scholarly journals Universality for conditional measures of the Bessel point process

2020 ◽  
Vol 10 (01) ◽  
pp. 2150012
Author(s):  
Leslie D. Molag ◽  
Marco Stevens
Keyword(s):  
Orthogonal Polynomial ◽  
Point Process ◽  
Real Line ◽  
Probability Measures ◽  
Conditional Measure ◽  
Positive Real ◽  
Bounded Interval ◽  
Positive Real Line ◽  
Orthogonal Polynomial Ensemble

The Bessel point process is a rigid point process on the positive real line and its conditional measure on a bounded interval [Formula: see text] is almost surely an orthogonal polynomial ensemble. In this paper, we show that if [Formula: see text] tends to infinity, one almost surely recovers the Bessel point process. In fact, we show this convergence for a deterministic class of probability measures, to which the conditional measure of the Bessel point process almost surely belongs.

Linear estimation of self-similar processes via Lamperti's transformation

2000 ◽  
Vol 37 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Carl J. Nuzman ◽  
H. Vincent Poor
Keyword(s):  
Brownian Motion ◽  
Closed Form ◽  
Real Line ◽  
Continuous Time ◽  
Stationary Processes ◽  
Linear Estimation ◽  
Positive Real ◽  
Self Similar ◽  
Infinite Intervals ◽  
Positive Real Line

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

scholarly journals Some Distributions on the Positive Real Line Which Have No Moments

1971 ◽  
Vol 25 (1) ◽  
pp. 46-47
Author(s):  
Peter A. Lachenbruch ◽  
Donna R. Brogan
Keyword(s):  
Real Line ◽  
Positive Real ◽  
Positive Real Line

scholarly journals A quasitopos containing CONV and MET as full subcategories

1988 ◽  
Vol 11 (3) ◽  
pp. 417-438 ◽  
Author(s):  
E. Lowen ◽  
R. Lowen
Keyword(s):  
Real Line ◽  
Metric Spaces ◽  
Limit Function ◽  
Positive Real ◽  
Convergence Spaces ◽  
Continuous Maps ◽  
Definition Of ◽  
Approach Spaces ◽  
Positive Real Line

We show that convergence spaces with continuous maps and metric spaces with contractions, can be viewed as entities of the same kind. Both can be characterized by a “limit function”λwhich with each filterℱassociates a mapλℱfrom the underlying set to the extended positive real line. Continuous maps and contractions can both be characterized as limit function preserving maps.The properties common to both the convergence and metric case serve as a basis for the definition of the category, CAP. We show that CAP is a quasitopos and that, apart from the categories CONV, of convergence spaces, and MET, of metric spaces, it also contains the category AP of approach spaces as nicely embedded subcategories.

Linear estimation of self-similar processes via Lamperti's transformation

2000 ◽  
Vol 37 (02) ◽  
pp. 429-452 ◽  
Author(s):  
Carl J. Nuzman ◽  
H. Vincent Poor
Keyword(s):  
Brownian Motion ◽  
Closed Form ◽  
Real Line ◽  
Continuous Time ◽  
Stationary Processes ◽  
Linear Estimation ◽  
Positive Real ◽  
Self Similar ◽  
Infinite Intervals ◽  
Positive Real Line

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

scholarly journals Renewal in Hawkes processes with self-excitation and inhibition

2020 ◽  
Vol 52 (3) ◽  
pp. 879-915
Author(s):  
Manon Costa ◽  
Carl Graham ◽  
Laurence Marsalle ◽  
Viet Chi Tran
Keyword(s):  
Real Line ◽  
Limit Theorems ◽  
Original Problem ◽  
Concentration Inequalities ◽  
Hawkes Process ◽  
Positive Real ◽  
Hawkes Processes ◽  
Exponential Moments ◽  
Cluster Representation ◽  
Positive Real Line

AbstractWe investigate the Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of such a point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows one to apply known results for Galton–Watson trees. We use renewal techniques to establish limit theorems for Hawkes processes that have reproduction functions which are signed and have bounded support. Notably, we prove exponential concentration inequalities, extending results of Reynaud-Bouret and Roy (2006) previously proven for nonnegative reproduction functions using a cluster representation no longer valid in our case. Importantly, we establish the existence of exponential moments for renewal times of M/G/$\infty$ queues which appear naturally in our problem. These results possess interest independent of the original problem.

scholarly journals On a Functional Equation Associated with (a,k)-Regularized Resolvent Families

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Carlos Lizama ◽  
Felipe Poblete
Keyword(s):  
Functional Equation ◽  
Real Line ◽  
Functional Equations ◽  
Bounded Operator ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Positive Real ◽  
Resolvent Family ◽  
Resolvent Families ◽  
Positive Real Line

Leta∈Lloc1(ℝ+)andk∈C(ℝ+)be given. In this paper, we study the functional equationR(s)(a*R)(t)-(a*R)(s)R(t)=k(s)(a*R)(t)-k(t)(a*R)(s), for bounded operator valued functionsR(t)defined on the positive real lineℝ+. We show that, under some natural assumptions ona(·)andk(·), every solution of the above mentioned functional equation gives rise to a commutative(a,k)-resolvent familyR(t)generated byAx=lim t→0+(R(t)x-k(t)x/(a*k)(t))defined on the domainD(A):={x∈X:lim t→0+(R(t)x-k(t)x/(a*k)(t))exists inX}and, conversely, that each(a,k)-resolvent familyR(t)satisfy the above mentioned functional equation. In particular, our study produces new functional equations that characterize semigroups, cosine operator families, and a class of operator families in between them that, in turn, are in one to one correspondence with the well-posedness of abstract fractional Cauchy problems.

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