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A functional limit theorem for additive functionals

Thibaud Taillefumier, Gordan Zitkovic

TL;DR

This work develops a general framework for the convergence of positive additive functionals of recurrent diffusions to Lévy subordinators, with explicit conditions and a resolvent-based Laplace-exponent formula. It furnishes a general convergence theorem and a diffusion-specific criterion that jointly govern when the additive functionals converge in $M_1$ to a subordinator. As a primary application, it analyzes Wright–Fisher and Feller diffusions under a joint vanishing regime for parameters, deriving a novel limiting subordinator whose Laplace exponent is $Φ(μ)= rac{γβ}{2}ig( oot 2 racc{ ig(1+ rac{4μ}{β^2}ig)^{1/2}-1}{1}ig)$ (inverse-Gaussian) for Feller and a Bessel-function–based expression for Wright–Fisher; the results give detailed structure of the jump measure, cumulants, and moment formulas. The neuroscience motivation connects these limits to time-changed models of synchrony in doubly-stochastic spiking activity, highlighting the practical relevance of the theory to neural systems.

Abstract

We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to Lévy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.

A functional limit theorem for additive functionals

TL;DR

This work develops a general framework for the convergence of positive additive functionals of recurrent diffusions to Lévy subordinators, with explicit conditions and a resolvent-based Laplace-exponent formula. It furnishes a general convergence theorem and a diffusion-specific criterion that jointly govern when the additive functionals converge in to a subordinator. As a primary application, it analyzes Wright–Fisher and Feller diffusions under a joint vanishing regime for parameters, deriving a novel limiting subordinator whose Laplace exponent is (inverse-Gaussian) for Feller and a Bessel-function–based expression for Wright–Fisher; the results give detailed structure of the jump measure, cumulants, and moment formulas. The neuroscience motivation connects these limits to time-changed models of synchrony in doubly-stochastic spiking activity, highlighting the practical relevance of the theory to neural systems.

Abstract

We study a general limiting framework for the convergence of sequences of additive functionals of diffusions to Lévy subordinators, and provide explicit sufficient conditions that both ensure convergence and characterize the law of the limit. As an application, we identify a novel limiting regime for Wright-Fisher and Feller diffusions in the reflecting case and describe the corresponding limiting subordinator. This work is motivated by, and has applications in, neuroscience, where reflected diffusions are used to parametrize synchrony in doubly-stochastic models of spiking activity.
Paper Structure (21 sections, 6 theorems, 104 equations)

This paper contains 21 sections, 6 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Our contributions
  4. A general convergence result
  5. The Wright-Fisher diffusion
  6. Neuroscientific motivation
  7. The structure of the paper.
  8. Sufficient conditions for convergence
  9. General convergence to a Lévy subordinator
  10. Convergence in a diffusion framework
  11. Condition \ref{['ttoz']} of Theorem \ref{['main-1']}.
  12. Condition \ref{['equi']} of Theorem \ref{['main-1']}
  13. Condition \ref{['reso']} of Theorem \ref{['main-1']}
  14. Wright-Fisher Diffusions
  15. The scaling regime.
  16. ...and 6 more sections

Key Result

Theorem 2.1

Suppose that the following conditions hold: Then the sequence $\{ A_n(t) \}_{t \in [0,\infty)}$ converges in law, under the Skorokhod's $M_1$-topology, to a Lévy subordinator whose Laplace exponent $\Phi(\mu)$ is given by

Theorems & Definitions (13)

  • Theorem 2.1
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • ...and 3 more