Ayan Bhattacharya

We prove the consistency of the Power-Law Fit PLFit method proposed by Clauset et al.(2009) 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.

Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric α-stable discrete parameter random field. We show that the power function converges to 1 as the sample-size increases to ∞ under various classes of alternatives having longer memory in the sense of Samorodnitsky (2004). Ergodic theory of nonsingular ℤ^d-actions plays a very important role in the design and analysis of our large sample test.

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten–Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the $n$th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of (J. Stat. Phys. 143 (3) (2011) 420–446) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165–170) in our framework. Nous étudions le comportement limite de processus ponctuels associés à la marche aléatoire branchante avec branchement surcritique et une loi de déplacement à variation régulière. Si le processus de branchement sous-jacent satisfait une condition de Kesten–Stigum, nous montrons que le processus ponctuel de la suite des déplacements changés d’échelle provenant de la $n$-ième génération converge faiblement vers un processus de Cox. En particulier, nous prouvons qu’une conjecture de (J. Stat. Phys. 143 (3) (2011) 420–446) reste valable dans ce contexte, nous étudions plusieurs questions soulevées dans leur article et retrouvons le résultat principal de (Z. Wahrsch. Verw. Gebiete 62 (2) (1983) 165–170) dans notre cadre.

Using the language of regular variation, we give a sufficient condition for a point process to be in the superposition domain of attraction of a strictly stable point process. This sufficient condition is then used to obtain an explicit representation of the weak limit of a sequence of point processes induced by a branching random walk with jointly regularly varying displacements. As a consequence, we extend the main result of Durrett (1983) and verify that two related predictions of Brunet and Derrida (2011) remain valid for this model.

We consider the limiting behaviour of the point processes associated with a branching random walk with supercritical branching mechanism and balanced regularly varying step size. Assuming that the underlying branching process satisfies Kesten-Stigum condition, it is shown that the point process sequence of properly scaled displacements coming from the n-th generation converges weakly to a Cox cluster process. In particular, we establish that a conjecture of Brunet and Derrida (2011) remains valid in this setup, investigate various other issues mentioned in their paper and recover the main result of Durrett (1983) in our framework.