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Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations. The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.
Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations. The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise.
随机序列由Tweedie分布控制，并通过展开箱的方法进行评估，在方差-均值幂律和幂律自相关之间表现出双条件关系。'''Wiener–Khinchin Theorem 维纳-辛钦定理'''进一步表明，在这些条件下，对于任何具有方差-均值幂律的序列，也会出现1/f噪声
The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">{{cite journal |last1=Jørgensen |first1=B. |last2=Martinez |first2=J. R. |last3=Tsao |first3=M. |year=1994 |title=Asymptotic behaviour of the variance function |journal=[[Scandinavian Journal of Statistics|Scand J Statist]] |volume=21 |issue=3 |pages=223–243 |jstor=4616314 }}</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model. Almost all distribution functions with finite [[cumulant|cumulant generating functions]] qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this [[Asymptotic expansion|asymptotic behavior]], and the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" />
The [[Tweedie distributions|'''Tweedie convergence theorem''']] provides a hypothetical explanation for the wide manifestation of fluctuation scaling and ''1/f'' noise.<ref name="Jørgensen1994">{{cite journal |last1=Jørgensen |first1=B. |last2=Martinez |first2=J. R. |last3=Tsao |first3=M. |year=1994 |title=Asymptotic behaviour of the variance function |journal=[[Scandinavian Journal of Statistics|Scand J Statist]] |volume=21 |issue=3 |pages=223–243 |jstor=4616314 }}</ref> It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a [[natural exponential family|variance function]] that comes within the [[Attractor|domain of attraction]] of a Tweedie model. Almost all distribution functions with finite [[cumulant|cumulant generating functions]] qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this [[Asymptotic expansion|asymptotic behavior]], and the Tweedie distributions become foci of convergence for a wide range of data types.<ref name="Kendal2011" />
The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.
The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types.
'''Tweedie Convergence Theorem Tweedie收敛定理'''为涨落标度和1/f噪声的广泛出现提供了一个假设性解释。本质上，它要求任何一个可以渐近地显示方差-均值幂律的指数弥散模型，需要在Tweedie模型的吸引域内表达一个方差函数。几乎所有具有有限累积母函数的分布函数都符合指数弥散模型，而大多数指数弥散模型都表现出这种形式的方差函数。因此，许多概率分布都有表达这种渐近行为的方差函数，而Tweedie分布成为了不同数据类型收敛的焦点。
Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />
Much as the [[central limit theorem]] requires certain kinds of random variables to have as a focus of convergence the [[normal distribution|Gaussian distribution]] and express [[white noise]], the Tweedie convergence theorem requires certain non-Gaussian random variables to express ''1/f'' noise and fluctuation scaling.<ref name="Kendal2011" />
Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.
Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling.
正如中心极限定理要求某些类型的随机变量以高斯分布为收敛焦点并表示白噪声一样，Tweedie收敛定理要求某些非高斯随机变量来表达1/f噪声和涨落标度。
===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->
===Cosmology 宇宙学===<!-- This section is linked from [[Cosmic inflation]] -->