scholarly journals Conditions for equality between Lyapunov and Morse decompositions

2015 ◽  
Vol 36 (4) ◽  
pp. 1007-1036 ◽  
Author(s):  
LUCIANA A. ALVES ◽  
LUIZ A. B. SAN MARTIN
Keyword(s):  
Probability Measure ◽  
Ergodic Theorem ◽  
Principal Bundle ◽  
Sufficient Conditions ◽  
Base Space ◽  
Morse Decomposition ◽  
Lyapunov Spectra ◽  
Multiplicative Ergodic Theorem ◽  
Morse Decompositions ◽  
Necessary And Sufficient

Let$Q\rightarrow X$be a continuous principal bundle whose group$G$is reductive. A flow${\it\phi}$of automorphisms of$Q$endowed with an ergodic probability measure on the compact base space$X$induces two decompositions of the flag bundles associated to$Q$: a continuous one given by the finest Morse decomposition and a measurable one furnished by the multiplicative ergodic theorem. The second is contained in the first. In this paper we find necessary and sufficient conditions so that they coincide. The equality between the two decompositions implies continuity of the Lyapunov spectra under perturbations leaving unchanged the flow on the base space.

scholarly journals A constructive view on ergodic theorems

2006 ◽  
Vol 71 (2) ◽  
pp. 611-623
Author(s):  
Bas Spitters
Keyword(s):  
Ergodic Theorem ◽  
Sufficient Conditions ◽  
Ergodic Measure ◽  
Constructive Mathematics ◽  
Ergodic Theorems ◽  
Invariant Functions ◽  
Almost Everywhere ◽  
The Mean ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

AbstractLet T be a positive L1-L∞ contraction. We prove that the following statements are equivalent in constructive mathematics.(1) The projection in L2, on the space of invariant functions exists:(2) The sequence (Tn)n∈N Cesáro-converges in the L2 norm:(3) The sequence (Tn)n∈N Cesáro-converges almost everywhere.Thus, we find necessary and sufficient conditions for the Mean Ergodic Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.As a corollary we obtain a constructive ergodic theorem for ergodic measure-preserving transformations.This answers a question posed by Bishop.

scholarly journals On Statistics of Permutations Chosen From the Ewens Distribution

2014 ◽  
Vol 23 (6) ◽  
pp. 889-913
Author(s):  
TATJANA BAKSHAJEVA ◽  
EUGENIJUS MANSTAVIČIUS
Keyword(s):  
Probability Measure ◽  
Sufficient Conditions ◽  
Random Permutation ◽  
Asymptotic Distributions ◽  
Additive Functions ◽  
Factorial Moments ◽  
Permutation Matrices ◽  
Poisson Limit ◽  
Number Of Cycles ◽  
Necessary And Sufficient

We explore the asymptotic distributions of sequences of integer-valued additive functions defined on the symmetric group endowed with the Ewens probability measure as the order of the group increases. Applying the method of factorial moments, we establish necessary and sufficient conditions for the weak convergence of distributions to discrete laws. More attention is paid to the Poisson limit distribution. The particular case of the number-of-cycles function is analysed in more detail. The results can be applied to statistics defined on random permutation matrices.

scholarly journals Necessary and sufficient conditions for a maximal ergodic theorem along subsequences

1987 ◽  
Vol 7 (2) ◽  
pp. 203-210 ◽  
Author(s):  
Roger L. Jones
Keyword(s):  
Ergodic Theorem ◽  
Maximal Function ◽  
Probability Space ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Ergodic Measure ◽  
Point Transformation ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractLet T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.

Regular variation of the tail of a subordinated probability distribution

1973 ◽  
Vol 5 (02) ◽  
pp. 308-327 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Probability Distribution ◽  
Probability Measure ◽  
Real Line ◽  
Regular Variation ◽  
Sufficient Conditions ◽  
The Real ◽  
Negative Order ◽  
Necessary And Sufficient

Let F be a probability measure on the real line and G = Σ C(k)Fk ∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

Regular variation of the tail of a subordinated probability distribution

1973 ◽  
Vol 5 (2) ◽  
pp. 308-327 ◽  
Author(s):  
A. J. Stam
Keyword(s):  
Probability Distribution ◽  
Probability Measure ◽  
Real Line ◽  
Regular Variation ◽  
Sufficient Conditions ◽  
The Real ◽  
Negative Order ◽  
Necessary And Sufficient

Let F be a probability measure on the real line and G = Σ C(k)Fk∗ the probability measure subordinate to F with subordinator C restricted to the nonnegative integers. Let V(x) vary regularly of order p for x→ ∞ and either (1) V(x) F[x, ∞)→ α ≧ 0 or (2) V(x) C[x, ∞)→ γ ≧ 0. If ρ > 1 and F(–∞, 0) = 0, necessary and sufficient in order that V(x) G[x, ∞)→b, is that both (1) and (2) hold for suitable α and γ. For 0 ≦ ρ ≦ 1 the conditions are of different type. For two-sided F a different situation arises and only sufficient conditions are found. An application to renewal moments of negative order is given.

scholarly journals A note on the equation λ * ρ * μ = ρ

1999 ◽  
Vol 59 (3) ◽  
pp. 421-426
Author(s):  
C. Robinson Edward Raja
Keyword(s):  
Topological Group ◽  
Probability Measure ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Probability Measures ◽  
Necessary And Sufficient

Let G be a Hausdorff topological group and μ and λ be probability measures on G. We prove necessary and sufficient conditions for the existence of a probability measure ρ such that λ * ρ * μ = ρ under certain conditions. We prove a similar result for probability measures on semigroups.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Budgeting in Socio-economic Development Strategies (Raising a Problem)

2008 ◽  
pp. 134-151
Author(s):  
A. Shastitko ◽  
M. Ovchinnikov
Keyword(s):  
Economic Policy ◽  
Economic Development ◽  
Social Change ◽  
Eastern Europe ◽  
Statistical Data ◽  
Sufficient Conditions ◽  
Transitional Period ◽  
Socio Economic Development ◽  
Necessary And Sufficient ◽  
Economic Development Strategies

The article proposes an approach to the analysis of social change and contributes to the clarification of concepts of economic policy. It deals in particular with the notion of "change of system". The author considers positive and normative aspects of the analysis of capitalist and socialist systems. The necessary and sufficient conditions for the system to be changed are introduced, their fulfillment is discussed drawing upon the historical and statistical data. The article describes both economic and political peculiarities of the transitional period in different countries, especially in Eastern Europe.

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