reflecting off either end.
The disk centers start at a random location within $[\frac{1}{2}, d-\...
- 151k
14
votes
0
answers
358
views
What is the asymptotic dynamics of the winning position in this game?
$n$ players indexed $1,2,...,n$ play a game of mock duel. The rules are simple: starting from player $1$, each player takes turns to act in the order $1,2,...,n,1,2,...$. In his turn, a player ...
- 2,619
13
votes
7
answers
2k
views
Finite-space dynamical systems
This question is quite open-ended, but I will formulate several sub-questions that I'll try to make precise. It is about finite-state dynamical system: start with a finite set $X$, with say $n$ ...
- 14.4k
12
votes
2
answers
3k
views
Is it fine to inquire about a paper that's been under review for around 9 months?
I have submitted a paper on applied probability in one of SIAM journals. The paper is under review for 9 months. I asked the editor 1 month ago about it, I was told that one review report has come and ...
11
votes
2
answers
928
views
Random circle rotations
Weyl's equidistribution theorem states that the orbit of a point on the circle under rotation by $\alpha$ becomes asymptotically equidistributed with respect to Lebesgue (Haar) measure whenever $\...
- 8,903
11
votes
1
answer
393
views
Growing a chain of unit-area triangles: Fills the plane?
Define a process to start with a unit-area equilateral triangle,
and at each step glue on another unit-area triangle.
$50$ ...
- 151k
11
votes
2
answers
2k
views
De Finetti's theorem, the pointwise ergodic theorem, and reverse martingales
De Finetti's theorem says that an exchangeable sequence of random variables $X_i$ is a mixture of i.i.d. random variables. In other words, if $\mu$ is a measure on $\mathbb{R}^\infty$ that is ...
- 6,287
11
votes
1
answer
1k
views
resampling over Bowen balls
Hello MO World
I'm working on a paper involving embedding your favourite measure-preserving transformation into a topological model (think Krieger generator theorem: embedding in a full shift) and ...
- 23.2k
10
votes
5
answers
509
views
Path length of ball on tilted, perforated plane
Imagine that an $\epsilon$-radius hole is punched in the plane centered
on every integer-coordinate point.
Now a point "ball" is dropped on the plane at a random spot $p$.
If $p$ has not already ...
- 151k
10
votes
2
answers
678
views
Irrational rotation - recurrence times
I consider the irrational rotation
$T_\alpha(x) = x + \alpha \text{ mod } 1$ for given irrational $\alpha \in [0,1]$. For a given open interval $A \subset [0,1]$ with length $|A|>0$, I consider the ...
- 103
10
votes
2
answers
559
views
Can Birkhoff's ergodic theorem for integrable functions easily be deduced from Birkhoff's ergodic theorem for bounded functions?
It seems to me that a considerably simpler proof [see below] of Birkhoff's ergodic theorem can be obtained for bounded observables than for more general $L^1$ observables. Therefore, I feel like it ...
- 708
10
votes
1
answer
494
views
Ping-pong progress through a quincunx
A quincunx or
Galton board consists of
staggered pegs from which ping-pong balls bounce and eventually display
a binomial / normal distribution in catch-bins. I am wondering if the
downward progress ...
- 151k
10
votes
2
answers
488
views
A functional equation involving the inverse function
$\newcommand\ep\epsilon\newcommand\R{\mathbb R}$Let $P$ denote the set of all continuous probability density functions (pdf's) $p$ on $\R$ vanishing at $\pm\infty$. Let us say that a pdf $p\in P$ is ...
- 128k
10
votes
0
answers
391
views
Question from an economist: solving a model of traders' behavior with expectations about the future values of the variable they are currently optimizing
Motivation
I am an economist writing a paper for an academic finance journal. My paper is about the behavior of currency traders, who choose the price at which they will sell currency today, based on ...
- 101
9
votes
2
answers
586
views
Fixed objects of the M endofunctor on category Meas
Consider the category $\operatorname{Meas}$ of measurable spaces: its objects are sets equipped with $\sigma$-algebras, and its morphisms are measurable functions between spaces.
As Gerald Edgar &...
