Rajat Subhra Hazra

Patterned Random Matrices and Method of Moments

We present a unified approach to limiting spectral distribution (LSD) of patterned matrices via the moment method. We demonstrate relatively short proofs for the LSD of common matrices and provide insight into the nature of different LSD and their interrelations. The method is flexible enough to be applicable to matrices with appropriate dependent entries, banded matrices, and matrices of the form Ap = 1 XX 0 where X is a p × n matrix with real entries and p ! 1 with n = n(p) ! 1 and p/n ! y with 0 � y < 1. This approach raises interesting questions about the class of patterns for which LSD exists and the nature of the possible limits. In many cases the LSD are not known in any explicit forms and so deriving probabilistic properties of the limit are also interesting issues.

Tail behavior of randomly weighted sums

Let {X t , t ≥ 1} be a sequence of identically distributed and pairwise asymptotically independent random variables with regularly varying tails, and let {Θ t , t ≥ 1} be a sequence of positive random variables independent of the sequence {X t , t ≥ 1}. We will discuss the tail probabilities and almost-sure convergence of X (∞) = ∑ t=1 ∞Θ t X t + (where X + = max{0, X}) and max1≤k<∞∑ t=1 k Θ t X t , and provide some sufficient conditions motivated by Denisov and Zwart (2007) as alternatives to the usual moment conditions. In particular, we illustrate how the conditions on the slowly varying function involved in the tail probability of X 1 help to control the tail behavior of the randomly weighted sums. Note that, the above results allow us to choose X 1, X 2,… as independent and identically distributed positive random variables. If X 1 has a regularly varying tail of index -α, where α > 0, and if {Θ t , t ≥ 1} is a positive sequence of random variables independent of {X t }, then it is known – which can also be obtained from the sufficient conditions in this article – that, under some appropriate moment conditions on {Θ t , t ≥ 1}, X (∞) = ∑ t=1 ∞Θ t X t converges with probability 1 and has a regularly varying tail of index -α. Motivated by the converse problems in Jacobsen, Mikosch, Rosiński and Samorodnitsky (2009) we ask the question: if X (∞) has a regularly varying tail then does X 1 have a regularly varying tail under some appropriate conditions? We obtain appropriate sufficient moment conditions, including the nonvanishing Mellin transform of ∑ t=1 ∞Θ t along some vertical line in the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.

From random matrices to long range dependence

Random matrices whose entries come from a stationary Gaussian process are studied. The limiting behavior of the eigenvalues as the size of the matrix goes to infinity is the main subject of interest in this work. It is shown that the limiting spectral distribution is determined by the absolutely continuous component of the spectral measure of the stationary process. This is similar to the situation where the entries of the matrix are i.i.d. On the other hand, the discrete component contributes to the limiting behavior of the eigenvalues after a different scaling. Therefore, this helps to define a boundary between short and long range dependence of a stationary Gaussian process in the context of random matrices.

Convergence of joint moments for independent random patterned matrices.

It is known that the joint limit distribution of independent Wigner matrices satisfies a very special asymptotic independence, called freeness. We study the joint convergence of a few other patterned matrices, providing a framework to accommodate other joint laws. In particular, the matricial limits of symmetric circulants and reverse circulants satisfy, respectively, the classical independence and the half independence. The matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy or explicit independence/dependence notions. Their limits are not independent, free or half independent.

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.

Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016. https://doi.org/10.1007/s00023-015-0433-x) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.

SPECTRAL PROPERTIES OF RANDOM TRIANGULAR MATRICES

We prove the existence of the limiting spectral distribution (LSD) of symmetric triangular patterned matrices and also establish the joint convergence of sequences of such matrices. For the particular case of the symmetric triangular Wigner matrix, we derive expression for the moments of the LSD using properties of Catalan words. The problem of deriving explicit formulae for the moments of the LSD does not seem to be easy to solve for other patterned matrices. The LSD of the non-symmetric triangular Wigner matrix also does not seem to be easy to establish.

Product of exponentials and spectral radius of random k-circulants

We consider n x n random k-circulant matrices with n → ∞ and k = k(n) whose input sequence {a l } l≥0 is independent and identically distributed (i.i.d.) random variables with finite (2 + δ) moment. We study the asymptotic distribution of the spectral radius, when n = k g + 1. For this, we first derive the tail behaviour of the g fold product of i.i.d. exponential random variables. Then using this tail behaviour result and appropriate normal approximation techniques, we show that with appropriate scaling and centering, the asymptotic distribution of the spectral radius is Gumbel. We also identify the centering and scaling constants explicitly.

Boolean convolutions and regular variation

In this article we study the influence of regularly varying probability measures on additive and multiplicative Boolean convolutions. We introduce the notion of Boolean subexponentiality (for additive Boolean convolution), which extends the notion of classical and free subexponentiality. We show that the distributions with regularly varying tails belong to the class of Boolean subexponential distributions. As an application we also study the behaviour of the Belinschi-Nica map. Breimans theorem study the classical product convolution between regularly varying measures. We derive an analogous result to Breimans theorem in case of multiplicative Boolean convolution. In proving these results we exploit the relationship of regular variation with different transforms and their Taylor series expansion.

Thick points for Gaussian free fields with different cut-offs

Massive and massless Gaussian free fields can be described as generalized Gaussian processes indexed by an appropriate space of functions. In this article we study various approaches to approximate these fields and look at the fractal properties of the thick points of their cut-offs. Under some sufficient conditions for a centered Gaussian process with logarithmic variance we study the set of thick points and derive their Hausdorff dimension. We prove that various cut-offs for Gaussian free fields satisfy these assumptions. We also give sufficient conditions for comparing thick points of different cut-offs.