scholarly journals WEAK CONVERGENCE OF COMPOUND PROBABILITY MEASURES ON UNIFORM SPACES

1999 ◽  
Vol 30 (4) ◽  
pp. 271-288
Author(s):  
JUN KAWABE
Keyword(s):  
Weak Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Transition Probabilities ◽  
Uniform Space ◽  
Probability Measures ◽  
Uniform Spaces ◽  
Convergence Theorems ◽  
Covariance Functions ◽  
Compound Probability

We obtain a convergence theorem of compound probability measures on a uniform space for a net of uniformly equicontinuous transition probabilities. This theorem contains convergence theorems of product or convolution measures. We also show that for Gaussian transition probabilities on a Hilbert spaces, our assumptions in the convergence theorem can be expressed in terms of mean and covariance functions.

scholarly journals UNIFORM TIGHTNESS FOR TRANSITION PROBABILITIES

1995 ◽  
Vol 26 (4) ◽  
pp. 283-298
Author(s):  
JUN KAWABE
Keyword(s):  
Weak Convergence ◽  
Transition Probabilities ◽  
Upper Semicontinuity ◽  
Fréchet Space ◽  
Topological Spaces ◽  
Probability Measures ◽  
Convergence Of Probability Measures ◽  
Set Valued Mappings ◽  
Strong Dual ◽  
Compound Probability

The aim of this paper is to give a notion of uniform tightness for transition probabilities on topological spaces, which assures the uniform tightness of compound probability measures. Then the upper semicontinuity of set-valued mappings are used in essence. As an important example, the uniform tightness for Gaussian transition probabilities on the strong dual of a nuclear real Frechet space is studied. It is also shown that some of our results contain well-known results concerning the uniform tightness and the weak convergence of probability measures.

Strong and weak convergence theorems for fixed points of a class of nonlinear mappings

2014 ◽  
Vol 64 (2) ◽  
Author(s):  
Changqun Wu ◽  
Zhiqiang Wei ◽  
Yu Li
Keyword(s):  
Weak Convergence ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Projection Methods ◽  
Convergence Theorems ◽  
Strong And Weak Convergence ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Total Asymptotically Nonexpansive Mappings ◽  
Weak Convergence Theorem

AbstractIn this paper, the class of total asymptotically nonexpansive mappings is considered. A weak convergence theorem of Mann-type iterative algorithm is established. Hybrid projection methods are considered for the class of total asymptotically nonexpansive mappings. Strong convergence theorems are also established in the framework of Hilbert spaces.

Weak and strong convergence theorems and a nonlinear ergodic theorem for N-generalized hybrid mappings

2017 ◽  
Vol 10 (01) ◽  
pp. 1750001
Author(s):  
Sattar Alizadeh ◽  
Fridoun Moradlou
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Ergodic Theorem ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Strong Convergence Theorem ◽  
Weak And Strong Convergence ◽  
Weak Convergence Theorem ◽  
Nonlinear Ergodic Theorem ◽  
Nonlinear Mappings

In this paper, assuming an appropriate condition, we prove that [Formula: see text]-generalized hybrid mappings are demiclosed in Hilbert spaces. Using this fact, we prove a weak convergence theorem of Ishikawa type for these nonlinear mappings. Also, a strong convergence theorem of Halpern–Ishikawa type and a nonlinear ergodic theorem for [Formula: see text]-generalized hybrid mappings have been proven in Hilbert spaces.

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

2021 ◽  
pp. 152-160
Author(s):  
Baoshuai Zhang ◽  
◽  
Ying Tian ◽  
Keyword(s):  
Variational Inequality ◽  
Weak Convergence ◽  
Hilbert Spaces ◽  
Iterative Algorithms ◽  
Common Element ◽  
Convergence Theorems ◽  
Strong And Weak Convergence ◽  
Nonexpansive Semigroups ◽  
General Variational Inequality ◽  
Mixed Equilibrium

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

scholarly journals Crossovers induced by discrete-time quantum walks

2011 ◽  
Vol 11 (9&10) ◽  
pp. 741-760
Author(s):  
Kota Chisaki ◽  
Norio Konno ◽  
Etsuo Segawa ◽  
Yutaka Shikano
Keyword(s):  
Weak Convergence ◽  
Random Walks ◽  
Discrete Time ◽  
Convergence Theorem ◽  
Continuous Time ◽  
Quantum Walk ◽  
Convergence Theorems ◽  
Final Time ◽  
Time Quantum ◽  
Weak Convergence Theorem

We consider crossovers with respect to the weak convergence theorems from a discrete-time quantum walk (DTQW). We show that a continuous-time quantum walk (CTQW) and discrete- and continuous-time random walks can be expressed as DTQWs in some limits. At first we generalize our previous study [Phys. Rev. A \textbf{81}, 062129 (2010)] on the DTQW with position measurements. We show that the position measurements per each step with probability $p \sim 1/n^\beta$ can be evaluated, where $n$ is the final time and $0<\beta<1$. We also give a corresponding continuous-time case. As a consequence, crossovers from the diffusive spreading (random walk) to the ballistic spreading (quantum walk) can be seen as the parameter $\beta$ shifts from 0 to 1 in both discrete- and continuous-time cases of the weak convergence theorems. Secondly, we introduce a new class of the DTQW, in which the absolute value of the diagonal parts of the quantum coin is proportional to a power of the inverse of the final time $n$. This is called a final-time-dependent DTQW (FTD-DTQW). The CTQW is obtained in a limit of the FTD-DTQW. We also obtain the weak convergence theorem for the FTD-DTQW which shows a variety of spreading properties. Finally, we consider the FTD-DTQW with periodic position measurements. This weak convergence theorem gives a phase diagram which maps sufficiently long-time behaviors of the discrete- and continuous-time quantum and random walks.

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