Weak Convergence and One-Sample Rank Statistics Under ϕ-mixing*

1975 ◽  
Vol 18 (4) ◽  
pp. 555-565
Author(s):  
K. L. Mehra
Keyword(s):  
Weak Convergence ◽  
Lebesgue Measure ◽  
Probability Space ◽  
Stationary Process ◽  
Mixing Condition ◽  
Absolutely Continuous ◽  
Rank Statistics ◽  
Finite Dimensional ◽  
Strictly Stationary Process

Let {Xi:i=1, 2,…} be a real strictly stationary process (defined on a probability space (Ω, A, P)) which has absolutely continuous finite dimensional distributions (with respect to Lebesgue measure) and satisfies the ϕ-mixing condition: Let and denote the sub-cr-fields generated, respectively, by {Xi:i≤k} and {Xi:i≥k+n}; then, for each k≥1 and n≥l, and together imply.

scholarly journals Shannon entropy for stationary processes and dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 447-480 ◽  
Author(s):  
D. HAMDAN ◽  
W. PARRY ◽  
J.-P. THOUVENOT
Keyword(s):  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Shannon Entropy ◽  
Gaussian Processes ◽  
Stationary Processes ◽  
Entropy Theory ◽  
Absolutely Continuous ◽  
Continuous Processes ◽  
Ergodic Processes ◽  
Finite Dimensional

AbstractWe consider stationary ergodic processes indexed by $\mathbb Z$ or $\mathbb Z^n$ whose finite-dimensional marginals have laws which are absolutely continuous with respect to Lebesgue measure. We define an entropy theory for these continuous processes, prove an analogue of the Shannon–MacMillan–Breiman theorem and study more precisely the particular example of Gaussian processes.

scholarly journals Condition for intersection occupation measure to be absolutely continuous

2020 ◽  
Vol 72 (9) ◽  
pp. 1304-1312
Author(s):  
X. Chen
Keyword(s):  
Stochastic Processes ◽  
Lebesgue Measure ◽  
Local Time ◽  
Minimal Condition ◽  
Occupation Measure ◽  
Absolutely Continuous ◽  
Intersection Local Time ◽  
Stationary Increments

UDC 519.21 Given the i.i.d. -valued stochastic processes with the stationary increments, a minimal condition is provided for the occupation measure to be absolutely continuous with respect to the Lebesgue measure on An isometry identity related to the resulting density (known as intersection local time) is also established.

Alternating projections and interpolation of stationary processes

1992 ◽  
Vol 29 (4) ◽  
pp. 921-931 ◽  
Author(s):  
Mohsen Pourahmadi
Keyword(s):  
Missing Values ◽  
Stationary Process ◽  
Dimensional Space ◽  
Stationary Processes ◽  
Finite Dimensional Space ◽  
Alternating Projections ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Explicit Formulae ◽  
Linear Interpolator

By using the alternating projection theorem of J. von Neumann, we obtain explicit formulae for the best linear interpolator and interpolation error of missing values of a stationary process. These are expressed in terms of multistep predictors and autoregressive parameters of the process. The key idea is to approximate the future by a finite-dimensional space.

The stochastic equation Yt+1 = AtYt + Bt with non-stationary coefficients

2001 ◽  
Vol 38 (1) ◽  
pp. 80-94 ◽  
Author(s):  
Ulrich Horst
Keyword(s):  
Global Convergence ◽  
Random Environment ◽  
Linear Relation ◽  
Stochastic Equation ◽  
Stationary Process ◽  
Sufficient Conditions ◽  
Convergence Result ◽  
Stationary Case ◽  
Finite Dimensional ◽  
Long Run

In this paper, we consider the stochastic sequence {Yt}t∊ℕ defined recursively by the linear relation Yt+1 = AtYt + Bt in a random environment which is described by the non-stationary process {(At, Bt)}t∊ℕ. We formulate sufficient conditions on the environment which ensure that the finite-dimensional distributions of {Yt}t∊ℕ converge weakly to the finite-dimensional distributions of a unique stationary process. If the driving sequence {(At, Bt)}t∊ℕ becomes stationary in the long run, then we can establish a global convergence result. This extends results of Brandt (1986) and Borovkov (1998) from the stationary to the non-stationary case.

