Heavy-Tailed Distributions

A Note on the Conditions for COS Convergence ArXiv ID: 2512.02745 “View on arXiv” Authors: Qinling Wang, Xiaoyu Shen, Fang Fang Abstract We study the truncation error of the COS method and give simple, verifiable conditions that guarantee convergence. In one dimension, COS is admissible when the density belongs to both L1 and L2 and has a finite weighted L2 moment of order strictly greater than one. We extend the result to multiple dimensions by requiring the moment order to exceed the dimension. These conditions enlarge the class of densities covered by previous analyses and include heavy-tailed distributions such as Student t with small degrees of freedom. ...

Multifractality and its sources in the digital currency market ArXiv ID: 2510.13785 “View on arXiv” Authors: Stanisław Drożdż, Robert Kluszczyński, Jarosław Kwapień, Marcin Wątorek Abstract Multifractality in time series analysis characterizes the presence of multiple scaling exponents, indicating heterogeneous temporal structures and complex dynamical behaviors beyond simple monofractal models. In the context of digital currency markets, multifractal properties arise due to the interplay of long-range temporal correlations and heavy-tailed distributions of returns, reflecting intricate market microstructure and trader interactions. Incorporating multifractal analysis into the modeling of cryptocurrency price dynamics enhances the understanding of market inefficiencies, may improve volatility forecasting and facilitate the detection of critical transitions or regime shifts. Based on the multifractal cross-correlation analysis (MFCCA) whose spacial case is the multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical tools for quantifying multifractality, in the present contribution a recently proposed method of disentangling sources of multifractality in time series was applied to the most representative instruments from the digital market. They include Bitcoin (BTC), Ethereum (ETH), decentralized exchanges (DEX) and non-fungible tokens (NFT). The results indicate the significant role of heavy tails in generating a broad multifractal spectrum. However, they also clearly demonstrate that the primary source of multifractality are temporal correlations in the series, and without them, multifractality fades out. It appears characteristic that these temporal correlations, to a large extent, do not depend on the thickness of the tails of the fluctuation distribution. These observations, made here in the context of the digital currency market, provide a further strong argument for the validity of the proposed methodology of disentangling sources of multifractality in time series. ...

Chaotic Bayesian Inference: Strange Attractors as Risk Models for Black Swan Events ArXiv ID: 2509.08183 “View on arXiv” Authors: Crystal Rust Abstract We introduce a new risk modeling framework where chaotic attractors shape the geometry of Bayesian inference. By combining heavy-tailed priors with Lorenz and Rossler dynamics, the models naturally generate volatility clustering, fat tails, and extreme events. We compare two complementary approaches: Model A, which emphasizes geometric stability, and Model B, which highlights rare bursts using Fibonacci diagnostics. Together, they provide a dual perspective for systemic risk analysis, linking Black Swan theory to practical tools for stress testing and volatility monitoring. ...

Fitting the seven-parameter Generalized Tempered Stable distribution to the financial data ArXiv ID: 2410.19751 “View on arXiv” Authors: Unknown Abstract The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S&P 500 and SPY ETF returns). For each index, the estimation results show that the six-parameter estimations are statistically significant except for the local parameter, $μ$. The goodness-of-fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson’s chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions. ...

Diversification for infinite-mean Pareto models without risk aversion ArXiv ID: 2404.18467 “View on arXiv” Authors: Unknown Abstract We study stochastic dominance between portfolios of independent and identically distributed (iid) extremely heavy-tailed (i.e., infinite-mean) Pareto random variables. With the notion of majorization order, we show that a more diversified portfolio of iid extremely heavy-tailed Pareto random variables is larger in the sense of first-order stochastic dominance. This result is further generalized for Pareto random variables caused by triggering events, random variables with tails being Pareto, bounded Pareto random variables, and positively dependent Pareto random variables. These results provide an important implication in investment: Diversification of extremely heavy-tailed Pareto profits uniformly increases investors’ profitability, leading to a diversification benefit. Remarkably, different from the finite-mean setting, such a diversification benefit does not depend on the decision maker’s risk aversion. ...

Characteristics of price related fluctuations in Non-Fungible Token (NFT) market ArXiv ID: 2310.19747 “View on arXiv” Authors: Unknown Abstract A non-fungible token (NFT) market is a new trading invention based on the blockchain technology which parallels the cryptocurrency market. In the present work we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market - even though young and governed by a somewhat different mechanisms of trading - shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators. ...