Decomposition of Random Variables and
                    Vectors

This paper argues that in the presence of intersectoral input-output linkages, microeconomic idiosyncratic shocks may lead to aggregate fluctuations. In particular, it shows that, as the economy becomes more disaggregated, the rate at which aggregate volatility decays is determined by the structure of the network capturing such linkages. Our main results provide a characterization of this relationship in terms of the importance of different sectors as suppliers to their immediate customers as well as their role as indirect suppliers to chains of downstream sectors. Such higher-order interconnections capture the possibility of "cascade effects" whereby productivity shocks to a sector propagate not only to its immediate downstream customers, but also indirectly to the rest of the economy. Our results highlight that sizable aggregate volatility is obtained from sectoral idiosyncratic shocks only if there exists significant asymmetry in the roles that sectors play as suppliers to others, and that the "sparseness" of the input-output matrix is unrelated to the nature of aggregate fluctuations.

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“…The following theorem is from Rotar (1975). A detailed treatment of the subject can be found in Chapter 9 of Linnik and Ostrovskiȋ (1977).…”

Section: Non-classical Central Limit Theoremsmentioning

confidence: 99%

The Network Origins of Aggregate Fluctuations

et al. 2011

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“…We also need some well-known results of the theory of entire functions. The proof of these theorems may be found in Chapters 1 and 5 of Levin's book (1964), Chapter 8 of Titchmarsh's book (1968), and Chapter 1-3 of the monograph of Linnik and Ostrovskii (1977).…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Independence of linear forms with random coefficients

Chistyakov

Götze

2006

Probab. Theory Relat. Fields

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We extend the classical Darmois-Skitovich theorem to the case where the linear forms L r1 = U 1 X 1 +· · ·+U n X n and L r2 = U n+1 X 1 +· · ·+U 2n X n have random coefficients U 1 , . . . , U 2n . Under minimal restrictions on the random coefficients we completely describe the distributions of the independent random variables X 1 , . . . , X n and U 1 , . . . , U 2n such that the linear forms L r1 and L r2 are independent.

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“…Proof: The first part of this property is a direct consequence of the maximum entropy property of the Gaussian distribution. The second part comes from Cramer's decomposition theorem [24], which states that if 1 X and 2 X are independent and their sum 1 2 X X + is Gaussian, then both 1 X and 2 X must also be Gaussian.…”

Section: Some Basic Properties Of Smee Criterionmentioning

confidence: 99%

On the Smoothed Minimum Error Entropy Criterion

Chen

Prı́ncipe

2012

Entropy

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Recent studies suggest that the minimum error entropy (MEE) criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations. In practice, however, one has to estimate the error entropy from the samples since in general the analytical evaluation of error entropy is not possible. By the Parzen windowing approach, the estimated error entropy converges asymptotically to the entropy of the error plus an independent random variable whose probability density function (PDF) corresponds to the kernel function in the Parzen method. This quantity of entropy is called the smoothed error entropy, and the corresponding optimality criterion is named the smoothed MEE (SMEE) criterion. In this paper, we study theoretically the SMEE criterion in supervised machine learning where the learning machine is assumed to be nonparametric and universal. Some basic properties are presented. In particular, we show that when the smoothing factor is very small, the smoothed error entropy equals approximately the true error entropy plus a scaled version of the Fisher information of error. We also investigate how the smoothing factor affects the optimal solution. In some special situations, the optimal solution under the SMEE criterion does not change with increasing smoothing factor. In general cases, when the smoothing factor tends to infinity, minimizing the smoothed error entropy will be approximately equivalent to minimizing error variance, regardless of the conditional PDF and the kernel.

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Decomposition of Random Variables and Vectors

Cited by 82 publications

References 0 publications

The Network Origins of Aggregate Fluctuations

The Network Origins of Aggregate Fluctuations

Independence of linear forms with random coefficients

On the Smoothed Minimum Error Entropy Criterion

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