The variance of a truncated mixed exponential process

1994 ◽  
Vol 31 (1) ◽  
pp. 167-179 ◽  
Author(s):  
Jaimie L. Hebert ◽  
John W. Seaman
Keyword(s):  
Explicit Expression ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Exponential Process ◽  
Original Variable ◽  
Mixed Exponential Process ◽  
Mixing Density

Mullooly (1988) provides sufficient conditions under which the variance of a left-truncated, non-negative random variable will be greater than the variance of the original variable. We consider this problem for the class of exponential mixtures, and provide an explicit expression for the inflation in variance in terms of the mixing density.

The variance of a truncated mixed exponential process

1994 ◽  
Vol 31 (01) ◽  
pp. 167-179
Author(s):  
Jaimie L. Hebert ◽  
John W. Seaman
Keyword(s):  
Explicit Expression ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Exponential Process ◽  
Original Variable ◽  
Mixed Exponential Process ◽  
Mixing Density

Mullooly (1988) provides sufficient conditions under which the variance of a left-truncated, non-negative random variable will be greater than the variance of the original variable. We consider this problem for the class of exponential mixtures, and provide an explicit expression for the inflation in variance in terms of the mixing density.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

scholarly journals Limit of 𝑝-Laplacian obstacle problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raffaela Capitanelli ◽  
Maria Agostina Vivaldi
Keyword(s):  
Asymptotic Behavior ◽  
Uniform Convergence ◽  
Explicit Expression ◽  
Positive Measure ◽  
Sufficient Conditions ◽  
Obstacle Problems ◽  
Dimensional Case ◽  
Asymptotic Behavior Of Solutions ◽  
One Dimensional ◽  
The One

AbstractIn this paper, we study asymptotic behavior of solutions to obstacle problems for p-Laplacians as {p\to\infty}. For the one-dimensional case and for the radial case, we give an explicit expression of the limit. In the n-dimensional case, we provide sufficient conditions to assure the uniform convergence of the whole family of the solutions of obstacle problems either for data f that change sign in Ω or for data f (that do not change sign in Ω) possibly vanishing in a set of positive measure.

Markov Chain Monte Carlo simulation of the distribution of some perpetuities

1999 ◽  
Vol 31 (01) ◽  
pp. 112-134 ◽  
Author(s):  
Jostein Paulsen ◽  
Arne Hove
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Lévy Process ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Levy Process ◽  
Present Value ◽  
Strong Markov

We study the present value Z ∞ = ∫0 ∞ e-X t- dY t where (X,Y) is an integrable Lévy process. This random variable appears in various applications, and several examples are known where the distribution of Z ∞ is calculated explicitly. Here sufficient conditions for Z ∞ to exist are given, and the possibility of finding the distribution of Z ∞ by Markov chain Monte Carlo simulation is investigated in detail. Then the same ideas are applied to the present value Z - ∞ = ∫0 ∞ exp{-∫0 t R s ds}dY t where Y is an integrable Lévy process and R is an ergodic strong Markov process. Numerical examples are given in both cases to show the efficiency of the Monte Carlo methods.

scholarly journals Resilient H-infinity filtering for networked nonlinear Markovian jump systems with randomly occurring distributed delay and sensor saturation

2021 ◽  
Vol 26 (2) ◽  
pp. 187-206
Author(s):  
Venkatesan Nithya ◽  
Rathinasamy Sakthivel ◽  
Yong Ren
Keyword(s):  
Sufficient Conditions ◽  
Distributed Delay ◽  
Random Variable ◽  
Markovian Jump Systems ◽  
Linear Matrix ◽  
Markovian Jump ◽  
Missing Measurements ◽  
Sensor Saturation ◽  
Jump Systems ◽  
Nonlinear Markovian Jump Systems

The H∞ filtering problem for a class of networked nonlinear Markovian jump systems subject to randomly occurring distributed delays, nonlinearities, quantization effects, missing measurements and sensor saturation is investigated in this paper. The measurement missing phenomenon is characterized via a random variable obeying the Bernoulli stochastic distribution. Moreover, due to bandwidth limitations, the measurement output is quantized using a logarithmic quantizer and then transmitted to the filter. Further, the output measurements are affected by sensor saturation since the communication links between the system and the filter are unreliable and is described by sector nonlinearities. The objective of this work is to design a quantized resilient filter that guarantees not only the stochastic stability of the augmented filtering error system but also a prespecified level of H∞ performance. Sufficient conditions for the existence of desired filter are established with the aid of proper Lyapunov–Krasovskii functional and linear matrix inequality approach together with stochastic analysis theory. Finally, a numerical example is presented to validate the developed theoretical results.

scholarly journals Sufficient conditions for the uniqueness of the maxima of the optimization problem in the framework of a stochastic model with priorities depending on one random variable

2020 ◽  
Vol 224 ◽  
pp. 01014
Author(s):  
N Neumerzhitskaia ◽  
S Uglich ◽  
T Volosatova
Keyword(s):  
Complex System ◽  
Stochastic Model ◽  
Objective Function ◽  
Optimization Problem ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Random Variable ◽  
Stationary Points

An objective function arising in the optimization problem of a quasilinear complex system with dependent priorities is considered. In the case of three priorities based on the results of one experiment, sufficient conditions are obtained for all stationary points of the objective function under consideration to be local maximum points.

scholarly journals When Does a Dual Matrix Have a Dual Generalized Inverse?

