regular-variation asymptotics

This study investigates how to distinguish whether large trades stem from private information or liquidity shocks driven by heavy-tailed distributions, and elucidates the impact of liquidity tail risk on price discovery and market microstructure. To this end, the authors develop a continuous-auction limit order book model under asymmetric information, where market makers observe only aggregate order flow and cannot differentiate informed trades from uninformed liquidity demands following a Student-t distribution. Innovatively treating the heavy-tailed nature of liquidity demand as a key state variable, they characterize equilibrium via a fixed-point equation for marginal cost scheduling and solve it using regular variation asymptotics within a tight class under tail control. The analysis reveals that heavy-tailed liquidity demand attenuates the convexity of price impact, slows the rate of information learning, and induces a regular variation law—governed by the tail index—for the price impact of large orders; notably, fundamental value remains asymptotically revealed even under a constant information arrival rate.

This study rigorously tests the universal validity of the Square Root Law (SRL) in nonlinear price impact—specifically, whether the exponent δ in the power-law relation (I propto Q^delta) between average price impact (I) and trade size (Q) is strictly 0.5. Leveraging eight years of high-resolution, order-level transaction data from the Tokyo Stock Exchange—including full market coverage and account-level granularity—we employ precise power-law estimation, robust cross-sectional validation across individual stocks and traders, and formal model rejection tests. Results demonstrate that δ converges to 0.5 with ±0.01 precision at both the single-stock and single-trader levels; this holds uniformly across the entire eight-year period and across the full liquidity spectrum. The study provides the first empirical confirmation of SRL’s statistically strict universality, decisively rejecting two prominent classes of non-universal theoretical models. It thus delivers definitive evidence for one of the rare high-precision universal laws in financial markets.

This paper identifies the systematic failure of classical statistical methods—including mean-based inference, principal component analysis (PCA), and asymptotic normality assumptions—under heavy-tailed distributions, particularly in medium-sample-size (medium-*n*) real-world settings where they are routinely misapplied. Methodologically, it challenges the uncritical adoption of Gaussian and stable-distribution assumptions and introduces the “Median Law” theoretical framework, which formalizes fundamental limitations under heavy tails: unreliable sample means, distorted empirical distributions, and degenerate principal components. The approach integrates extreme value theory, generalized stable distribution modeling, robust parametric estimation, and pre-asymptotic analysis. Empirical validation draws on counterexamples from finance, economics, and psychology, supplemented by cross-disciplinary case studies. The core contribution is a foundational rethinking of uncertainty quantification and causal inference: it demonstrates that many canonical “cognitive biases” are, in fact, rational inferences under heavy-tailed probability structures—thereby advocating a paradigm shift in statistical practice from idealized asymptotics to empirically grounded probabilistic modeling.

Under extreme market events, market efficiency declines, and existing multivariate extreme-value models fail to capture asymmetries in upper- versus lower-tail dependence. Method: This paper proposes the Efficient Tail Hypothesis (ETH), redefining market efficiency through extreme-value theory. It innovatively constructs a regularly varying model defined over the full space ℝᵈ and introduces the Directional Tail Dependence (DTD) measure, establishing the first market efficiency testing framework tailored to extreme behavior—integrating regular variation analysis, multivariate extreme-value statistics, permutation testing, and high-dimensional tail visualization. Contribution/Results: Empirical analysis reveals statistically significant violations of ETH in China’s futures markets, uncovering robust arbitrage opportunities. Concurrently, the study releases a novel high-frequency derivatives dataset, addressing a critical gap in market microstructure research.

This study challenges the conventional view that liquidity, supply, and demand are fundamental economic variables, arguing instead that they emerge from the geometric structure induced by order book observations. By modeling the market as an expanding relational system devoid of predefined metrics, time, or price coordinates, and applying spectral embedding of the graph Laplacian to obtain a one-dimensional projection, the authors derive a price-like coordinate and a corresponding liquidity distribution. Remarkably, this approach reproduces canonical order book regularities without invoking assumptions about agent behavior. Using high-frequency Level II data from U.S. equities, the research demonstrates the cross-asset universality of a cumulative gamma-shaped liquidity profile, with information criteria confirming its superior fit compared to existing models.

This study addresses the challenge of precisely characterizing dynamic liquidity recovery and nonlinear price impact in market impact modeling. Based on the generalized Langevin equation, it simulates latent trader activation and liquidity pool evolution. Methodologically, this work achieves the first exact non-approximate Markovian lift, deriving the intermediate impact regime without presupposing a square-root law while rigorously proving the non-negativity of round-trip costs. The results reveal the volatility-dependent nature of the square-root impact mechanism and demonstrate that liquidity depletion compresses this intermediate regime. Furthermore, distinct memory spectra yield significantly different responses following large trades. These findings establish a theoretical foundation for model calibration.

This study clarifies the fundamental distinction and coexistence mechanism between long memory in trade signs and the square-root price impact law under event time versus physical (calendar) time. By employing a coupled discrete reaction–diffusion model augmented with non-uniformly sampled event times and meta-order source terms, and assuming a locally linear order book with constant participation rate execution, the authors formulate the dynamics via Volterra integral equations and heavy-tailed Pareto distributions. They demonstrate that long memory arises from sign correlations in event time, whereas the square-root law reflects market resilience constrained by the distribution of waiting times in physical time. The model successfully reproduces the empirically observed concave impact trajectory and the proportionality between transient impact and the square root of meta-order size, while elucidating how non-uniform event timing modulates the impact profile in calendar time.

研究在资产价格遵循幂律轨迹变动时，如何通过凯利公式确定最优投资比例，并利用比特币数据验证模型中波动率指数的稳定性。

This study addresses the nonparametric characterization of second-moment dependence structures in continuous-time multivariate asset price processes. Building on high-frequency returns, the authors construct local volatility estimators and introduce, for the first time, the “realized volatility copula” statistic to nonparametrically estimate the empirical copula of latent stochastic volatilities. They establish infill asymptotic consistency under both fixed and expanding time horizons and derive a functional central limit theorem for the empirical process of time-invariant marginal copulas subject to measurement error. Simulations demonstrate that the method accurately approximates the true volatility copula even with moderate sampling frequencies and short sample periods, while goodness-of-fit tests exhibit well-controlled size and high power. Empirically, the analysis reveals that the dependence between U.S. equity and Treasury futures volatilities is well captured by a Gumbel copula.

This study addresses the bias inherent in tail index estimation for heavy-tailed distributions by proposing a novel estimator that integrates bias correction with empirical likelihood. The method uniquely combines bias correction techniques within an empirical likelihood framework to yield a more accurate and stable estimator, accompanied by rigorous asymptotic theory. Simulation experiments demonstrate that the proposed approach significantly outperforms existing methods in finite samples, while empirical analyses on real-world data further confirm its practical effectiveness and applicability.