arXiv subjects

Ayan Bhattacharya

Publications and source records attributed to Ayan Bhattacharya.

12 recordsLinked to original sources

arXiv2024

Branching random walk and log-slowly varying tails

We study a branching random walk with independent and identically distributed, heavy tailed displacements. The offspring law is supercritical and satisfies the Kesten-Stigum condition. We treat the case when the law of the displacements does not lie in the max-domain of attraction of an extreme value distribution. Hence, the classical extreme value theory, which is often deployed in this kind of models, breaks down. We show that if the tails of the displacements are such that the absolute value of the logarithm of the tail is a slowly varying function, one can still effectively analyse the extremes of the process. More precisely, after a non-linear transformation the extremes of the branching random walk process converge to a cluster Cox process.

math.PR↗

arXiv2023

Extreme positions of regularly varying branching random walk in random and time-inhomogeneous environment

In this article, we consider a Branching Random Walk on the real line. The genealogical structure is assumed to be given through a supercritical branching process in the i.i.d. environment and satisfies the Kesten-Stigum condition. The displacements coming from the same parent are assumed to have jointly regularly varying tails. Conditioned on the survival of the underlying genealogical tree, we prove that the appropriately normalized (normalization depends on the quenched size of the $n$-th generation) maximum among positions at the $n$-th generation converges weakly to a scale-mixture of Frechét random variable. Furthermore, we derive the weak limit of the point processes composed of appropriately scaled positions at the $n$-th generation and show that the limit point process is a member of the randomly scaled scale-decorated Poisson point processes. Hence, an analog of the predictions by Brunet and Derrida (2011) holds. We have obtained an explicit description of the limit point process. This description captures the influence of the environment in the joint asymptotic behavior of the extreme positions. We show that the law of the clusters in the limit depends on the time-reversed environment. The asymptotic distribution of the normalized rightmost position is derived as a consequence. This approach (based on weak convergence of extremal processes) can not be adapted when the genealogical structure is given through a supercritical Branching Process in a time-Inhomogeneous Environment (BPIE) (due to lack of structural regularity in the genealogical structure). We provide a simple example where the point processes do not converge weakly. The tightness of the point processes holds in this example though the point processes do not (weakly) converge (due to having different subsequential weak limits). This phenomenon is not yet known in the literature.

math.PR↗

arXiv2022

Arbitrage from a Bayesian's Perspective

This paper builds a model of interactive belief hierarchies to derive the conditions under which judging an arbitrage opportunity requires Bayesian market participants to exercise their higher-order beliefs. As a Bayesian, an agent must carry a complete recursion of priors over the uncertainty about future asset payouts, the strategies employed by other market participants that are aggregated in the price, other market participants' beliefs about the agent's strategy, other market participants beliefs about what the agent believes their strategies to be, and so on ad infinitum. Defining this infinite recursion of priors -- the belief hierarchy so to speak -- along with how they update gives the Bayesian decision problem equivalent to the standard asset pricing formulation of the question. The main results of the paper show that an arbitrage trade arises only when an agent updates his recursion of priors about the strategies and beliefs employed by other market participants. The paper thus connects the foundations of finance to the foundations of game theory by identifying a bridge from market arbitrage to market participant belief hierarchies.

econ.TH↗

arXiv2022

Large deviations of extremes in branching random walk with regularly varying displacements

In this article, we consider a branching random walk on the real-line where displacements coming from the same parent have jointly regularly varying tails. The genealogical structure is assumed to be a supercritical Galton-Watson tree, satisfying Kesten-Stigum condition. We study the large deviations of the extremal process, formed by the appropriately normalized positions in the $n$-th generation and show that the large extreme-positions form clusters in the limit. As a consequence of this, we also study the large deviations of the maximum among positions at the $n$-th generation.

math.PR↗

arXiv2022

Persistence of heavy-tailed sample averages: principle of infinitely many big jumps