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1386
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Explicit Expressions

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

scholarly journals Distributed Robust Filtering for Wireless Sensor Networks with Markov Switching Topologies and Deception Attacks

Sensors ◽  
2020 ◽  
Vol 20 (7) ◽  
pp. 1948
Author(s):  
Fengzeng Zhu ◽  
Xu Liu ◽  
Jiwei Wen ◽  
Linbo Xie ◽  
Li Peng
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Sufficient Conditions ◽  
Random Variable ◽  
Wireless Sensor ◽  
Switching Topology ◽  
Homogeneous Markov Chain ◽  
External Interference ◽  
Reduced Order ◽  
Switching Topologies

This paper is concerned with the distributed full- and reduced-order l 2 - l ∞ state estimation issue for a class of discrete time-invariant systems subjected to both randomly occurring switching topologies and deception attacks over wireless sensor networks. Firstly, a switching topology model is proposed which uses homogeneous Markov chain to reflect the change of filtering networks communication modes. Then, the sector-bound deception attacks among the communication channels are taken into consideration, which could better characterize the filtering network communication security. Additionally, a random variable obeying the Bernoulli distribution is used to describe the phenomenon of the randomly occurring deception attacks. Furthermore, through an adjustable parameter E, we can obtain full- and reduced-order l 2 - l ∞ state estimator over sensor networks, respectively. Sufficient conditions are established for the solvability of the addressed switching topology-dependent distributed filtering design in terms of certain convex optimization problem. The purpose of solving the problem is to design a distributed full- and reduced-order filter such that, in the presence of deception attacks, stochastic external interference and switching topologies, the resulting filtering dynamic system is exponentially mean-square stable with prescribed l 2 - l ∞ performance index. Finally, a simulation example is provided to show the effectiveness and flexibility of the designed approach.

A theorem on the cumulative product of independent random variables

1959 ◽  
Vol 55 (4) ◽  
pp. 333-337 ◽  
Author(s):  
Harold Ruben
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Product Theorem ◽  
Image Position ◽  
Introductory Discussion ◽  
Necessary And Sufficient ◽  
Complex Valued

1. Introductory discussion and summary. Consider a sequence {ui} of independent real or complex-valued random variables such that E(ui) = 1, and a sequence of mutually exclusive events S1, S2,…, such that Si depends only on u1, u2, …,ui, with ΣP(Sj) = 1. Define the random variable n = n(u1, u2,…) = m when Sm occurs. We shall obtain the necessary and sufficient conditions under whichreferred to as the product theorem.

scholarly journals LEBESGUE STRUCTURE OF DISTRIBUTION OF GENERALIZED BERNOULLI CONVOLUTIONS OF SPECIAL TYPE

2020 ◽  
pp. 15-18
Author(s):  
O. Makarchuk ◽  
K. Salnik
Keyword(s):  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Special Kind ◽  
Random Variable ◽  
Random Series ◽  
Bernoulli Convolution ◽  
Bernoulli Convolutions ◽  
Lévy’S Theorem ◽  
Distribution Matrix ◽  
Necessary And Sufficient

The paper deals with the problem of deepening the Jessen-Wintner theorem for generalized Bernoulli convolutions of a special kind. The main attention is paid to the case when the terms of a random series acquire three values: 0, 1, 2. In the case when the probability that the term of a random series becomes 2 is 0, the corresponding generalized Bernoulli convolutions coincide with classic Bernoulli convolutions, which were actively studied domestic scientists (Pratsovyty M., Turbin G., Torbin G., Honcharenko Ya., Baranovsky O., Savchenko I. and others) as well as foreign researchers (Erdos P., Peres Y., Schlag W, Solomyak B., Albeverio S. and others). The problem of deepening the Jessen-Wintner theorem concerning the necessary and sufficient conditions for the distribution of a probably convergent random series with discrete additions to each of the three pure types, is extremely difficult to formulate and is not completely solved even for classical Bernoulli convolutions. The results of the study are a deepening in relation to the analysis of the Lebesgue structure of random series formed by s-expansions of real numbers. In the case when the corresponding Bernoulli convolution is generated by the sequence 3-n, we have a random variable with independent triple digits, which was studied by scientists in different directions: Lebesgue structure (Chaterji S., Marsaglia G.), topological-metric structure of the distribution spectrum (Pratsovityi M., Turbin G.), fractal analysis of the distribution carrier (Pratsovyty M., Torbin G.), asymptotic properties of the characteristic function at infinity (Honcharenko Ya., Pratsovyty M., Torbin G.). The paper presents certain sufficient conditions for the absolute continuity and singularity of the distribution, with certain restrictions on the stochastic distribution matrix and the asymptotics of the values of the random terms of the series. In the case when the Lebesgue measure of the set of realizations of the generalized Bernoulli convolution is different from zero, it is possible together with Levy's theorem to formulate criteria for belonging of the Bernoulli convolution distribution to each of the three pure Lebesgue types, namely: purely discrete, purely continuous or purely singular.

Sign in / Sign up

Export Citation Format

Share Document