All Questions
Tagged with pr.probability ergodic-theory
165 questions
4
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1
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227
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Event of positive probability occurs infinitely often in stationary ergodic sequence
Setup:
Suppose $X = \{X_n\}_{n\in\mathbb{Z}}$ is a stationary ergodic proces on the real line and let $A = \prod_{n\in\mathbb{Z}}A_n$ be a Borel measurable set such that
$$
P(X \in A) = P\left(X_n\in ...
- 479
4
votes
1
answer
363
views
Maximal ergodic inequality
A map $f: X \to X$ preserves an ergodic probability $\mu$, i.e., $\mu \circ f^{-1}=\mu$ and for any $\phi: X \to \mathbb{R}$ with $\int \phi d\mu=0$,
$$\frac{1}{n} \sum_{i \le n} \phi \circ f^i \to 0 \...
- 43
4
votes
1
answer
340
views
On the spectrum of stationary Gaussian process
What is the condition for ergodicity, weakly mixing, and strongly mixing properties of Gaussian process in terms of its spectrum?
In a similar way let us consider a stationary vector valued Gaussian ...
- 421
4
votes
1
answer
106
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Two-side deviations for ergodic sums
Let $(X,\mu)$ be a probability space and $f\colon (X,\mu)\to (X,\mu)$ be an ergodic automorphism. Let $\phi\in L^\infty(X,\mu)$ be such that $\int\phi d\mu=0$.
Suppose that for $\mu$-a.e. $x\in X$, ...
- 1,060
4
votes
1
answer
302
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Almost sure stability of a scalar, nonautonomous, nonlinear SDE
I asked this problem on MSE some while ago, but it has stubbornly resisted any attempts at solving it.
Maybe there is someone here who can either close the gap in one of the existing answers or has ...
- 1,675
4
votes
1
answer
288
views
Radon-Nikodym derivative of the group action on the Furstenberg-Poisson boundary of lamplighter groups
Let $G_d$ be the Lamplighter group $G_d = \mathbb{Z}^d \wr \mathbb{Z}_2 $ and $\Gamma =\{(\bar{\eta},\tilde{0}),(\bar{0},\tilde{e_1}), \cdots,(\bar{0},\tilde{e_d})\}$ be the generator set of $G_d$ (...
- 93
4
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1
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245
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Operator version of Birkhoff ergodic theorem
Suppose that $(\Omega,\mathcal{E},P)$ is a probability space and suppose that we have a measurable operator $T:\Omega\to\Omega$.
Recall that $T$ is said to be egodic if:
$T$ is measure preserving: ...
- 267
4
votes
2
answers
201
views
Uniform convergence of averages for stationary ergodic process
Let $\{X_t, t\in\mathbb R\}$ be a well-behaved$^*$ stationary ergodic process.
I'm interested in the uniform convergence of averages:
$$
\sup_{|x|\le R_n} \left|\frac1{2n}\int_{x-n}^{x+n} X_t dt - \...
- 1,533
4
votes
1
answer
352
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Measure of the rate of convergence for filtration and conditional expectations
This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.
Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that ...
- 869
4
votes
1
answer
222
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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
446
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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
337
views
Support of bivariate joint distribution of stationary and ergodic sequence
Let $\{X_t\}_{t\in \mathbb{N}}$ be a strictly stationary and ergodic sequence of real valued random variables and let the support of $X_1$ equal $[-1,1]$. Can the support of $(X_1,X_2)$ equal the unit ...
- 43
4
votes
1
answer
213
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Practical way to check for geometric convergence
Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution.
When I measure the convergence rate ...
- 101
4
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0
answers
116
views
Convergence in probability results with still open point-wise versions
In ergodic theory and more generally in stochastic processes, often convergence in probability results precede convergence almost-surely results in quite a few years. Classical examples include the ...
- 41
4
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0
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200
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Asymptotic behavior of a dynamical system of density functions
On September 24, 2022, I asked the question below on Mathematics Stack Exchange, linked here:
Link to question on Mathematics Stack Exchange.
I received two up-votes, but no comments or answer. I ...
- 41
4
votes
0
answers
95
views
When the Jacobian of unstable measure converges
Let $T:X \to X$ be a hyperbolic map on the compact metric space $X$. Hyperbolicity means that $T$ has local stable and unstable sets with uniform exponential bounds, which satisfy a local product ...
- 1,043
4
votes
0
answers
98
views
Weighted distribution of irrational rotation
Let $\theta\in [0,1]\setminus\mathbb{Q}$. Let $\alpha_0=\theta$ and $\alpha_1=1$. Let $0<p_0<1$ and $p_1=1-p_0$. For a finite word $I=(i_1, i_2, \dots, i_n)\in \{0,1\}^n$, denote by $I'=(i_1, ...
- 675
4
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0
answers
405
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Reference request: stationary measures as convex combinations of ergodic measures
Does anyone know a good reference for the fact that a stationary probability measure is a convex combination of the stationary and ergodic probability measures?
