arXiv subjects

Pradeep Vishwakarma

Publications and source records attributed to Pradeep Vishwakarma.

10 recordsLinked to original sources

arXiv2026

Revisiting Toeplitz and Hankel random matrices via $*$-convergence of circulant-type matrices

We establish the joint $*$-convergence of a random circulant matrix and a specific deterministic diagonal matrix. We also show that the empirical spectral distributions of skew-circulant and left skew-circulant random matrices converge weakly a.s.~to complex Gaussian and symmetrized Rayleigh distributions, respectively. The $*$-convergence of symmetric Toeplitz and Hankel random matrices is well known. So is the weak convergence of their random spectrum. However, not much is known about the limits. We exploit the connections of circulant, reverse circulant, and left skew-circulants with the Hankel and Toeplitz matrices, to show the $*$-convergence of the random symmetric Toeplitz matrix to the sum of two non-commutative self-adjoint variables, each having a real Gaussian distribution. A similar result holds for the non-symmetric Toeplitz matrix, but the variables are not self-adjoint and have a complex Gaussian distribution. The random Hankel matrix is shown to converge in $*$-distribution to a sum of two self-adjoint variables, each of which has a symmetrized Rayleigh distribution. As a consequence of these results, we also obtain a different proof of the convergence of the empirical spectral distribution of symmetric Toeplitz and Hankel matrices, and a slightly different way of expressing the moments of the limit spectral distribution.

math.PR↗

arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗

arXiv2026

Time-changed multiparameter Poisson processes and martingales

We consider a multiparameter extension of the homogeneous Poisson counting process, namely, the multiparameter Poisson process (MPP). We derive its various distributional properties. Also, we consider an integral of the MPP and analyze its asymptotic distribution. Thereafter, we investigate three time-changed variants of the MPP, where the time-changing components are multivariate subordinator and inverse subordinators with both dependent and independent marginals. Later, we obtain some properties of the multiparameter martingales, which are then used to derive multiparameter martingale characterizations for the MPP and one of its time-changed variants.

math.PR↗

arXiv2026

Convergence of random sums in non-commutative probability

Random sums of independent random variables have been extensively studied in classical probability theory. We consider random sums of self-adjoint variables from a non-commutative probability space, and establish several $*$-convergence results. In particular, we show that the joint $*$-convergence of the standardized random sum of identically distributed self-adjoint variables and the standardized stopping random variable (rv) is equivalent to the convergence of all moments of the stopping rv together with the convergence of the ratio of its mean to its variance. We obtain central limit theorems for the random sums of free, independent and half independent self-adjoint variables with both deterministic and random scaling. Furthermore, we derive some scaling $*$-convergence limits for randomly indexed self-adjoint variables.

math.PR↗

arXiv2026

Patterned matrices with random walk entries

It is well known that the weak limit of a suitably scaled continuous-time random walk (CTRW) is the Brownian motion. We investigate the convergence of certain patterned random matrices whose entries are independent CTRWs and their time-changed versions, in a non-commutative probability framework. For the Wigner link function, the limits are free Brownian motion and its time-changed version driven by an inverse stable subordinator. For the symmetric circulant and the circulant with CTRW entries, we use their explicit eigenvalue expressions to define some empirical processes that converge weakly to a Brownian motion and a complex Brownian motion, respectively. For matrices with iid entries, and for elliptic matrices, the algebraic limits are equal in $*$-distribution to processes whose marginals are circular and elliptic variables, respectively. A random time-changed variant of these results is also established.

math.PR↗

arXiv2026

Skellam Random Fields and Their Fractional Variants

We study some Skellam-type spatial point processes. As a particular case, we consider a Skellam random field (SRF) on the positive quadrant of the plane, which is a two parameter Lévy process with rectangular increments. A weak convergence result is obtained for the SRF. The Riemann-Liouville integral of the SRF over finite rectangles is analyzed. We derive a scaled compound Poisson field characterization for the Riemann integral of the SRF. Also, an explicit expression of its characteristic function is obtained. Later, we consider three fractional variants of the two parameter SRF. Their point probabilities, associated governing equations, and various other distributional properties are analyzed.

math.PR↗

arXiv2026

Some Spatial Point Processes of Poisson Family

Spatial Poisson point processes on finite-dimensional Euclidean space provide fundamental mathematical tools for modeling random spatial point patterns. In this paper, we introduce and analyze several Poisson-type spatial point processes. In particular, we propose and study a point process, namely, the generalized Poisson random field (GPRF), in which more than one point can be observed with positive probability, within a rectangular region having infinitesimal Lebesgue measure. A thinning of the GPRF into independent GPRFs with reduced rate parameters is discussed. Furthermore, we consider these processes indexed by the positive quadrant of the plane and analyze their fractional variants. Various distributional properties of these processes and related governing differential equations are obtained. Later, we define and analyze a spatial Skellam-type point process via GPRF. Moreover, a fractional variant of it in the two parameter case is studied in detail.

math.PR↗

arXiv2025

Skellam Processes via Multiparameter Poisson Process

We introduce a mltiparameter version of Skellam point process via multiparameter Poisson processes. Its distributional properties are studied in detail. Its compound representation is derived for a particular case. Also, its Riemann integral over a rectangle in $\mathbb{R}^M_+$, $M\ge1$ is introduced and a closed expression for its characteristic function is obtained. Later, we introduce a different version of multiparameter Skellam process, and derive a weak convergence result for it. Moreover, a two parameter fractional Skellam process is discussed.

math.PR↗

arXiv2025

On Two Parameter Time-Changed Poisson Random Fields with Drifts

We study the composition of bivariate Lévy process with bivariate inverse subordinator. The explicit expressions for its dispersion and auto correlation matrices are obtained. Also, the time-changed two parameter Lévy processes with rectangular increments are studied. We introduce some time-changed variants of the Poisson random field in plane with and without drift, and derive the associated fractional differential equations for their distributions. Later, we consider some time-changed Lévy processes where the time-changing components are two parameter Poisson random fields with drifts. Moreover, two parameter coordinatewise semigroup operators associated with some of the introduced processes are discussed.

math.PR↗

arXiv2025

On a Fractional Variant of Linear Birth-Death Process

We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.

math.PR↗