We consider the sample average of a centered random walk in $\mathbb{R}^d$ with regularly varying step size distribution. For the first exit time from a compact convex set $A$ not containing the origin, we show that its tail is of lognormal type. Moreover, we show that the typical way for a large exit time to occur is by having a number of jumps growing logarithmically in the scaling parameter.

math.PR↗

arXiv2020

Consistency of the PLFit estimator for power-law data

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.

math.ST↗

arXiv2019

Slower variation of the generation sizes induced by heavy-tailed environment for geometric branching

Motivated by seminal paper of Kozlov et al.(1975) we consider in this paper a branching process with a geometric offspring distribution parametrized by random success probability $A$ and immigration equals $1$ in each generation. In contrast to above mentioned article, we assume that environment is heavy-tailed, that is $\log A^{-1} (1-A)$ is regularly varying with a parameter $α>1$, that is that ${\bf P} \Big( \log A^{-1} (1-A) > x \Big) = x^{-α} L(x)$ for a slowly varying function $L$. We will prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of distribution of the population at $n$-th generation which gets even heavier with $n$ increasing. Precisely, in this work, we prove that asymptotic tail ${\bf P}(Z_l \ge m)$ of $l$-th population $Z_l$ is of order $ \Big(\log^{(l)} m \Big)^{-α} L \Big(\log^{(l)} m \Big)$ for large $m$, where $\log^{(l)} m = \log \ldots \log m$. The proof is mainly based on Tauberian theorem. Using this result we also analyze the asymptotic behaviour of the first passage time $T_n$ of the state $n \in \mathbb{Z}$ by the walker in a neighborhood random walk in random environment created by independent copies $(A_i : i \in \mathbb{Z})$ of $(0,1)$-valued random variable $A$. This version differs from the final version as it contains an alternative proof for the tail behavior for generation sizes which is not very sharp (lacks constant) but completely avoids arguments based on Tauberian theorem. This proof may be of an independent interest.

math.PR↗

arXiv2018

Extremes of Multi-type Branching Random Walks: Heaviest Tail Wins

We consider a branching random walk on a multi($Q$)-type, supercritical Galton-Watson tree which satisfies Kesten-Stigum condition. We assume that the displacements associated with the particles of type $Q$ have regularly varying tails of index $α$, while the other types of particles have lighter tails than that of particles of type $Q$. In this article, we derive the weak limit of the sequence of point processes associated with the positions of the particles in the $n^{th}$ generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process (SScDPPP) using the tools developed in \cite{bhattacharya:hazra:roy:2016}. As a consequence, we shall obtain the asymptotic distribution of the position of the rightmost particle in the $n^{th}$ generation.

math.PR↗

arXiv2018

A note on randomly scaled scale-decorated Poisson point processes

Randomly scaled scale-decorated Poisson point process is introduced recently in Bhattacharya et al. [2017] where it appeared as weak limit of a sequence of point processes in the context of branching random walk. In this article, we obtain a characterization for these processes based on scaled-Laplace functional. As a consequence, we obtain a characterization for strictly $α$-stable point process (also known as scale-decorated Poisson point process) based on scaled-Laplace functional . a connection with randomly shifted decorated Poisson point process is obtained. The tools and approach used e very similar to those in Subag and Zeitouni [2015].

math.PR↗

arXiv2017

A large sample test for the length of memory of stationary symmetric stable random fields via nonsingular $\mathbb{Z}^d$-actions

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 one as the sample-size increases to infinity under various classes of alternatives having longer memory in the sense of Samorodnitsky(2004). Ergodic theory of nonsingular $\mathbb{Z}^d$-actions play a very important role in the design and analysis of our large sample test.

math.PR↗

arXiv2016

Branching Random Walks, Stable Point Processes and Regular Variation

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.

math.PR↗

arXiv2016

Point process convergence for branching random walks with regularly varying steps

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.

math.PR↗