I have found some references for the ...
- 710
4
votes
0
answers
282
views
Markov operators and existence of ergodic measures
My question refers to the yesterday's question (see here)
of John Learner and goes as follows:
Can we deduce the existence of an ergodic measure if we know that an invariant measure exists, but the ...
- 141
3
votes
2
answers
163
views
Questions about some properties of random probabilities and random expectations
Let $(\Omega, \mathcal{A}, \mathbb P)$ be a probability space with $\mathcal{A}$ countably generated, and let $P: \mathcal{A} \times \Omega \to [0,1]$ be a random probability measure. By that I mean $...
- 869
3
votes
2
answers
250
views
Existence of a positive measurable set with disjoint preimage under iterated transformation
Let $(X,\mathcal B,\mu)$ be a atomless probability measure space and $T:X\to X$ be a non-singular transformation such that $\mu\left({x\in X: T^n(x)=x}\right)=0$ for every $n\ge 1$. Let $A\in \mathcal ...
- 105
3
votes
2
answers
265
views
Can one realize this as an ergodic process?
Consider the lattice $\mathbb Z^2$ and take iid random variables $Y_e$ on all edges $e$ of the graph.
We then define random variables $X_i:=\sum_{e \text{ adjacent to } i}Y_e.$
In other words: For ...
user135480
3
votes
1
answer
343
views
Positive and Null recurrence of Markov Chains on a General State Space
Suppose $X_n$ is an irreducible, aperiodic and Harris recurrent Markov chain. It is well known that in this case, $X_n$ has a stationary distribution $\pi$.
Are there any conditions that are ...
- 339
3
votes
2
answers
224
views
Measures with superexponential moments on finitely generated groups
Let $\Gamma$ be an infinite finitely generated group and let $\nu$ be a measure on $\Gamma$ which generates a transient random walk. I was reading this paper, and the authors prove many of their ...
3
votes
1
answer
190
views
Quantitative version of ergodic theorem in Markov chains
Consider an irreducible Markov chain $X_t$ with finite state space $E$, and unique invariant measure $\pi$. Fix a function $V:E\to\mathbb R$ such that $E_\pi[V]=0$. The ergodic theorem tells us that, ...
- 59
3
votes
1
answer
372
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Attractors in random dynamics
Let $\Delta$ be the interval $[-1,1]$, then we can consider the probability space $(\Delta , \mathcal{B}(\Delta),\nu)$, where $\mathcal{B}(\Delta)$ is the Borel $\sigma$-algebra and $\nu$ is equal ...
- 333
3
votes
2
answers
194
views
A Really Simple Stochastic Dynamic Billiard
Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, $c>0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t),z(t))$ be the position at time $t$ of a point which moves ...
- 640
3
votes
1
answer
127
views
A Simple Stochastic Dynamic Billiard
Consider the following stochastic dynamical system.
Fix $a > 0$, $b > 0$, and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the ...
- 640
3
votes
1
answer
194
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Dynamics of a random stretch map
Notation: Here $S^1$ denotes the circle, which we view as the unit sphere in $\mathbb C$. We equip the circle with its natural length metric.
Let $\{\epsilon_n\}_{n \geq 1}$ be iid uniformly ...
- 6,215
3
votes
1
answer
307
views
"Ergodic theorem" for Markov kernels
Consider a discrete time Markov chain $(X_t)$ on a finite state space $\mathcal{S}$, with transition matrix $P$. Assume that the chain admits a stationary distribution $\pi$, which I will identify ...
3
votes
3
answers
394
views
When is the minimal Martin boundary closed?
Let $\Gamma$ be a finitely generated group and $\mu$ a symmetric measure of finite support on $\Gamma$. Let $\partial_{M}\Gamma$ be the Martin boundary of $(\Gamma,\mu)$ and let $\partial^{min}_{M}\...
- 2,964
3
votes
1
answer
295
views
Finitarily Markovian Finite Factors of Bernoulli Schemes
By processes, I mean discrete, stationary stochastic processes, that is $(X,\mathcal{U},\mu,T)$ where $X$ is the set of doubly infinite sequences of some alphabet $A$, $\mathcal{U}$ is the $\sigma$-...
- 325
3
votes
0
answers
92
views
What dynamical properties should we expect from systems satisfying statistical ones?
Some results on probability theory can be generalized to more abstract ones in ergodic theory, for example:
the strong law of large numbers can be seen as a particular case of Birkhoff's ergodic ...
3
votes
0
answers
153
views
Metropolis-Hastings sampling as a group action
Suppose that you have a topological space $\Omega \subset \mathbb R^n$ accompanied a measure $\mu$ and you're running an iterative sampling algorithm like Metropolis-Hastings. To sample you choose a ...
