Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities
نویسندگان
چکیده
Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.
منابع مشابه
The Local Limit Theorem: A Historical Perspective
The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...
متن کاملPortfolio Modeling with Heavy Tailed Random Vectors
Since the work of Mandelbrot in the 1960’s there has accumulated a great deal of empirical evidence for heavy tailed models in finance. In these models, the probability of a large fluctuation falls off like a power law. The generalized central limit theorem shows that these heavy-tailed fluctuations accumulate to a stable probability distribution. If the tails are not too heavy then the varianc...
متن کاملRobust Queueing Theory
We propose an alternative approach for studying queues based on robust optimization. We model the uncertainty in the arrivals and services via polyhedral uncertainty sets which are inspired from the limit laws of probability. Using the generalized central limit theorem, this framework allows to model heavy-tailed behavior characterized by bursts of rapidly occurring arrivals and long service ti...
متن کاملCentral limit theorems for linear statistics of heavy tailed random matrices
We show central limit theorems (CLT) for the linear statistics of symmetric matrices with independent heavy tailed entries, including entries in the domain of attraction of α-stable laws and entries with moments exploding with the dimension, as in the adjacency matrices of Erdös-Rényi graphs. For the second model, we also prove a central limit theorem of the moments of its empirical eigenvalues...
متن کاملNeed for Most Accurate Discrete Approximations Explains Heavy-Tailed Distributions
In many practical situations, we encounter Gaussian distributions, for which the distribution tails are light – in the sense that as the value increases, the corresponding probability density tends to 0 very fast. There are many theoretical explanations for the Gaussian distributions and for similar light-tail distributions. In practice, however, we often encounter heavy-tailed distributions, i...
متن کامل