[Paper Review] Renewal theory for extremal Markov sequences of the Kendall type

B. H. Jasiulis-Gołdyn, Karolina Naskręt+2 more|arXiv (Cornell University)|Mar 29, 2018

Financial Risk and Volatility Modeling10 references3 citations

TL;DR

This paper establishes renewal theory for extremal Markov sequences driven by Kendall random walks, using generalized convolutions and regularly varying functions. It proves a Blackwell-type theorem and an elementary renewal theorem for the Kendall convolution, showing that the renewal function grows asymptotically as $ R(t) o rac{2}{m( heta)} t^ heta $ when the step distribution has a finite $ heta $-th moment, with convergence rates governed by the Williamson transform and tail behavior.

ABSTRACT

The paper deals with renewal theory for a class of extremal Markov sequences connected with the Kendall convolution. We consider here some particular cases of the Wold processes associated with generalized convolutions. We prove an analogue of the Fredholm theorem for all regular generalized convolutions algebras. Using regularly varying functions we prove a Blackwell theorem for renewal processes defined by Kendall random walks.

Motivation & Objective

• To develop renewal theory for extremal Markov sequences generated by Kendall random walks.
• To extend classical renewal theorems—particularly Fredholm and Blackwell—to the context of generalized convolutions.
• To analyze the asymptotic behavior of renewal processes under the Kendall convolution using regularly varying functions.
• To characterize the scaling limits of renewal counting processes and their moments via the Williamson transform.
• To establish precise asymptotic expressions for the renewal function and interarrival time distributions under heavy-tailed step laws.

Proposed method

• The paper employs generalized convolutions, particularly the Kendall convolution, to define a non-classical random walk structure where the sum of independent unit steps follows a Pareto-type distribution.
• It uses the Williamson transform as a key analytical tool to characterize the distribution of the renewal process and to derive asymptotic properties of the renewal function.
• The analysis relies on the theory of regularly varying functions to handle heavy-tailed distributions and to derive scaling limits for the renewal process.
• The Fredholm equation for renewal measures is proven in the generalized convolution setting: $ m = \nu + \nu \diamond m $, ensuring existence and uniqueness of the renewal measure.
• Blackwell-type theorems are derived by analyzing the difference $ R(t+h) - R(t) $, showing convergence to $ \frac{2h}{m(\theta)} $ under appropriate moment conditions.
• The paper uses weak convergence techniques to derive limit distributions for the normalized renewal process, such as $ S_n / U(n) \Rightarrow Z^{-1/(α-θ)} $, where $ Z $ has a compound distribution.

Experimental results

Research questions

• RQ1 How can the classical Fredholm and Blackwell renewal theorems be extended to generalized convolutions, particularly the Kendall convolution?
• RQ2 What is the asymptotic behavior of the renewal function $ R(t) = \sum_{n=0}^\infty F^{\diamond n}(t) $ when the unit step distribution has a regularly varying tail?
• RQ3 What is the limiting distribution of the normalized renewal counting process $ N(t) $ under the Kendall random walk?
• RQ4 How does the Williamson transform facilitate the analysis of renewal processes under generalized convolutions?
• RQ5 What are the exact scaling limits and convergence rates for $ R(t) $ and $ R(t+h) - R(t) $ when the step distribution has a finite $ \theta $-th moment?

Key findings

• The elementary renewal theorem holds for Kendall random walks: if $ m(\theta) = E[T_1^\theta] < \infty $, then $ \lim_{t \to \infty} \frac{R(t)}{t^\theta} = \frac{2}{m(\theta)} $.
• A Blackwell-type theorem is established: $ \lim_{t \to \infty} \left( \frac{R(t+h)}{(t+h)^{\theta-1}} - \frac{R(t)}{t^{\theta-1}} \right) = \frac{2h}{m(\theta)} $ for $ h > 0 $.
• For unit steps with tail $ \overline{F}(x) \sim x^{-\beta} $, the renewal function satisfies $ R(t) \sim \frac{(\alpha - \beta)(\alpha + \beta)}{\alpha^2} t^{-\beta} $ when $ \alpha > \beta $.
• When $ \alpha = \beta $, the asymptotic behavior is $ \alpha t^{-\alpha} \log t \cdot R(t) \to 2 $, and $ R(t+h) - R(t) \sim \frac{2h}{\alpha t^{\alpha-1} \log t} $.
• The normalized renewal process $ S_n / U(n) $ converges in distribution to $ Z^{-1/(α-θ)} $, where $ Z $ has a compound distribution with characteristic function $ \Phi_Z(v) = (1 - iv)^{-2} $.
• The counting process $ N(t) $ satisfies $ \alpha t^{-\beta} N(t) / \theta \to Z $ in distribution, with $ \theta = \alpha - \beta $, and $ U(y) \sim (y \log y)^{1/\alpha} $ in the $ \alpha = \beta $ case.