Aggregating heavy-tailed random vectors: from finite sums to Lévy processes

Paper: arXiv 2301.10423

Authors: Bikramjit Das, Vicky Fasen-Hartmann

Abstract

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of “one large jump’’, be it for finite sums, random sums, or, Lévy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive quadrant, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities which were rendered asymptotically negligible under classical regular variation, despite the “one large jump’’ asymptotics. We also discover that depending on the structure of the tail event of concern, the tail behavior of the aggregates may be characterized by more than a single large jump. Eventually, we illustrate a similar phenomenon for multivariate regularly varying Lévy processes, establishing as well a relationship between multivariate regular variation of a Lévy process and multivariate regular variation of its Lévy measure on different subcones.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical exploration of multivariate regular variation and heavy-tailed random vectors, extending classical ‘one large jump’ principles to subcones. However, it is entirely theoretical with no empirical backtesting or real-world financial data applications included in the provided text.

Research Flowchart

  flowchart TD
    A["Research Goal: Analyze tail behavior of heavy-tailed aggregates beyond standard 'one large jump'"] --> B["Inputs: Multivariate regularly varying random vectors & Lévy processes on subcones"]
    B --> C["Methodology: Restrict regular variation to various subcones of the positive quadrant"]
    C --> D["Analysis: Compute tail probabilities of finite sums, random sums, and Lévy processes"]
    D --> E["Finding 1: Aggregates inherit multivariate regular variation on subcones, revealing hidden tail risks"]
    D --> F["Finding 2: Tail behavior can be characterized by MULTIPLE large jumps depending on event structure"]
    D --> G["Finding 3: Established equivalence between Lévy process regular variation and its Lévy measure on subcones"]