3
votes
1
answer
233
views
A subadditive maximal ergodic theorem
Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $\tau:\Omega\to\Omega$ be a measurable map on $(\Omega,\mathcal A)$ with $\operatorname P\circ\:\tau^{-1}=\operatorname P$, $Y_n:\...
- 167
3
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0
answers
188
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Invariant subspaces of Markov operators
I am currently working on some kind of graph theoretic problem and the following question came up:
Suppose you have a Markov operator $T$ on $\ell^\infty$, that is a positive, bounded operator such ...
- 381
3
votes
0
answers
123
views
Maximal ergodic theorem on some dyadic intervals
What we refer to maximal ergodic theorem in this thread is the following: let $\left(\Omega,\mathcal F,\mu\right)$ be a probability space and let $T\colon\Omega\to \Omega$ be a measurable and measure ...
- 3,958
3
votes
0
answers
95
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Empirically random, quickly multiplicable matrices
I have encountered a need for fast computation of a transformation $Ax$ where $A\in \mathbb{C}^{K\times N},\ K\sim 10^7,\ N\sim 10^3$ is designed, and $x\in \mathbb{C}^N$ has iid $\mathcal{CN}(0,1)$ ...
3
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0
answers
157
views
Question about martin boundaries of random walks induced on transient subgroups
Suppose $\Gamma$ is a discrete, finitely generated, non-amenable group, and
consider a random walk given by a measure $\mu$.
Assume the measure is symmetric, finitely generated, and the support of
$\...
- 2,964
3
votes
0
answers
209
views
On the decay of correlations of an ergodic sequence over the set $X_{0}=0$
The following question arose while I was trying to explore possible further extensions of a CLT by Liverani which I mentioned here already (see this link, I can tell you more details upon request). It ...
- 486
3
votes
0
answers
157
views
Pointwise convergence of ergodic averages of unconventional conditional expectations
Let $(X_i,Y_i)_{i\in\mathbb{Z}}$ be a finite-valued stationary process whose $\sigma$-algebra of tail events is trivial. Let $\mathcal{F}_n^m$ be the $\sigma$-algebra generated by $X_n,\dots,X_m$ ($n,...
- 191
3
votes
0
answers
95
views
Best convergence rate for convolutions on $\mathbb{Z}_p$
Suppose, that we have sequence of i.i.d variables $X_1,\ldots,X_n$ taking values in $\mathbb{Z}_p$ such that $d_{TV}(X_1,U) < \delta$.
How fast, in terms of $\delta$ and $n$ does the sum $X_1+\...
- 696
2
votes
4
answers
610
views
How to generalize normal number theorem
The Borel number theorem states that with respect to Lebesgue measure, almost all real numbers are normal numbers. It is sometimes stated in the context of the compact interval $[0,1]$, where one ...
- 121
2
votes
2
answers
242
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iid random operator and its spectrum
consider an insteresting question:
given Banach Space $ \mathcal{B}$, independent identical distribution random operator on $ \mathcal{B}$: $ (T_i)_{i \ge 1} $, where operator space is endowed with ...
- 553
2
votes
1
answer
1k
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Given a probability \mu, can we always find a transformation T s.t. \mu is T-invariant?
It is true that, under some conditions, given a measure-preserving transformation $T$, we can always construct a $T$-invariant probability. I am wondering whether we can do a converse. See Parry's ...
2
votes
1
answer
159
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Can we show that this transition semigroup preserves a certain Wasserstein space?
Let $E$ be a separable $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{\gamma\:\in\:C^1([0,\:1],\:E)\\ \gamma(0)\:=\:x\\ \gamma(1)\:=\:y}}\int_0^1v\left(\gamma(...
- 167
2
votes
1
answer
409
views
Existence and uniqueness of a stationary measure
This same question was also posted on MSE https://math.stackexchange.com/questions/3327007/existence-and-uniqueness-of-a-stationary-measure.
Recently I have posted the following question on MO ...
- 333
2
votes
1
answer
1k
views
Understanding measure-preserving transformation [closed]
Given measure space $(S, \mathcal{S}, \mu)$, and measurable function $\phi: S \to S$. $\phi$ is measure-preserving if $\forall A \in \mathcal{S}, \mu(A) = \mu(\phi^{-1}(A))$. My confusion is that why ...
- 23
2
votes
1
answer
200
views
Measurable isomorphism between two non-totally ergodic systems
Suppose $(X,\mathcal A,\mu,T)$ is a finite measure-preserving system. Then we define a new measure system $(X^{(K)},\mathcal A^{(K)},\mu^{(K)},T^{(K)})$ defined by $X^{(K)}=X\times \{1,2,...,K\}$ for ...
- 319
2
votes
2
answers
557
views
trivial map on $\sigma-$algebra $\mod{}0$ is trivial
Hi everyone!
I am currently studying the basic theory of measurable actions and need the following result, which I am not able to prove myself. It is stated without a proof, so probably it should not ...
- 23