Questions tagged [mixing]
For questions about mixing in ergodic or probability theory.
138
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Weak law of large numbers, strong mixing
Let $\{X_n\}_{n\in \mathbb{N}}$ be a strong mixing sequence (no stationarity) with exponentially decaying mixing rate. Further assume that $X_n$ has uniformly bounded forth moments. Does $$\frac{1}{n}\...
- 11
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1
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36
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How can I check that the following map is not topologically mixing?
Let me consider the matrix $$ M=\begin{bmatrix}
1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 \\
0 & 0 & 1 & 1 \\
0 & 0 & 1 & 1
\end{bmatrix} $$ We consider the ...
- 1,930
6
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1
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136
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Equivalence of the definitions of exactness and mixing
Let $f:X \to X$ be a continuous map, where $X$ is a compact metric space. We say that $f$ is expanding if there are constants $\lambda >1$ and $\delta_0 > 0$ such that, for all $x, y\in X$, $d(f(...
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40
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Product of coordinates of alpha mixing random vector.
I have the lemma that for any measurable function and $\alpha$-mixing process $\{x_t\}$, where $x_t$ can be a sequence of random vectors or just univariate random variables. Then $\{f(x_t)\}$, is also ...
- 281
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Any hyperbolic automorphism of $\mathbb{T}^n$ is mixing.
We know a hyperbolic toral automorphism is defined to be an automorphism of $\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n$ induced by integer matrix in $\text{GL}(n,\mathbb{Z})$ which has no eigenvalue of ...
- 61
1
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1
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23
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Functions of uniform-mixing processes
Let $\{X_t\}_{t=1}^\infty$ be a stationary process which is uniform mixing with coefficients $\phi_r$ such that $\sum_{r=}^\infty\phi_r<\infty$. Consider the periodogram for $\omega\in[-\pi,\pi]$
$$...
- 199
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21
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Bounds on $K\left(\mathcal{L}(X_t,X_{t+s})||\mathcal{L}(X_t)\otimes \mathcal{L}(X_{t+s})\right)$ for a diffusion process $(X_t)_{t\geq 0}$?
There are many results giving bounds on the $\beta$-mixing coefficients for diffusion processes (see Proposition 1 in [1] for example). For an homogeneous and stationary process, it means upper ...
- 11
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266
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Equivalent definition of weakly mixing
I found this exercise online some time ago, unfortunately I cannot remember where. I fail to see how to prove one of the implications, maybe someone here has an idea.
Let $(X, \mathcal{F}, \mu, T)$ ...
- 1,271
1
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1
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129
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Central Limit Theorem with Mixing
I would like to apply a CLT theorem to a sequence of random variables $X_1,X_2,X_3, \ldots $ which are "almost independent". Specifically, I want to apply a CLT theorem that replaces the ...
- 153
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116
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Are continuous mixtures of the gamma distribution identifiable with respect to the scale parameter?
Consider the two-parameter Gamma($\alpha$,$\beta$) distribution with PDF
$$f(x|\alpha,\beta) = \frac{\beta^\alpha x^{\alpha - 1} \exp(-\beta x)}{\Gamma(\alpha)}, \quad x>0, \alpha>0, \beta>0,$...
- 81
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19
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Rokhlin's paper on category of mixing transformations
I want to read the proof of the theorem due to Rokhlin that the set of mixing transformations is of first category among measure preserving transformations. The paper is supposed to be there in a ...
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1
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118
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Mixing time of discrete random walk on N-cycle
I would like to know the mixing time of a discrete random walk on an N-cycle which moves to the right or left with probability $\frac{1}{2}$. I read in a paper that this mixing time is
$$M_\epsilon \...
- 147
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2
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123
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Applications where the mixing time of a Markov chain is important
I was wondering what the importance of the mixing time of a Markov chain is. I understand that the mixing time is a measure of how long it takes the distribution to approach the stationary ...