- 8,512
9
votes
1
answer
359
views
Relaxation of notion of positive definite function
A function $f:\mathbb{R}\to\mathbb{R}$ is called positive definite (in the semigroup sense) if for all $n\geq 1$ and $x_1,\ldots,x_n\in\mathbb{R}$ pairwise different the matrix $(f(x_i+x_j))_{i,j=1}^n$...
- 3,031
8
votes
3
answers
255
views
Random reflections unexpectedly produce banded distributions
Start with $p_1$ a random point on the origin-centered unit circle $C$.
At step $i$, select a random point $q_i$ on $C$, and a random mirror line
$M_i$ through $q_i$, and reflect $p_i$ in $M_i$ to ...
- 151k
8
votes
0
answers
157
views
Pursuit-evasion with many slow pursuers
Question: Suppose that intelligent pursuers with speed $v<1$ are randomly scattered on the plane with area density $1/r$ ($r>0$ is distance from the origin). If you start at the origin ...
- 7,545
7
votes
2
answers
335
views
Wait time to grid network disconnection with failing edges
Let $G_n$ be an $n \times n$ planar toroidal grid graph, with each node
connected to its four neighbors, with the top row connected to the bottom,
and the right column connected to the left.
Suppose ...
- 151k
7
votes
1
answer
319
views
Why aren't operator semigroups studied from a dynamical perspective?
Often times one talks about iterating a continuous map to get discrete topological dynamics, or having a 1-parameter family of continuous maps to get continuous topological dynamics.
When studying ...
- 277
7
votes
1
answer
253
views
Are all quasi-regular points on Polish spaces generic points?
Let $X$ be a Polish space and $T\colon X\to X$ be a continuous map. We say that a point $x\in X$ is quasi-regular if for every bounded continous function $\varphi\colon X\to\mathbb{R}$ the sequence $...
- 1,658
7
votes
2
answers
321
views
Random suborbits of a rotation
Let $u_n = x + n\alpha \pmod 1$ with $\alpha$ irrational. We know that $(u_n)_{n \geq 0}$ is dense in $\mathbb{R}/\mathbb{Z}$ (equivalently $(u_n)_{n \geq 0}$ visits every open interval infinitely ...
- 2,560
7
votes
1
answer
274
views
Uniqueness of stationary measures for $(G,\mu)$ boundaries
Let $G$ be a countable group acting minimally by homeomorphisms on a compact Hausdorff space $X$ and $\mu$ be a probability measure on $G$ whose support generates $G$ as a semigroup.
Let $\nu$ is a $\...
- 481
7
votes
0
answers
299
views
Generalized Skorokhod spaces
Skorokhod spaces of càdlàg functions are an extremely useful setting to describe stochastic processes. I'd like to understand the Skorokhod topology from a pure topological point of view, without ...
- 8,512
6
votes
1
answer
611
views
The "Chaos Game" as a particular series of i.i.d. random variables
Fix a parameter $\alpha\in(0,1)$ and take an i.i.d. sequence $X_0,X_1,\ldots$ of $\mathbb{R}^n$ valued random variables. Construct the limiting random variable
$X_\infty = (1-\alpha)\sum_{k=0}^\infty ...
- 646
6
votes
2
answers
3k
views
What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)
Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
- 1,026
6
votes
1
answer
819
views
Birkhoff ergodic theorem for dynamical systems driven by a Wiener process
At the risk of asking a stupid question I have the following problem.
Suppose I have a measure preserving dynamical system $(X, \mathcal{F}, \mu, T_s)$, where
$X$ is a set
$\mathcal{F}$ is a sigma-...
- 617
5
votes
1
answer
389
views
Is a random circle rotation weak mixing almost surely?
Consider the random circle rotation $x \to x + Z \text{ mod 1}$ on $([0, 1], \text{Lebesgue})$ where at each rotation, $Z$ is uniformly distributed on $[0, 1]$ and independent of previous rotations.
...
- 6,155
5
votes
1
answer
600
views
Spectrum of a generic integral matrix.
My collaborators and I are studying certain rigidity properties of hyperbolic toral automorphisms.
These are given by integral matrices A with determinant 1 and without eigenvalues on the unit circle....
