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Limit Theorems for Network Dependent Random Variables

Denis Kojevnikov, Vadim Marmer () and Kyungchul Song

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Abstract: This paper considers a general form of network dependence where dependence between two sets of random variables becomes weaker as their distance in a network gets longer. We show that such network dependence cannot be embedded as a random field on a lattice in a Euclidean space with a fixed dimension when the maximum clique increases in size as the network grows. This paper applies Doukhan and Louhichi (1999)'s weak dependence notion to network dependence by measuring dependence strength by the covariance between nonlinearly transformed random variables. While this approach covers examples such as strong mixing random fields on a graph and conditional dependency graphs, it is most useful when dependence arises through a large functional-causal system of equations. The main results of our paper include the law of large numbers, and the central limit theorem. We also propose a heteroskedasticity-autocorrelation consistent variance estimator and prove its consistency under regularity conditions. The finite sample performance of this latter estimator is investigated through a Monte Carlo simulation study.

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Date: 2019-03
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