- 147
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29
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Is $Y_t = f(X_t, \beta_t)$ $\rho$-mixing if $X_t$ and $\beta_t$ are independent and each of them are $\rho$-mixing?
There is a post which says $Y_t = f(X_t, \beta_t)$ is strong mixing if $X_t$ and $\beta_t$ are independent and both strong mixing. Does this statement have any reference (book or paper)? In addition ...
- 11
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67
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Is this probability distribution part of a wider class of distributions?
The Hyperbolic distribution is characterized by a single log change of coordinates on the PDF yielding a hyperbola.
What if we impose a double log change of coordinates on the PDF yielding a hyperbola?...
- 850
4
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1
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174
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Strong mixing of a function of strong mixing and convergence sequence
Suppose the process $X = \left\{X_{t}:t\in Z\right\}$ is strong mixing with the coefficient $\alpha(j) \rightarrow 0$ defined as
\begin{equation}
\alpha(j)=\sup_T\sup_{1\leq k\leq T-j}\sup\left\{\...
- 464
2
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0
answers
24
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Random walk on a graph whose vertices are $k$-tuples of distinct $n$-bit strings
I would appreciate any pointer or help with the following random walk. Consider a $k\times n$ $0$-$1$ matrix A, in which the following operation is performed repeatedly:
Two different columns $C_i$ ...
- 31
1
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1
answer
101
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What exactly is the probabilistic/intuitive interpretation of weak mixing of all orders? Why $A_0,\varphi^{\ast n}A_1,\varphi^{\ast 2n}A_2...$?
$\newcommand{\dlim}{\operatorname{Dlim}}\newcommand{\fix}{\operatorname{fix}}\newcommand{\1}{\mathbf{1}}$I am asking about the interpretations of certain ergodic dynamics definitions, citing this text ...
- 32.3k
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86
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Random walk of $k$ particles on a $n$-dimensional hypercube
I would appreciate your help with the following, if possible.
Consider an $n$-dimensional hypercube where nodes are connected by an edge if they differ in a single bit. There are $k=poly(n)$ particles,...
- 31
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0
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113
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Mixing by the cat map : proof?
Let $f : \mathbb{T}^2 \to \mathbb{T}^2$ be the "cat map" defined by $f(x,y)=(2x+y \mod 1,x+y\mod 1)$.
How can I prove that $f$ is exponentially mixing on $\mathbb{T}^2$ endowed with the ...
- 127
1
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2
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167
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Prove this $\phi$ mixing inequality
Am trying to understand Theorem 2.2 in Serfling (1968):
Proposition. Let $(\Omega,\mathcal A,P)$ be a probability space and $\mathcal F$ a sub-$\sigma$-algebra of $\mathcal A$. Let $X$ be a real ...
- 4,236
3
votes
1
answer
90
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How to show this $\phi$ mixing inequality?
Let $(\Omega,\mathcal A,P)$ be a probability space and $\mathcal F$ a sub-$\sigma$-algebra of $\mathcal A$. Am trying to show that
$$ |P(A|\mathcal F)-P(A)|\leq \phi(\mathcal A,\mathcal F) \quad \quad ...
- 4,236
1
vote
0
answers
31
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Limit theorem for a conditional distribution on stationary mixing fields
For a project, I am looking into central limit theorems for stationary mixing random fields. I found many results for the asymptotic distribution of partial sums, such as that of Bolthausen (1982). ...
- 23
1
vote
1
answer
43
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A continuous distribution that "mixes easily" with the uniform.
I want to consider a mixture of uniform distributions between $0$ and $t-U$ such that the $U$ is itself drawn from some mixing distribution. I want to pick a distribution such that the PDF (and ...
- 6,421
4
votes
1
answer
299
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Exponential convergence to equilibrium for finite continuous-time Markov chains
Setting: Consider a continuous-time Markov chain on a finite state space $\mathcal Z$ with irreducible (constant) generator matrix $L\in\mathbb R^{|\mathcal Z| \times |\mathcal Z|}$. Let $\rho_t\in\...
