Series expansions for the all-time maximum of α-stable random walks

2016 ◽  
Vol 48 (3) ◽  
pp. 744-767
Author(s):  
Clifford Hurvich ◽  
Josh Reed
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Queueing Theory ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Series Expansions ◽  
Stable Random Variables ◽  
The Mean ◽  
The Stability

AbstractWe study random walks whose increments are α-stable distributions with shape parameter 1<α<2. Specifically, assuming a mean increment size which is negative, we provide series expansions in terms of the mean increment size for the probability that the all-time maximum of an α-stable random walk is equal to 0 and, in the totally skewed-to-the-left case of skewness parameter β=-1, for the expected value of the all-time maximum of an α-stable random walk. Our series expansions generalize previous results for Gaussian random walks. Key ingredients in our proofs are Spitzer's identity for random walks, the stability property of α-stable random variables, and Zolotarev's integral representation for the cumulative distribution function of an α-stable random variable. We also discuss an application of our results to a problem arising in queueing theory.

scholarly journals Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (4) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

Nondecreasing Lower Bound on the Poisson Cumulative Distribution Function for z Standard Deviations Above the Mean

2013 ◽  
Vol 50 (04) ◽  
pp. 909-917
Author(s):  
M. Bondareva
Keyword(s):  
Distribution Function ◽  
Lower Bound ◽  
Cumulative Distribution Function ◽  
Random Variable ◽  
Step Function ◽  
Cumulative Distribution ◽  
Control Parameters ◽  
Poisson Random Variable ◽  
The Mean ◽  
Standard Deviations

In this paper we discuss a nondecreasing lower bound for the Poisson cumulative distribution function (CDF) at z standard deviations above the mean λ, where z and λ are parameters. This is important because the normal distribution as an approximation for the Poisson CDF may overestimate or underestimate its value. A sharp nondecreasing lower bound in the form of a step function is constructed. As a corollary of the bound's properties, for a given percent α and parameter λ, the minimal z is obtained such that, for any Poisson random variable with the mean greater or equal to λ, its αth percentile is at most z standard deviations above its mean. For Poisson distributed control parameters, the corollary allows simple policies measuring performance in terms of standard deviations from a benchmark.

scholarly journals On the control of the difference between two Brownian motions: a dynamic copula approach

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Thomas Deschatre
Keyword(s):  
Cumulative Distribution Function ◽  
Survival Function ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Gaussian Copula ◽  
Brownian Motions ◽  
Dynamic Copula ◽  
Copula Approach ◽  
The Difference ◽  
The Right

AbstractWe propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for is the same for Markovian pairs and all pairs of Brownian motions, that is with φ being the cumulative distribution function of a standard Gaussian random variable.

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

scholarly journals SYSTEM RELIABILITY AND WEIGHTED LATTICE POLYNOMIALS

2008 ◽  
Vol 22 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Alexander Dukhovny ◽  
Jean-Luc Marichal
Keyword(s):  
System Reliability ◽  
Cumulative Distribution Function ◽  
Probability Generating Function ◽  
Joint Probability ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Special Cases ◽  
Indicator Variables ◽  
Lattice Polynomials ◽  
Lattice Polynomial

The lifetime of a system of connected units under some natural assumptions can be represented as a random variable Y defined as a weighted lattice polynomial of random lifetimes of its components. As such, the concept of a random variable Y defined by a weighted lattice polynomial of (lattice-valued) random variables is considered in general and in some special cases. The central object of interest is the cumulative distribution function of Y. In particular, numerous results are obtained for lattice polynomials and weighted lattice polynomials in the case of independent arguments and in general. For the general case, the technique consists in considering the joint probability generating function of “indicator” variables. A connection is studied between Y and order statistics of the set of arguments.

scholarly journals Statistical Properties of SNR for Compressed Measurements

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Yulong Gao ◽  
Yanping Chen ◽  
Linxiao Su
Keyword(s):  
Numerical Simulation ◽  
Signal Processing ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Statistical Properties ◽  
Cumulative Distribution ◽  
Statistical Characteristics ◽  
Communication Signal ◽  
The Mean ◽  
Simulation Results

Some basic statistical properties of the compressed measurements are investigated. It is well known that the statistical properties are a foundation for analyzing the performance of signal detection and the applications of compressed sensing in communication signal processing. Firstly, we discuss the statistical properties of the compressed signal, the compressed noise, and their corresponding energy. And then, the statistical characteristics of SNR of the compressed measurements are calculated, including the mean and the variance. Finally, probability density function and cumulative distribution function of SNR are derived for the cases of the Gamma distribution and the Gaussian distribution. Numerical simulation results demonstrate the correctness of the theoretical analysis.

scholarly journals Detection of Giant pulses from pulsar PSR B0950+08

2012 ◽  
Vol 8 (S291) ◽  
pp. 502-504
Author(s):  
T. V. Smirnova
Keyword(s):  
Distribution Function ◽  
Flux Density ◽  
Power Law ◽  
Pulse Energy ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Maximum Peak ◽  
Giant Pulses ◽  
Peak Flux ◽  
The Mean

AbstractWe investigated pulse intensities of PSR B0950+08 at 112 MHz at various longitudes (phases) and detected very strong pulses exceeding the amplitude of the mean profile by more than one hundred times. The maximum peak flux density of a recorded pulse is 15240 Jy, and the energy of this pulse exceeds the mean pulse energy by a factor of 153. The analysis shows that the cumulative distribution function (CDF) of pulse intensities at the longitudes of the main pulse is described by a piece-wise power law, with a slope changing from n=−1.25 ± 0.04 to n=−1.84 ± 0.07 at I≥600 Jy. The CDF for pulses at the longitudes of the precursor has a power law with n=−1.5 ± 0.1. Detected giant pulses from this pulsar have the same signature as giant pulses of other pulsars.

Investigations on a New Failure Model for Electromigration

1991 ◽  
Vol 225 ◽  
Author(s):  
J. Kitchin ◽  
J. R. Lloyd
Keyword(s):  
Activation Energy ◽  
Grain Boundary ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Failure Model ◽  
Line Segments ◽  
Standard Normal ◽  
The Mean ◽  
Image Position ◽  
Line P

A new statistical model for electromigration failure in fine-line thin film conductors is developed in [1] by extending [2] to incorporate the statistics of microstructure and concomitant variations in the activation energy for grain boundary diffusion. The resulting distribution of time t to failure is well-approximated bywhere N is the number of line segments that are potential independent failure elements, 1μ and σ are, respectively, the mean and standard deviation of an assumed normal distribution of activation energy in grain boundaries, γ is a scaling constant, T is in degrees Kelvin, j is the current density in the line, p1 is the fraction of line segments for which exactly one grain boundary occurs between two blocking boundaries, and Φ is the standard normal cumulative distribution function. (Note: .)

scholarly journals Classes of random walks on temporal networks with competing timescales

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Julien Petit ◽  
Renaud Lambiotte ◽  
Timoteo Carletti
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Stationary State ◽  
Mathematical Analysis ◽  
Random Walk Model ◽  
Temporal Networks ◽  
Underlying Network ◽  
The Mean ◽  
Rich Spectrum ◽  
General Stochastic

Abstract Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the co-existence of different timescales in the system. Here, we introduce random walks on general stochastic temporal networks allowing for lasting interactions, with up to three competing timescales. We then compare the mean resting time and stationary state of different models. We also discuss the accuracy of the mathematical analysis depending on the random walk model and the structure of the underlying network, and pay particular attention to the emergence of non-Markovian behaviour, even when all dynamical entities are governed by memoryless distributions.

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