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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2601.04490 |
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| _version_ | 1866914239892946944 |
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| author | Petrosyan, Armen |
| author_facet | Petrosyan, Armen |
| contents | Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central tendency accuracy. In this paper, we introduce a Weighted Kolmogorov Metric tailored for financial time series with sub-cubic moments ($\mathbb{E}|X|^{2+δ}<\infty$). By incorporating an exhaustion function $h(x)$ that mechanically downweights extreme tail noise, we prove that we can restore the optimal Gaussian convergence rate of $O(n^{-1/2})$ even for Pareto and Student-t distributions common in Crypto and FX markets. We provide a complete proof using a core/tail truncation scheme and establish the optimal tuning of the weight parameter $q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04490 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Restoring Convergence in Heavy-Tailed Risk Models: A Weighted Kolmogorov Approach for Robust Backtesting Petrosyan, Armen Probability Standard risk metrics used in model validation, such as the Kolmogorov-Smirnov distance, fail to converge at practical rates when applied to high-frequency financial data characterized by heavy tails (infinite skewness). This creates a "noise barrier" where valid risk models are rejected due to tail events irrelevant to central tendency accuracy. In this paper, we introduce a Weighted Kolmogorov Metric tailored for financial time series with sub-cubic moments ($\mathbb{E}|X|^{2+δ}<\infty$). By incorporating an exhaustion function $h(x)$ that mechanically downweights extreme tail noise, we prove that we can restore the optimal Gaussian convergence rate of $O(n^{-1/2})$ even for Pareto and Student-t distributions common in Crypto and FX markets. We provide a complete proof using a core/tail truncation scheme and establish the optimal tuning of the weight parameter $q$. |
| title | Restoring Convergence in Heavy-Tailed Risk Models: A Weighted Kolmogorov Approach for Robust Backtesting |
| topic | Probability |
| url | https://arxiv.org/abs/2601.04490 |