- 559
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1
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26
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What Variance equation is this textbook using?
I'm doing a question on a Gamma/Poisson mix, which turns into a negative binomial. What I don't understand is the answer to the question - most of the answer makes sense to me (normal approximation ...
- 7
2
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answers
83
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Expected node probability of given degree after finite random walk
Let $G=(V,E)$ be a connected, non-bipartite graph with $n$ nodes and $m$ edges. Consider a random walk of length $t$ on $G$ that starts at a uniform random node.
The sequence of nodes in the random ...
1
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0
answers
28
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Why is a completely random measure mixing?
Let $S_x$ denote an operator on $\mathcal{M}_{\chi}^{\#}$ by $S_x\xi(\cdot)=\xi(\cdot+x)$, where $\xi$ is a random measure. Here, $\mathcal{M}_{\chi}^{\#}$ refers to the space of boundedly finite ...
- 109
3
votes
1
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74
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(Exponential) Mixing for Gauss map - going from cylinders to intervals
I'm trying to understand the proof of the mixing property of the Gauss map from the paper - 'Some metrical theorems in number theory' and I'm getting confused by the logic in a step.
The Gauss map $T$ ...
- 41
2
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0
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46
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Modelling dependence under strong mixing conditions
Assume that the sequence $\{X_i\}_{i\in\mathbb N^*}$ of random variables on $(\Omega,\mathcal A,P)$ is strongly mixing. Denote by $\mathcal F_i^k=\sigma(X_t, i\leq t\leq k)$ a sigma-algebra generated. ...
- 2,577
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69
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Proof writing clarification.
Problem
Let $X_n$ be a sequence of mixing random variables. If there is $a>1$ such that the strong mixing coefficient satisfies $$\alpha(s)\leq Cs^{-a}\quad (1)$$ ($C>0$ is a generic positive ...
- 2,577
1
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1
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35
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Counting samples that are different in the given percentage of elements
Suppose there are n unique elements. A mix is a set of m elements pulled from the elements without replacement, so that there are $ C^{m}_{n} $ possible mixes in total. Suppose one has all such mixes. ...
- 113
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62
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Mixing inequality in the paper "The absence of mixing in special flows over rearrangements of segments"
I can't verify an inequality in the paper: https://link.springer.com/content/pdf/10.1007/BF02110361.pdf
The inequality is at the end of the second page in the proof of the second theorem and is shown ...
1
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0
answers
33
views
Spatial covariance decomposition
$\{\mathbf{Z}_{\mathbf{i}}\}_{\mathbf{i} \in Z^{d}}$ is a strictly stationary process, following an isotropic short memory dependence with the covariance matrix given by
$\big\{ C_{k,l}(\lVert \...
- 11
2
votes
1
answer
102
views
Bounding a probability using a strong mixing property
Definition. Let $\{X_t, t>0\}$ be a sequence of real random variables on $(\Omega,\mathcal A,P)$ and denote by $\mathcal F_i^k=\sigma(X_t, i\leq t\leq k)$ the sigma-algebra generated. For any ...
- 2,577
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1
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62
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Mixing time of a lifted Markov Chain.
Let $(X_t)$ be a Markov chain and $(Y_t)$ a lifted chain of $(X_t)$, that is, the lifted chain splits each state in several states.
$t_O$ denotes the mixing time of the original chain;
$t_L$ denotes ...
- 1,276
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1
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110
views
Help with a covariance inequality
Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of real-valued random variables. In a proof of a theorem, the author starts with
$$\lvert Cov(X_i,X_j)\rvert\leq \int_0^{\alpha_{\lvert i-j\rvert}}[Q_i(u)]^2+...
- 2,577
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310
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Covariance and indicator function relations
Let $\mathcal A$ and $\mathcal B$ be two (sub) sigma-algebras of some probability space. If $A\in\mathcal A$ and $B\in\mathcal B$, why the following equality holds:
$$Cov(1_A-1_{A^c},1_B)=E((P(B\...
