scholarly journals On the dependence of the component counting process of a uniform random variable

2021 ◽  
Vol 26 (none) ◽  
Author(s):  
Joseph Squillace
Keyword(s):  
Counting Process ◽  
Random Variable ◽  
Uniform Random Variable

scholarly journals REGULARIZATION OF NON-NORMAL MATRICES BY GAUSSIAN NOISE—THE BANDED TOEPLITZ AND TWISTED TOEPLITZ CASES

2019 ◽  
Vol 7 ◽  
Author(s):  
ANIRBAN BASAK ◽  
ELLIOT PAQUETTE ◽  
OFER ZEITOUNI
Keyword(s):  
Toeplitz Matrices ◽  
Random Variable ◽  
Singular Values ◽  
Equivalence Theorem ◽  
Empirical Measure ◽  
Uniform Random Variable ◽  
Normal Matrices ◽  
Nilpotent Matrix ◽  
The Matrix ◽  
Triangular Toeplitz Matrix

We consider the spectrum of additive, polynomially vanishing random perturbations of deterministic matrices, as follows. Let$M_{N}$be a deterministic$N\times N$matrix, and let$G_{N}$be a complex Ginibre matrix. We consider the matrix${\mathcal{M}}_{N}=M_{N}+N^{-\unicode[STIX]{x1D6FE}}G_{N}$, where$\unicode[STIX]{x1D6FE}>1/2$. With$L_{N}$the empirical measure of eigenvalues of${\mathcal{M}}_{N}$, we provide a general deterministic equivalence theorem that ties$L_{N}$to the singular values of$z-M_{N}$, with$z\in \mathbb{C}$. We then compute the limit of$L_{N}$when$M_{N}$is an upper-triangular Toeplitz matrix of finite symbol: if$M_{N}=\sum _{i=0}^{\mathfrak{d}}a_{i}J^{i}$where$\mathfrak{d}$is fixed,$a_{i}\in \mathbb{C}$are deterministic scalars and$J$is the nilpotent matrix$J(i,j)=\mathbf{1}_{j=i+1}$, then$L_{N}$converges, as$N\rightarrow \infty$, to the law of$\sum _{i=0}^{\mathfrak{d}}a_{i}U^{i}$where$U$is a uniform random variable on the unit circle in the complex plane. We also consider the case of slowly varying diagonals (twisted Toeplitz matrices), and, when$\mathfrak{d}=1$, also of independent and identically distributed entries on the diagonals in$M_{N}$.

scholarly journals A Randomly Weighted Minimum Arborescence with a Random Cost Constraint

2021 ◽  
Author(s):  
Alan M. Frieze ◽  
Tomasz Tkocz
Keyword(s):  
Dual Problem ◽  
Random Variable ◽  
Uniform Random Variable ◽  
Asymptotic Value ◽  
Optimum Weight ◽  
Range Of Values

We study the minimum spanning arborescence problem on the complete digraph [Formula: see text], where an edge e has a weight We and a cost Ce, each of which is an independent uniform random variable Us, where [Formula: see text] and U is uniform [Formula: see text]. There is also a constraint that the spanning arborescence T must satisfy [Formula: see text]. We establish, for a range of values for [Formula: see text], the asymptotic value of the optimum weight via the consideration of a dual problem.

scholarly journals Formalization of the Standard Uniform random variable

2007 ◽  
Vol 382 (1) ◽  
pp. 71-83 ◽  
Author(s):  
Osman Hasan ◽  
Sofiène Tahar
Keyword(s):  
Random Variable ◽  
Uniform Random Variable

scholarly journals On Some Compound Random Variables Motivated by Bulk Queues

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Time Of Arrival ◽  
Uniform Random Variable ◽  
Wide Range ◽  
Bulk Queue ◽  
Probability Mass ◽  
Number Of Customers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (04) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.

scholarly journals On increasing-failure-rate random variables

2005 ◽  
Vol 42 (03) ◽  
pp. 797-809 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar ◽  
Zegang Zhu
Keyword(s):  
Markov Chain ◽  
Failure Rate ◽  
Renewal Process ◽  
Counting Process ◽  
Random Number ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Increasing Failure Rate ◽  
Random Threshold

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

scholarly journals On increasing-failure-rate random variables

2005 ◽  
Vol 42 (3) ◽  
pp. 797-809 ◽  
Author(s):  
Sheldon M. Ross ◽  
J. George Shanthikumar ◽  
Zegang Zhu
Keyword(s):  
Markov Chain ◽  
Failure Rate ◽  
Renewal Process ◽  
Counting Process ◽  
Random Number ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Random Variable ◽  
Increasing Failure Rate ◽  
Random Threshold

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

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