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.

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.

scholarly journals Mean Value theorems for multiplicative functions bounded in mean α-power, α >1

1980 ◽  
Vol 29 (2) ◽  
pp. 177-205 ◽  
Author(s):  
P. D. T. A. Elliott
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Mean Value ◽  
Necessary And Sufficient Conditions ◽  
Multiplicative Functions ◽  
The Real ◽  
Mean Value Theorems ◽  
Necessary And Sufficient

AbstractOn analogy with functions if Lebesuge class Lα on the real line the author considers those multiplicative arthmetic functions which are bounded in mean α>1. Necessary and sufficient conditions are obtained in order that they should have a mean-value, zero or non-zero. An application is made to Ramanujan's τ-function.

scholarly journals The ordering on permutations induced by continuous maps of the real line

1987 ◽  
Vol 7 (2) ◽  
pp. 155-160 ◽  
Author(s):  
Chris Bernhardt
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Ordering ◽  
The Real ◽  
Continuous Maps ◽  
Necessary And Sufficient ◽  
Natural Way

AbstractContinuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

scholarly journals Exponential dichotomy for evolution families on the real line

2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Adina Luminiţa Sasu
Keyword(s):  
Real Line ◽  
Exponential Dichotomy ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Evolution Family ◽  
The Real ◽  
Evolution Families ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for uniform exponential dichotomy of evolution families in terms of the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X)). We show that the admissibility of the pair(Lp(ℝ,X),Lq(ℝ,X))is equivalent to the uniform exponential dichotomy of an evolution family if and only ifp≥q. As applications we obtain characterizations for uniform exponential dichotomy of semigroups.

Characterizations of symmetric cones by means of the basic relative invariants of homogeneous cones

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Hideto Nakashima
Keyword(s):  
Sufficient Conditions ◽  
Rational Functions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Symmetric Cones ◽  
The Real ◽  
Tube Domains ◽  
Homogeneous Cone ◽  
Necessary And Sufficient ◽  
Relative Invariants

AbstractIn this paper, we give necessary and sufficient conditions for a homogeneous cone Ω to be symmetric in two ways. One is by using the multiplier matrix of Ω, and the other is in terms of the basic relative invariants of Ω. In the latter approach, we need to show that the real parts of certain meromorphic rational functions obtained by the basic relative invariants are always positive on the tube domains over Ω. This is a generalization of a result of Ishi and Nomura [Math. Z. 259 (2008), 604–674].

Multifractal formalism for the inverse of random weak Gibbs measures

2019 ◽  
Vol 20 (04) ◽  
pp. 2050024
Author(s):  
Zhihui Yuan
Keyword(s):  
Probability Measure ◽  
Lebesgue Measure ◽  
Real Line ◽  
Gibbs Measures ◽  
Borel Probability Measure ◽  
Cantor Set ◽  
Multifractal Formalism ◽  
The Real ◽  
Random Dynamics ◽  
Borel Probability

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

Sufficient conditions for long-range count dependence of stationary point processes on the real line

2001 ◽  
Vol 38 (2) ◽  
pp. 570-581 ◽  
Author(s):  
Rafał Kulik ◽  
Ryszard Szekli
Keyword(s):  
Long Range ◽  
Real Line ◽  
Stationary Point ◽  
Point Processes ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Second Moment ◽  
The Real ◽  
Palm Measure ◽  
Stationary Point Processes

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

scholarly journals Representation of functions as the Post-Widder inversion operator of generalized functions

1984 ◽  
Vol 7 (2) ◽  
pp. 371-396 ◽  
Author(s):  
R. P. Manandhar ◽  
L. Debnath
Keyword(s):  
Representation Theory ◽  
Function Space ◽  
Fundamental Theorem ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Representation Theorems ◽  
The Real ◽  
Inversion Operator ◽  
Inversion Theory ◽  
Necessary And Sufficient

A study is made of the Post-Widder inversion operator to a class of generalized functions in the sense of distributional convergence. Necessary and sufficient conditions are proved for a given function to have the representation as therth operate of the Post-Widder inversion operator of generalized functions. Some representation theorems are also proved. Certain results concerning the testing function space and its dual are established. A fundamental theorem regarding the existence of the real inversion operator (1.6) withr=0is proved in section4. A classical inversion theory for the Post-Widder inversion operator with a few other theorems which are fundamental to the representation theory is also developed in this paper.

Uniform conditional variability ordering of probability distributions

1985 ◽  
Vol 22 (03) ◽  
pp. 619-633 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Probability Distribution ◽  
Real Line ◽  
Convex Functions ◽  
Probability Distributions ◽  
Stochastic Order ◽  
Queueing Network ◽  
Network Models ◽  
Scalar Multiplication ◽  
Closed Queueing Network ◽  
The Real

Variability orderings indicate that one probability distribution is more spread out or dispersed than another. Here variability orderings are considered that are preserved under conditioning on a common subset. One density f on the real line is said to be less than or equal to another, g, in uniform conditional variability order (UCVO) if the ratio f(x)/g(x) is unimodal with the model yielding a supremum, but f and g are not stochastically ordered. Since the unimodality is preserved under scalar multiplication, the associated conditional densities are ordered either by UCVO or by ordinary stochastic order. If f and g have equal means, then UCVO implies the standard variability ordering determined by the expectation of all convex functions. The UCVO property often can be easily checked by seeing if f(x)/g(x) is log-concave. This is illustrated in a comparison of open and closed queueing network models.

Probability measures with trivial Stam groups

1982 ◽  
Vol 91 (3) ◽  
pp. 477-484
Author(s):  
Gavin Brown ◽  
William Mohan
Keyword(s):  
Probability Measure ◽  
Real Number ◽  
Real Line ◽  
Probability Measures ◽  
The Real ◽  
Interesting Observation ◽  
Image Position ◽  
Positive Integers

Let μ be a probability measure on the real line ℝ, x a real number and δ(x) the probability atom concentrated at x. Stam made the interesting observation that eitheror else(ii) δ(x)* μn, are mutually singular for all positive integers n.

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