- 2,577
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106
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Mixing property of stationary Process, only finite dimensional marginal distributions needed??
Let $X=(X_t)_{t\in\mathbb{R}}$ be a stationary process, i.e. the shifted process
$X^s=(X_t^s)_{t\in\mathbb{R}}=(X_{t+s})_{t\in\mathbb{R}}$ has the same distribution as $X=X^0$, and assume that $X$ has ...
- 359
4
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247
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Can a Markov chain converge faster than a geometric rate?
In Theorem 4.9 of the book Markov Chains and Mixing Times by Levin & Peres, there is a convergence theorem for ergodic Markov chains which states that there exist constants $C > 0$ and $\alpha \...
- 469
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102
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Lower bound on mixing time to lazy random walk on complete graphs
Let $K_n$ be complete graph with $n$ vertices
and $$d^{\infty }(t) := \max_{x,y\, \in V} \left|\frac{P^{t}(x,y)}{\pi(y)}-1\right|$$
and $$t^{\infty}_{\text{mix}} := \min \{t \geq 0 : d^{\infty }(t) \...
- 21
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vote
1
answer
140
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Markov chain mixing in finite time
In Theorem 4.9 of the book Markov Chains and Mixing Times by Levin & Peres, there is a convergence theorem for ergodic Markov chains which states that there exist constants $C > 0$ and $\alpha \...
- 469
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196
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Meaning of Exact Transformation and $K$-Automorphism in the context of Ergodic Theory/Mixing
I am reading MAGIC$010$ Ergodic Theory course. In this course's lecture $4$ notes, it is mentioned that
$1)$ Let $T$ be an exact transformation of the probability space $(X,B,μ)$ .Then $T$ is strong-...
- 3,348
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57
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Triangle array multiplied by an indicator function is also strong mixing (proof verification)
Say that a triangle array $\{x_{n,i}:1\leq i\leq n, n\geq 1\}$ of random variables on some probability space $\{\Omega, \mathcal{F}, P\}$ is strong mixing if the coefficients
$$\alpha_n(j)=\sup_{1\leq ...
- 2,577
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votes
1
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74
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What is the relationship between $P(\theta a + (1-\theta)b > 1)$ and $P(a > 1)$, $P(b > 1)$?
Suppose random value $c$ is a mixture of $a$ and $b$
$$c = \theta a + (1-\theta)b$$
What is the relationship of $P(c > 1)$ and $P(a>1)$, $P(b>1)$, is it the following equation?
$$
P(c > 1) ...
- 5
1
vote
1
answer
146
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Bounding the expectation of random variables' product.
Let $\{X_i\}$ be a sequence of strongly mixing random variables, not necessarily (strict) stationary. Assume that $E\lvert X_i\rvert^4\leq C<\infty$ and that there is $0<a<1$ such that the ...
- 2,577
1
vote
1
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270
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Lower bound of mixing time time of two glued complete graphs
I am reading the "Markov Chains and mixing time 1st edition" by Levin and Peres, I got stuck on exercise $6.7$ and have a hard time to understand its solution.
Here is the description direct from ...
- 23
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0
answers
42
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Mixing time of combined Markov Chains
Let $P_1,...,P_k$ denote the transition matrices of $k$ irreducible, aperiodic Markov Chains on finite state spaces $S_1,...,S_k$ all of them being disjoint. Hence, there are stationary distributions $...
- 1,027
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178
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Mixing time of a biased random walk on a line segment
The mixing time $t_{mix}$ of a $\beta$ biased random walk on a line segment $\{0,1,\cdots n\}$ is given by $\beta^{-1}n+O(\sqrt{n})$. I am not interested in the proof but an understanding of the ...
- 3,284
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82
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Counterexample to Davydov inequality for strictly stationary mixing sequence
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strongly mixing, strictly stationary sequence of random variables. If we assume $\mathbb{E}[|X_i|^{2+\epsilon}]<\infty$ for some $\epsilon>0$ and if we have ...
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