A Family of Generalized Riesz Products

1996 ◽  
Vol 48 (2) ◽  
pp. 302-315 ◽  
Author(s):  
A. H. Dooley ◽  
S. J. Eigen
Keyword(s):  
Lebesgue Measure ◽  
Spectral Measure ◽  
Probability Space ◽  
Rank One ◽  
Almost All ◽  
Riesz Products

AbstractGeneralized Riesz products similar to the type which arise as the spectral measure for a rank-one transformation are studied. A condition for the mutual singularity of two such measures is given. As an application, a probability space of transformations is presented in which almost all transformations are singular with respect to Lebesgue measure.

scholarly journals Littlewood-Paley theory on Gaussian spaces

1988 ◽  
Vol 109 ◽  
pp. 47-61 ◽  
Author(s):  
Jürgen Potthoff
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Euclidean Spaces ◽  
General Measure ◽  
Finite Dimensional

In this article we prove a number of inequalities of Littlewood-Paley-Stein (LPS) type for functions on general Gaussian spaces (s. below).In finite dimensional Euclidean spaces (with Lebesgue measure) the power of such inequalities has been demonstrated in Stein’s book [12]. In his second book [13], Stein treats other spaces too: also the situation of a general measure space (X, μ). However the latter case is too general to allow for a rich class of inequalities (cf. Theorem 10 in [13]).

scholarly journals S-Subgradient Projection Methods with S-Subdifferential Functions for Nonconvex Split Feasibility Problems

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1517
Author(s):  
Jinzuo Chen ◽  
Mihai Postolache ◽  
Yonghong Yao
Keyword(s):  
Weak Convergence ◽  
Projection Method ◽  
Split Feasibility Problem ◽  
Projection Methods ◽  
Feasibility Problem ◽  
Split Feasibility Problems ◽  
Finite Dimensional ◽  
Subgradient Projection ◽  
Weak Convergence Theorem ◽  
Split Feasibility

In this paper, the original C Q algorithm, the relaxed C Q algorithm, the gradient projection method ( G P M ) algorithm, and the subgradient projection method ( S P M ) algorithm for the convex split feasibility problem are reviewed, and a renewed S P M algorithm with S-subdifferential functions to solve nonconvex split feasibility problems in finite dimensional spaces is suggested. The weak convergence theorem is established.

Tensor Bogolyubov representations of the renormalized square of white noise (RSWN) algebra

2019 ◽  
Vol 22 (04) ◽  
pp. 1950025
Author(s):  
Habib Rebei ◽  
Luigi Accardi ◽  
Hajer Taouil
Keyword(s):  
White Noise ◽  
Lebesgue Measure ◽  
Absolutely Continuous ◽  
First Order ◽  
Commutation Relations ◽  
Hyperbolic Sine ◽  
Arbitrary Complex ◽  
Borel Functions ◽  
Complex Valued

We introduce the quadratic analog of the tensor Bogolyubov representation of the CCR. Our main result is the determination of the structure of these maps: each of them is uniquely determined by two arbitrary complex-valued Borel functions of modulus [Formula: see text] and two maps of [Formula: see text] into itself whose inverses induce transformations that map the Lebesgue measure [Formula: see text] into measures [Formula: see text] absolutely continuous with respect to it. Furthermore, the Radon–Nikodyn derivatives [Formula: see text], of these measures with respect to [Formula: see text], must satisfy the relation [Formula: see text] for [Formula: see text]-almost every [Formula: see text]. This makes a surprising bridge with the hyperbolic sine and cosine defining the structure of usual (i.e. first-order) Bogolyubov transformations. The reason of the surprise is that the linear and quadratic commutation relations are completely different.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

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