- 4,188
5
votes
1
answer
140
views
Do distinct idempotent measures on finite binary systems have distinct supports?
Suppose that $(S,*)$ is a finite set equipped with a binary operation. Extend the binary operation to the vector space $V$ with basis $S$.
The set of probability measures on $S$, viewed as a compact ...
- 3,547
5
votes
1
answer
348
views
"strongly mixing" action on dimers?
In Local Statistics of Lattice Dimers we study a nice familiar object, domino tilings in the plane extending out to infinity.
His paper is going to discuss the frequency of various "motifs" in ...
- 22.8k
5
votes
0
answers
183
views
Central limit theorem versus entropy in dynamical systems context
A dynamical system $(S^1,T, \mu)$, $T_* \mu=\mu$, $T$ ergodic, $S^1$ is circle. Assume it has central limit theorem.
Want to know the relation between its measure-theoretic entropy $h_{\mu}(T)$ and ...
- 553
5
votes
0
answers
143
views
Smoothing properties of convolutions of $P^1(\mathbb{R})$ by $SL(2,\mathbb{R})$
Consider the action of $SL_2(\mathbb R)$ on real projective space $P^1(\mathbb R)$; given $A \in SL_2(\mathbb R)$ and $\alpha \in P^1(\mathbb R)$ we write $A . \alpha \in P^1(\mathbb R)$ for this ...
- 485
5
votes
0
answers
81
views
What statistical data/quantities are known about the time spent by a generic orbit of an ergodic system in a fixed set?
By the ergodic theorem, we know that for almost every point, the average time spent by an orbit in a set is equal to the relative measure of that set.
What other information about that time can we ...
user76098
5
votes
0
answers
127
views
First return time in an interval for N particles rotating on the circle at constant random speeds
Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...
5
votes
0
answers
221
views
Quasicompactness of transfer operators associated to IID matrix products
Let $P^1$ denote one-dimensional real projective space, and for each $A \in GL(2,\mathbb{R})$ let $\overline{A}$ denote the homeomorphism of $P^1$ induced by $A$. I am currently reading a paper which ...
- 6,206
4
votes
2
answers
2k
views
On Mathematical Foundations of Football
Football (soccer) is arguably one of the most unpredictable sports. Countless variables play a role in determining the outcome of a certain football match. Due to the high complexity of the entire set ...
4
votes
1
answer
277
views
Shannon entropy of $p(x)(1-p(x))$ is no less than entropy of $p(x)$
If $p(x)$ is a discrete probabilistic density function, one could construct another discrete probabilistic density function proportional to $p(x)[1-p(x)]$ with a corresponding partition function to ...
- 191
4
votes
1
answer
173
views
Expectation and Dependence
Probably a very simple question, I think I'm looking for a general statement that says 'if one decreases the independence between two processes then the expected value of the maximum of these two ...
- 717
4
votes
1
answer
222
views
Is there a version of the Return Times Theorem for Dunford-Schwartz operators?
Bourgain's "Return Times Theorem" establishes that if $(\Omega_{j},\mathcal{F}_{j},\mathbb{P}_{j},T_{j})$ ($j=1,2$) are measure-preserving Dynamical systems (i.e. $(\Omega_{j},\mathcal{F}_{j},\mathbb{...
- 486
4
votes
1
answer
426
views
A process of repeated convolution and conditioning and the resulting sequence of probability distributions
I am interested in the following procedure that yields a sequence $D_1,D_2,\ldots,$ of probability distributions over $\mathbb{R}^n$.
Let $D_1$ be the $n$-dimensional Gaussian distribution with ...
- 1,016
4
votes
1
answer
446
views
Birkhoff ergodic theorem for ergodic Markov processes
This question was previously posted on MSE.
This question might be easy but I am really stuck on it.
Let $M$ be compact metric space and $\mathcal B(M)$ the Borel $\sigma$-algebra of M. Consider the ...
- 333
4
votes
1
answer
473
views
A question on random walks on semisimple groups
Let $G$ be a connected semisimple Lie group without compact factor, $\mu$ be a Borel probability measure on $G$ such that the group generated by $\mathrm{supp}(\mu)$ is Zariski dense in $G$. For ...
- 41