Showing papers by "Palaniappan Vellaisamy published in 2019"

Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse

Abstract: In this paper, we study the fractional Poisson process (FPP) time-changed by an independent Levy subordinator and the inverse of the Levy subordinator, which we call TCFPP-I and TCFPP-II, respectively. Various distributional properties of these processes are established. We show that, under certain conditions, the TCFPP-I has the long-range dependence property, and also its law of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal process and its waiting time distribution is identified. The bivariate distributions of the TCFPP-II are derived. Some specific examples for both the processes are discussed. Finally, we present simulations of the sample paths of these processes.

On the convolution of Mittag–Leffler distributions and its applications to fractional point processes

Abstract: We obtain the distribution of the sum of independent Mittag–Leffler (ML) random variables which are not necessarily identically distributed. Firstly we discuss the corresponding known result for in...

Non-homogeneous space-time fractional Poisson processes

Abstract: The space-time fractional Poisson process (STFPP), defined by Orsingher and Poilto (2012), is a generalization of the time fractional Poisson process (TFPP) and the space fractional Poisson...

On Distributions of Certain State-Dependent Fractional Point Processes

Abstract: We obtain the explicit expressions for the state probabilities of various state-dependent versions of fractional point processes. The inversion of the Laplace transforms of the state probabilities of such processes is rather cumbersome and involved. We employ the Adomian decomposition method to solve the difference-differential equations governing the state probabilities of these state-dependent processes. The distributions of some convolutions of the Mittag-Leffler random variables are derived as special cases of the obtained results.

On large deviations for sums of discrete m-dependent random variables

Abstract: The ratio P(Sn=x)/P(Zn=x) is investigated for three cases: (a) when Sn is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and Zn ...

Analysis of intersections of trajectories of systems of linear fractional differential equations.

Abstract: This article deals with trajectorial intersections in systems of linear fractional differential equations. We propose a classification of intersections of trajectories into three classes: (a) trajectories intersecting at the same time (IST), (b) trajectories intersecting at different times (IDT), and (c) self-intersections of a trajectory. We prove a generalization of the separation theorem for the case of linear fractional systems. This result proves the existence of the IST. Based on the presence of the IST, systems are further classified into two types, Type I and Type II systems, which are analyzed further for the IDT. Self-intersections in a fractional trajectory can be regular such as constant solution or limit-cycle behavior, or they can be irregular such as cusps or nodes. We give necessary and sufficient conditions for a trajectory to be regular.

On large deviations for sums of discrete m-dependent random variables

Abstract: The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.

Probabilistic Proofs of Some Beta-Function Identities.

Abstract: Using a probabilistic approach, we derive some interesting identities involving beta functions. These results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.

Exponential order statistics and some combinatorial identities

Abstract: It is known that the k-th (1≤k≤n) order statistic from the unit exponential distribution can be represented as a sum of independent exponential random variables. We present a proof of this ...

Adomian Decomposition Method and Fractional Poisson Processes: A Survey

Abstract: This paper gives a survey of recent results related to the applications of the Adomian decomposition method (ADM) to certain fractional generalizations of the homogeneous Poisson process. First, we briefly discuss the ADM and its advantages over existing methods. As applications, this method is employed to obtain the state probabilities of the time fractional Poisson process (TFPP), space fractional Poisson process (SFPP) and Saigo space–time fractional Poisson process (SSTFPP). Usually, the Laplace transform technique is used to obtain the state probabilities of fractional processes. However, for certain state-dependent fractional Poisson processes, the Laplace transform method is difficult to apply, but the ADM method could be effectively used to obtain the state probabilities of such processes.

Evaluation of missing data mechanisms in two and three dimensional incomplete tables

Abstract: The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies.