scholarly journals Conditionally identically distributed species sampling sequences

2010 ◽  
Vol 42 (02) ◽  
pp. 433-459 ◽  
Author(s):  
Federico Bassetti ◽  
Irene Crimaldi ◽  
Fabrizio Leisen
Keyword(s):  
Random Variables ◽  
Random Sequences ◽  
Generalized Poisson ◽  
Sampling Sequence ◽  
New Type ◽  
Conditional Identity ◽  
Sampling Sequences

In this paper the theory of species sampling sequences is linked to the theory of conditionally identically distributed sequences in order to enlarge the set of species sampling sequences which are mathematically tractable. The conditional identity in distribution (see Berti, Pratelli and Rigo (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper a class of random sequences, called generalized species sampling sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, two types of generalized species sampling sequence that are conditionally identically distributed are introduced and studied: the generalized Poisson-Dirichlet sequence and the generalized Ottawa sequence. Some examples are discussed.

scholarly journals Conditionally identically distributed species sampling sequences

2010 ◽  
Vol 42 (2) ◽  
pp. 433-459 ◽  
Author(s):  
Federico Bassetti ◽  
Irene Crimaldi ◽  
Fabrizio Leisen
Keyword(s):  
Random Variables ◽  
Random Sequences ◽  
Generalized Poisson ◽  
Sampling Sequence ◽  
New Type ◽  
Conditional Identity ◽  
Sampling Sequences

In this paper the theory of species sampling sequences is linked to the theory of conditionally identically distributed sequences in order to enlarge the set of species sampling sequences which are mathematically tractable. The conditional identity in distribution (see Berti, Pratelli and Rigo (2004)) is a new type of dependence for random variables, which generalizes the well-known notion of exchangeability. In this paper a class of random sequences, called generalized species sampling sequences, is defined and a condition to have conditional identity in distribution is given. Moreover, two types of generalized species sampling sequence that are conditionally identically distributed are introduced and studied: the generalized Poisson-Dirichlet sequence and the generalized Ottawa sequence. Some examples are discussed.

scholarly journals Fully degenerate Bell polynomials associated with degenerate Poisson random variables

2021 ◽  
Vol 19 (1) ◽  
pp. 284-296
Author(s):  
Hye Kyung Kim
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Combinatorial Identities ◽  
Bell Polynomials ◽  
Poisson Random Variable ◽  
Central Moments ◽  
Special Polynomials ◽  
Special Numbers And Polynomials ◽  
New Type ◽  
Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

scholarly journals New-type Hoeffding’s inequalities and application in tail bounds

2021 ◽  
Vol 5 (1) ◽  
pp. 248-261
Author(s):  
Pingyi Fan ◽  
Keyword(s):  
Information Processing ◽  
Random Variables ◽  
Order Moment ◽  
First Order ◽  
Hoeffding’S Inequalities ◽  
New Type ◽  
Hoeffding’S Inequality ◽  
Hoeffding Inequality ◽  
Moments Of Random Variables ◽  
Refined Version

It is well known that Hoeffding's inequality has a lot of applications in the signal and information processing fields. How to improve Hoeffding's inequality and find the refinements of its applications have always attracted much attentions. An improvement of Hoeffding inequality was recently given by Hertz [<a href="#1">1</a>]. Eventhough such an improvement is not so big, it still can be used to update many known results with original Hoeffding's inequality, especially for Hoeffding-Azuma inequality for martingales. However, the results in original Hoeffding's inequality and its refined version by Hertz only considered the first order moment of random variables. In this paper, we present a new type of Hoeffding's inequalities, where the high order moments of random variables are taken into account. It can get some considerable improvements in the tail bounds evaluation compared with the known results. It is expected that the developed new type Hoeffding's inequalities could get more interesting applications in some related fields that use Hoeffding's results.

Limsup theorems and a uniform LIL for asymptotically negatively associated random sequences

2012 ◽  
Vol 49 (2) ◽  
pp. 223-235
Author(s):  
Yong-Kab Choi ◽  
Kyo-Shin Hwang
Keyword(s):  
Random Variables ◽  
Partial Sums ◽  
Random Sequences ◽  
Negatively Associated ◽  
Iterated Logarithm ◽  
Uniform Law

In this paper we establish some limsup theorems and a generalized uniform law of the iterated logarithm (LIL) for the increments of the partial sums of a strictly stationary and asymptotically negatively associated (ANA) sequence of random variables.

Goodness-of-fit tests for random sequences incorporating several components

2017 ◽  
Vol 25 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Tetiana O. Ianevych ◽  
Yuriy V. Kozachenko ◽  
Viktor B. Troshki
Keyword(s):  
Covariance Function ◽  
Goodness Of Fit ◽  
Random Sequence ◽  
Random Variables ◽  
Gaussian Random Variables ◽  
Random Sequences ◽  
Goodness Of Fit Tests

AbstractIn this paper we have constructed the goodness-of-fit tests incorporating several components, like expectation and covariance function for identification of a non-centered univariate random sequence or auto-covariances and cross-covariances for identification of a centered multivariate random sequence. For the construction of the corresponding estimators and investigation of their properties we utilized the theory of square Gaussian random variables.

scholarly journals Nonparametric density estimators based on nonstationary absolutely regular random sequences

1996 ◽  
Vol 9 (3) ◽  
pp. 233-254 ◽  
Author(s):  
Michel Harel ◽  
Madan L. Puri
Keyword(s):  
Central Limit ◽  
Markov Processes ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Density Estimator ◽  
Random Sequences ◽  
Arma Processes ◽  
Density Estimators

In this paper, the central limit theorems for the density estimator and for the integrated square error are proved for the case when the underlying sequence of random variables is nonstationary. Applications to Markov processes and ARMA processes are provided.

scholarly journals THE QUANTUM DECOMPOSITION OF RANDOM VARIABLES WITHOUT MOMENTS

2013 ◽  
Vol 16 (02) ◽  
pp. 1350012 ◽  
Author(s):  
LUIGI ACCARDI ◽  
HABIB REBEI ◽  
ANIS RIAHI
Keyword(s):  
Hilbert Spaces ◽  
Random Variables ◽  
Random Variable ◽  
Quantum Probability ◽  
Fock Spaces ◽  
Infinitely Divisible ◽  
New Type ◽  
First Moment ◽  
Finite Moments

The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.

scholarly journals Extremal clustering in non-stationary random sequences

Extremes ◽  
2021 ◽  
Author(s):  
Graeme Auld ◽  
Ioannis Papastathopoulos
Keyword(s):  
Asymptotic Distribution ◽  
Long Range ◽  
Extreme Values ◽  
Random Variables ◽  
Limiting Distribution ◽  
Long Range Dependence ◽  
Dependence Structure ◽  
Stationary Sequences ◽  
Random Sequences ◽  
Extremal Clustering

AbstractIt is well known that the distribution of extreme values of strictly stationary sequences differ from those of independent and identically distributed sequences in that extremal clustering may occur. Here we consider non-stationary but identically distributed sequences of random variables subject to suitable long range dependence restrictions. We find that the limiting distribution of appropriately normalized sample maxima depends on a parameter that measures the average extremal clustering of the sequence. Based on this new representation we derive the asymptotic distribution for the time between consecutive extreme observations and construct moment and likelihood based estimators for measures of extremal clustering. We specialize our results to random sequences with periodic dependence structure.

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