[Paper Review] Renewal shot noise processes in the case of slowly varying tails

Zakhar Kabluchko, Alexander Marynych|arXiv (Cornell University)|May 9, 2016

Financial Risk and Volatility Modeling20 references3 citations

TL;DR

This paper establishes the weak convergence of renewal shot noise processes with slowly varying inter-arrival time tails, showing that after a non-linear scaling involving the inverse of the slowly varying function, the process converges in finite-dimensional distributions to a transformed inverse extremal process. The key result identifies the limiting process as a time-changed inverse stable subordinator, with marginals distributed as exponential random variables scaled by power functions.

ABSTRACT

We investigate weak convergence of renewal shot noise processes in the case of slowly varying tails of the inter-shot times. We show that these processes, after an appropriate non-linear scaling, converge in the sense of finite-dimensional distributions to an inverse extremal process.

Motivation & Objective

• To analyze the asymptotic behavior of renewal shot noise processes when the inter-arrival times have slowly varying tails.
• To address a gap in the literature concerning the weak convergence of such processes under non-exponential, heavy-tailed inter-arrival distributions.
• To establish a functional limit theorem for these processes using non-linear scaling based on the inverse of the slowly varying function.
• To characterize the limiting process as a time-changed inverse extremal process with explicit marginal and increment distributions.

Proposed method

• The authors use the functional limit theorem for random walks with slowly varying tails, specifically the convergence of $ n^{-1}L(S_{[n\cdot]}) $ to an extremal process $ m(\cdot) $ in the $ M_1 $-topology.
• They apply a non-linear time change using the inverse function $ L^{\leftarrow} $, transforming the time scale to $ t \mapsto L^{\leftarrow}(tu) $, which normalizes the heavy-tailed behavior.
• The shot noise process $ Y(t) $ is scaled by $ t \cdot h(L^{\leftarrow}(t)) $, where $ h \circ L^{\leftarrow} $ is regularly varying with index $ \alpha $, ensuring proper normalization.
• The convergence is proven via weak convergence of finite-dimensional distributions by decomposing the integral into two parts: one over intervals where the function is bounded away from the jump times, and another over neighborhoods of potential jumps.
• The proof relies on the continuity of the first-passage time mapping in the $ M_1 $-topology and the self-similarity of the extremal process.
• An auxiliary lemma establishes that any slowly varying function can be replaced by a strictly increasing, continuous version without altering the asymptotic behavior.

Experimental results

Research questions

• RQ1 What is the limiting behavior of renewal shot noise processes when the inter-arrival times have slowly varying tails?
• RQ2 How should the time and amplitude scales be transformed to obtain a non-degenerate weak limit?
• RQ3 What is the nature of the limiting process in terms of its finite-dimensional distributions and path properties?
• RQ4 How does the scaling function $ L^{\leftarrow} $ influence the convergence and the structure of the limit?
• RQ5 Can the limiting process be explicitly characterized, and what are the distributions of its increments and marginals?

Key findings

• The finite-dimensional distributions of the scaled renewal shot noise process converge to those of $ (u_1^\alpha m^{\leftarrow}(u_1), \dots, u_n^\alpha m^{\leftarrow}(u_n)) $, where $ m^{\leftarrow} $ is the generalized inverse of the extremal process.
• The limiting process $ m^{\leftarrow}(u) $ has independent increments and marginal distributions that are exponential: $ \mathbb{P}\{m^{\leftarrow}(u) > v\} = e^{-v/u} $.
• The convergence holds under a non-linear time change $ t \mapsto L^{\leftarrow}(tu) $, which accounts for the slowly varying tail of the inter-arrival time distribution.
• The limiting process is almost surely continuous at every fixed $ u \geq 0 $, a consequence of the path properties of the extremal process.
• The convergence is established in the sense of weak convergence of finite-dimensional distributions, with the limit process being a time-changed inverse stable subordinator.
• The result extends the classical theory of shot noise convergence to the case of slowly varying tails, where linear normalization fails and non-linear scaling is essential.