Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Almost sure stability of maxima

1973 ◽  
Vol 10 (02) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n≧ 1} unbounded above setMn= max {X1,X2, …,Xn}. When do normalizing constantsbnexist such thatMn/bn→1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for alland in this casebn∼F–1(1 – 1/n) Necessary and sufficient conditions for lim supn→∞,Mn/bn= l >1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn= 1 a.s. except whenl= 1. When theXnare r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (02) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima M n of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants B n such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions π i=1 m H i (x). The effect of each factor H i (x) on the behavior of maxima from π i=1 m H i is analyzed. Under a mild regularity condition, B n can be chosen to be the maximum of the m quantiles of order (1 - n -1) of the H's.

A note on roots of Markov shifts

1976 ◽  
Vol 13 (03) ◽  
pp. 591-596
Author(s):  
S. M. Rudolfer
Keyword(s):  
Markov Chain ◽  
Shift Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Markov Shifts ◽  
Necessary And Sufficient

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.

A note on roots of Markov shifts

1976 ◽  
Vol 13 (3) ◽  
pp. 591-596
Author(s):  
S. M. Rudolfer
Keyword(s):  
Markov Chain ◽  
Shift Operator ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Markov Shifts ◽  
Necessary And Sufficient

Let Tv be the two-sided shift operator associated with a finite Markov chain of period v; Using results of Krengel and Michel and Adler, Shields and Smorodinsky, necessary and sufficient conditions for the existence of an rth root of Tv are obtained. In particular, if the Markov chain is irreducible, then Tv has an rth root when and only when (r, v) = 1.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

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 Multiresolution analysis and wavelets on Vilenkin groups

2008 ◽  
Vol 21 (3) ◽  
pp. 309-325 ◽  
Author(s):  
Yury Farkov
Keyword(s):  
Multiresolution Analysis ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Partition Of Unity ◽  
Necessary And Sufficient Conditions ◽  
Refinable Functions ◽  
Vilenkin Groups ◽  
Orthogonal Wavelets ◽  
The Stability ◽  
Necessary And Sufficient

This paper gives a review of multiresolution analysis and compactly sup- ported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of 'integer shifts' of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

Sign in / Sign up

Export Citation Format

Share Document