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On a measurable space, let $m_n, n=1, \dots, $ be a sequence measures on it and $\mu$ is another measure.

let $Z := \{ϕ∈L^2(\mu): ∫ ϕ\,d\mu = 0 \}$.

The first type of convergence of $m_i$'s is defined as $$ \lim_{n \to \infty} \sup_{\phi\in Z:\,\|\phi\|_2=1} \int \phi \, dm_n =0. $$ This is motivated by the definition of a stochastic process being ρ-mixing.

The second type of convergence of $m_i$'s is defined as $$ \lim_{n \to \infty} \sup_{\phi\in Z:\,\|\phi\|_\infty=1} \int \phi \, dm_n =0. $$ This is motivated by the definition of a stochastic process being α-mixing.

The third type of convergence of $m_i$'s is defined as $$ \lim_{n\to \infty} \sup_{0\leq\phi\leq1} \Big| \int \phi \, dm_n - \int \phi \, dQ \Big| = 0 $$ This is motivated by the definition of a stochastic process being β-mixing. A side question: what does $0\leq\phi\leq1$ really means?

Are there names for these types of convergence of measures? I would like to know about their relations with other types of convergence of a sequence of measures, such as weak and weak* convergence.

I have tried to read Chen, Xiaohong; Hansen, Lars Peter; Carrasco, Marine (2010). "Nonlinearity and temporal dependence". Journal of Econometrics 155 (2): 155–169. cited in the Wikipedia article for the three types of mixing, and also traced Mixing Conditions for Markov Chains Yu. A. Davydov Theory Probab. Appl. 18-2 (1974) cited in that paper for the definitions of β-mixing. But I failed to see answers to my above questions.

Thanks!

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1 Answer 1

I think that $\rho$-convergence is $L^2$ convergence to zero. For simplicity assume that the measures $m_n$ are absolutely continuous with respect to $\mu$, with Radon-Nikodym density $q_n$. Then you want $$ \int q_n^2 d \mu \rightarrow 0. $$ By Holder's inequality this implies the conergence you give, and if this $L^2$ convergence doesn't hold, you can construct a $\phi$ to violate your condition.

$\alpha$-mixing looks like $L^1$ convergence to zero. This is $$ \int q_n d \mu \rightarrow 0. $$ We don't need absolute values since $q_n$ is nonnegative. A lot of this is related to duals of $L^p$ spaces. The dual of $L^2$ is $L^2$, while the dual of $L^\infty$ is $L^1$.

In your definition of $\beta$-mixing, what is $Q$.

Note, for example that $\rho$-mixing is stronger than weak convergence in $L^2$. That would be equivalent to saying that for each $\phi\in L^2$, $$ \int \phi q_n d \mu \rightarrow 0, $$ without saying anything about convergence holding uniformly over all such $\phi$ with $||\phi||_2 = 1$.

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Thanks! In the definition of β-mixing, Q is the measure $M_n$'s converge to. –  Tim Dec 1 '12 at 4:40
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