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Papers by Bohan Chen

Research paper thumbnail of Consistency of the PLFit estimator for power-law data

arXiv (Cornell University), Feb 17, 2020

We prove the consistency of the Power-Law Fit (PLFit) method proposed by Clauset et al. [5] to es... more We prove the consistency of the Power-Law Fit (PLFit) method proposed by Clauset et al. [5] to estimate the power-law exponent in data coming from a distribution function with regularly-varying tail. In the complex systems community, PLFit has emerged as the method of choice to estimate the power-law exponent. Yet, its mathematical properties are still poorly understood. The difficulty in PLFit is that it is a minimum-distance estimator. It first chooses a threshold that minimizes the Kolmogorov-Smirnov distance between the data points larger than the threshold and the Pareto tail, and then applies the Hill estimator to this restricted data. Since the number of order statistics used is random, the general theory of consistency of power-law exponents from extreme value theory does not apply. Our proof consists in first showing that the Hill estimator is consistent for general intermediate sequences for the number of order statistics used, even when that number is random. Here, we call a sequence intermediate when it grows to infinity, while remaining much smaller than the sample size. The second, and most involved, step is to prove that the optimizer in PLFit is with high probability an intermediate sequence, unless the distribution has a Pareto tail above a certain value. For the latter special case, we give a separate proof.

Research paper thumbnail of Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

Mathematics of Operations Research, 2019

We propose a class of strongly efficient rare-event simulation estimators for random walks and co... more We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with se...

Research paper thumbnail of Importance sampling of heavy-tailed iterated random functions

Advances in Applied Probability, 2018

We consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn... more We consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn}n∈ℕis a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} whenxis large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψnsuch that, for largex, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

Research paper thumbnail of Simulation of Stochastic Volterra Equations Driven by Space–Time Lévy Noise

The Fascination of Probability, Statistics and their Applications, 2015

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equa... more In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods L p-and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.

Research paper thumbnail of Consistency of the PLFit estimator for power-law data

arXiv (Cornell University), Feb 17, 2020

We prove the consistency of the Power-Law Fit (PLFit) method proposed by Clauset et al. [5] to es... more We prove the consistency of the Power-Law Fit (PLFit) method proposed by Clauset et al. [5] to estimate the power-law exponent in data coming from a distribution function with regularly-varying tail. In the complex systems community, PLFit has emerged as the method of choice to estimate the power-law exponent. Yet, its mathematical properties are still poorly understood. The difficulty in PLFit is that it is a minimum-distance estimator. It first chooses a threshold that minimizes the Kolmogorov-Smirnov distance between the data points larger than the threshold and the Pareto tail, and then applies the Hill estimator to this restricted data. Since the number of order statistics used is random, the general theory of consistency of power-law exponents from extreme value theory does not apply. Our proof consists in first showing that the Hill estimator is consistent for general intermediate sequences for the number of order statistics used, even when that number is random. Here, we call a sequence intermediate when it grows to infinity, while remaining much smaller than the sample size. The second, and most involved, step is to prove that the optimizer in PLFit is with high probability an intermediate sequence, unless the distribution has a Pareto tail above a certain value. For the latter special case, we give a separate proof.

Research paper thumbnail of Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

Mathematics of Operations Research, 2019

We propose a class of strongly efficient rare-event simulation estimators for random walks and co... more We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with se...

Research paper thumbnail of Importance sampling of heavy-tailed iterated random functions

Advances in Applied Probability, 2018

We consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn... more We consider the stationary solutionZof the Markov chain {Zn}n∈ℕdefined byZn+1=ψn+1(Zn), where {ψn}n∈ℕis a sequence of independent and identically distributed random Lipschitz functions. We estimate the probability of the event {Z>x} whenxis large, and develop a state-dependent importance sampling estimator under a set of assumptions on ψnsuch that, for largex, the event {Z>x} is governed by a single large jump. Under natural conditions, we show that our estimator is strongly efficient. Special attention is paid to a class of perpetuities with heavy tails.

Research paper thumbnail of Simulation of Stochastic Volterra Equations Driven by Space–Time Lévy Noise

The Fascination of Probability, Statistics and their Applications, 2015

In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equa... more In this paper we investigate two numerical schemes for the simulation of stochastic Volterra equations driven by space-time Lévy noise of pure-jump type. The first one is based on truncating the small jumps of the noise, while the second one relies on series representation techniques for infinitely divisible random variables. Under reasonable assumptions, we prove for both methods L p-and almost sure convergence of the approximations to the true solution of the Volterra equation. We give explicit convergence rates in terms of the Volterra kernel and the characteristics of the noise. A simulation study visualizes the most important path properties of the investigated processes.