scholarly journals Recurrence properties of Processes with stationary independent increments

1964 ◽  
Vol 4 (2) ◽  
pp. 223-228 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Random Walk ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Increments ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

Let X1, X2,…Xn, … be independent and identically distributed random variables, and write . In [2] Chung and Fuchs have established necessary and sufficient conditions for the random walk {Zn} to be recurrent, i.e. for Zn to return infinitely often to every neighbourhood of the origin. The object of this paper is to obtain similar results for the corresponding process in continuous time.

scholarly journals On the law of the iterated logarithm in the infinite variance case

1980 ◽  
Vol 30 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. A. Maller
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Infinite Variance ◽  
Iterated Logarithm ◽  
The Law ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

AbstractThe main purpose of the paper is to give necessary and sufficient conditions for the almost sure boundedness of (Sn – αn)/B(n), where Sn = X1 + X2 + … + XmXi being independent and identically distributed random variables, and αnand B(n) being centering and norming constants. The conditions take the form of the convergence or divergence of a series of a geometric subsequence of the sequence P(Sn − αn > a B(n)), where a is a constant. The theorem is distinguished from previous similar results by the comparative weakness of the subsidiary conditions and the simplicity of the calculations. As an application, a law of the iterated logarithm general enough to include a result of Feller is derived.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (02) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of N n , the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

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.

Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes

1982 ◽  
Vol 19 (01) ◽  
pp. 29-38 ◽  
Author(s):  
H.-J. Schuh
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Probabilistic Proof ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.

Sums of i.i.d. random variables and an application to the explosion criterion for markov branching processes

1982 ◽  
Vol 19 (1) ◽  
pp. 29-38 ◽  
Author(s):  
H.-J. Schuh
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Probabilistic Proof ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for in terms of , where Sn is the sum of n i.i.d. random variables with values in]0, ∞[, and A ≧ 0. We use these results to give a probabilistic proof of the ‘explosion criterion' for continuous-time Markov branching processes, which is usually shown analytically.

scholarly journals Asymptotic normality for the number of records from general distributions

2011 ◽  
Vol 43 (2) ◽  
pp. 422-436 ◽  
Author(s):  
Raul Gouet ◽  
F. Javier López ◽  
Gerardo Sanz
Keyword(s):  
Asymptotic Normality ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
General Distribution ◽  
Continuous Components ◽  
General Distributions ◽  
Necessary And Sufficient ◽  
Independent And Identically Distributed

We provide necessary and sufficient conditions for the asymptotic normality of Nn, the number of records among the first n observations from a sequence of independent and identically distributed random variables, with general distribution F. In the case of normality we identify the centering and scaling sequences. Also, we characterize distributions for which the limit is not normal in terms of their discrete and continuous components.

On the Lp convergence of sums of independent random variables

1977 ◽  
Vol 82 (3) ◽  
pp. 439-446 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Independent Random Variables ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet {Xnj, 1 ≤ j ≤ kn} be independent random variables with zero means and satisfying . Let p ≥ 1. We prove thatif and only if, for all ε > 0,and we use this result to obtain necessary and sufficient conditions for the Lp convergence of sums of non-negative, independent random variables.

The λ-classification of continuous-time birth-and-death processes

2003 ◽  
Vol 35 (04) ◽  
pp. 1111-1130 ◽  
Author(s):  
Andrew G. Hart ◽  
Servet Martínez ◽  
Jaime San Martín
Keyword(s):  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Birth And Death Processes ◽  
Positive Recurrent ◽  
Necessary And Sufficient

We study the λ-classification of absorbing birth-and-death processes, giving necessary and sufficient conditions for such processes to be λ-transient, λ-null recurrent and λ-positive recurrent.

Positive fractional continuous-time linear systems with singular pencils

2012 ◽  
Vol 60 (1) ◽  
pp. 9-12 ◽  
Author(s):  
T. Kaczorek
Keyword(s):  
Linear Systems ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Descriptor Systems ◽  
State Variables ◽  
Algebraic Constraints ◽  
Necessary And Sufficient ◽  
Time Linear

Positive fractional continuous-time linear systems with singular pencils A method for checking the positivity and finding the solution to the positive fractional descriptor continuous-time linear systems with singular pencils is proposed. The method is based on elementary row and column operations of the fractional descriptor systems to equivalent standard systems with some algebraic constraints on state variables and inputs. Necessary and sufficient conditions for the positivity of the fractional descriptor systems are established.

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