University of Calcutta (informally known as Calcutta University or CU) is a collegiate public state university located in Kolkata, West Bengal, India.&nbsp;Within India it is recognized as a &quot;Five-Star University&quot; and accredited &quot;A&quot; Grade by National Assessment and Accreditation Council.

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The University of Calcutta has secured the Fifth position among the Universities of India in the prestigious &quot;Indian University Ranking 2019&quot; list, released by the NIRF of the Ministry of Human Resource Development, Government of India.

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It was established on 24 January 1857, and was one of the first institutions in Asia to be established as a multidisciplinary and Western-style university. Its alumni and faculty include several heads of state, heads of government, social reformers, prominent artists, only academy award winner and Dirac medal winner in India, many Fellows of the Royal Society and five Nobel laureates as of 2019 which is highest in South Asia.

University Of Calcutta

Asymptotic tests for identical distribution of responses in two independent sets of Likert-type scale data using latent variable models are developed. The proposed tests are compared with regard to asymptotic relative efficiency with the commonly used two-sample t-test and Mann–Whitney U test. Robustness and unbiasedness of the tests are also established. © 2021 Taylor & Francis Group, LLC.

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Statistics

Communications in Statistics - Theory and Methods

Large-sample tests for comparing Likert-type scale data

INTERNATIONAL STATISTICAL REVIEW

Introduction to Probability

Theorem 2.2 of Majumdar (Sankhyā 67:670-673, 2005) obtained two conditions that are necessary and jointly sufficient for weak convergence in M(H). In this note we significantly relax the second condition while maintaining the joint sufficiency. © 2013, Indian Statistical Institute.

Sankhya: The Indian Journal of Statistics

On weak convergence in M(H): Relaxation of a sufficient condition

Theorem 2.2 of Majumdar (Sankhyā 67:670–673, 2005) obtained two conditions that are necessary and jointly sufficient for weak convergence in M(H). In this note we significantly relax the second condition while maintaining the joint sufficiency. © 2013, Indian Statistical Institute.

Sankhya A

On weak convergence in M(H): relaxation of a sufficient condition

We consider stochastic processes (Mt)t ≥ 0 for which the class V of time-space harmonic functions is rich enough to yield the Markov property for the process. In particular, we prove that denseness for all t ≥ 0 of Vt {colon equals} {f (t, {dot operator})} {divides} f ∈ V} in Lp (μt) for any p ≥ 1, where μt denotes the law of Mt, is sufficient to guarantee the Markov property. We use this to improve upon a result of [Sengupta, Arindam, 2000. Time-space harmonic polynomials martingales for continuous-time processes and an extension. Journal of Theoretical Probability 13 (4), 951-976] concerning p-harmonizability, describe two new methods for constructing time-space harmonic polynomials and apply them to get some interesting examples. © 2008 Elsevier B.V. All rights reserved.

Statistics and Probability Letters

Markov processes, time-space harmonic functions and polynomials

We study the properties of sums of lower records from a distribution on [0, ∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0, ∞) originate as infinite sums of lower records.

Journal of Applied Probability

Convergence of lower records and infinite divisibility

A time-space harmonic polynomial for a continuous-time process X=(Xt:t≥0) is a two-variable polynomial P such that (P(t,Xt):t≥0) is a martingale for the natural filtration of X. Motivated by Lévy’s characterisation of Brownian motion and Watanabe’s characterisation of the Poisson process, we look for classes of processes with reasonably general path properties in which a characterisation of those members whose laws are determined by a finite number of such polynomials is available. We exhibit two classes of processes, the first containing the Lévy processes, and the second a more general class of additive processes, with this property and describe the respective characterisations. © 2001 Applied Probability Trust.

Fulltext

Electronic Journal of Probability

Finitely polynomially determined lévy processes

A time-space harmonic polynomial for a stochastic process M = (Mt) is a polynomial P in two variables such that P(t, Mt) is a martingale. In this paper, we investigate conditions for the existence of such polynomials of each degree in the second, "space," argument. We also describe various properties a sequence of time-space harmonic polynomials may possess and the interaction of these properties with distributional properties of the underlying process. Thus, continuous-time conterparts to the results of Goswami and Sengupta,(2) where the analoguous problem in discrete time was considered, are derived. A few additional properties are also considered. The resulting properties of the process include independent increments, stationary independent increments and semi-stability. Finally, a generalization to a "measure" proposed by Hochberg(3) on path space is obtained.

Journal of Theoretical Probability

Time-Space Harmonic Polynomials for Continuous-Time Processes and an Extension

Extending an idea of Spitzer [2], a way to compute the moments of the time of escape from (-N,L) by a symmetric simple random walk is exhibited. It is shown that all these moments depend polynomially on L and N.

Proceedings of the Indian Academy of Sciences: Mathematical Sciences

Moments of escape times of random walk

We investigate, for a given martingale M={Mn: n≥0}, the conditions for the existence of polynomials P(·,·) of two variables, "time" and "space," and of arbitrary degree in the latter, such that {P(n, Mn)} is a martingale for the natural filtration of M. Denoting by ℘ the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span ℘. A complete necessary and sufficient condition is obtained in the case when M has independent increments. For general M, we obtain a necessary condition which entails, under mild additional hypotheses, that M is necessarily Markovian. Considering a slightly more general class of polynomials than ℘ we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set ℘ determines the law of the martingale in a certain sense. © 1995 Plenum Publishing Corporation.