Asymptotic Normality

Insights into Tail-Based and Order Statistics ArXiv ID: 2511.04784 “View on arXiv” Authors: Hamidreza Maleki Almani Abstract Heavy-tailed phenomena appear across diverse domains –from wealth and firm sizes in economics to network traffic, biological systems, and physical processes– characterized by the disproportionate influence of extreme values. These distributions challenge classical statistical models, as their tails decay too slowly for conventional approximations to hold. Among their key descriptive measures are quantile contributions, which quantify the proportion of a total quantity (such as income, energy, or risk) attributed to observations above a given quantile threshold. This paper presents a theoretical study of the quantile contribution statistic and its relationship with order statistics. We derive a closed-form expression for the joint cumulative distribution function (CDF) of order statistics and, based on it, obtain an explicit CDF for quantile contributions applicable to small samples. We then investigate the asymptotic behavior of these contributions as the sample size increases, establishing the asymptotic normality of the numerator and characterizing the limiting distribution of the quantile contribution. Finally, simulation studies illustrate the convergence properties and empirical accuracy of the theoretical results, providing a foundation for applying quantile contributions in the analysis of heavy-tailed data. ...

Consistent Estimation of the High-Dimensional Efficient Frontier ArXiv ID: 2409.15103 “View on arXiv” Authors: Unknown Abstract In this paper, we analyze the asymptotic behavior of the main characteristics of the mean-variance efficient frontier employing random matrix theory. Our particular interest covers the case when the dimension $p$ and the sample size $n$ tend to infinity simultaneously and their ratio $p/n$ tends to a positive constant $c\in(0,1)$. We neither impose any distributional nor structural assumptions on the asset returns. For the developed theoretical framework, some regularity conditions, like the existence of the $4$th moments, are needed. It is shown that two out of three quantities of interest are biased and overestimated by their sample counterparts under the high-dimensional asymptotic regime. This becomes evident based on the asymptotic deterministic equivalents of the sample plug-in estimators. Using them we construct consistent estimators of the three characteristics of the efficient frontier. It it shown that the additive and/or the multiplicative biases of the sample estimates are solely functions of the concentration ratio $c$. Furthermore, the asymptotic normality of the considered estimators of the parameters of the efficient frontier is proved. Verifying the theoretical results based on an extensive simulation study we show that the proposed estimator for the efficient frontier is a valuable alternative to the sample estimator for high dimensional data. Finally, we present an empirical application, where we estimate the efficient frontier based on the stocks included in S&P 500 index. ...