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1

Ergodic and fixed point theorems for sequences and nonlinear mappings in a Hilbert space

Rouhani B. D.

Demonstratio Mathematica

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2018

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Vol. 51, nr 1

27--36

EN

In this paper, we introduce the notion of 2-generalized hybrid sequences, extending the notion of nonexpansive and hybrid sequences introduced and studied in our previous work [Djafari Rouhani B., Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. thesis, Yale University, 1981; and other published in J. Math. Anal. Appl., 1990, 2002, and 2014; Nonlinear Anal., 1997, 2002, and 2004], and prove ergodic and convergence theorems for such sequences in a Hilbert space H. Subsequently, we apply our results to prove new fixed point theorems for 2-generalized hybrid mappings, first introduced in [Maruyama T., Takahashi W., Yao M., Fixed point and mean ergodic theorems for new nonlinear mappings in Hilbert spaces, J. Nonlinear Convex Anal., 2011, 12, 185-197] and further studied in [Lin L.-J., Takahashi W., Attractive point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 2012, 16, 1763-1779], defined on arbitrary nonempty subsets of H.

2

On positive definite and stationary sequences with respect to polynomial hypergroups

Lasser R.

Journal of Applied Analysis

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2011

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Vol. 17, nr 2

207-230

EN

We study bounded positive definite double sequences which are stationary with respect to a polynomial hypergroup structure generated by (Rn(t)) ∈nNo. Connected with bounded positive definite and Rn-stationary double sequences is an Rn-stationary sequence of elements in a Hilbert space. We derive an ergodic theorem for such Rn-stationary sequences and we give a complete characterization of the space of multipliers defined by such an Rn-stationary sequence. Further we give examples of bounded positive definite double sequences.

3

Data-driven score test of fit for conditional distribution in the GARCH (1, 1) model

Inglot T., Stawiarski B.

Probability and Mathematical Statistics

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2005

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Vol. 25, Fasc. 2

331--362

EN

A data-driven score test for a conditional distribution in the GARCH (1, 1) model is proposed. Conditional distribution assumption is verified by a score test, obtained from nesting the null density into an exponential family and then choosing the dimension of this exponential family by a score-based selection rule. A simulation study, which is provided, shows good empirical behaviour of the proposed test, outperforming in most cases the behaviour of competitive tests.

4

Almost sure limit theorems for semi-selfsimilar processes

Fazekas I., Rychlik Z.

Probability and Mathematical Statistics

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2005

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Vol. 25, Fasc. 2

241--255

EN

An integral analogue of the almost sure limit theorem is presented for semi-selfsimilar processes. In the theorem, instead of a sequence of random elements, a continuous time random process is involved; moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the theorem is applied to obtain almost sure limit theorems for semistable processes. Discrete versions of the above theorems are proved. In particular, the almost sure functional limit theorem is obtained for semistable random variables.