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On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP

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  • Cremers, Heinz
  • Kadelka, Dieter

Abstract

The weak convergence of certain functionals of a sequence of stochastic processes is investigated. The functionals under consideration are of the form f[phi](x) = [integral operator] [phi] (t, x(t))[mu](dt). The main result is as follows: If a sequence is weakly tight in a certain sense, and, in addition, the finite dimensional distributions of the processes converge weakly, then this implies weak convergence of the functionals (f[phi]1([xi]n),..., f[phi]m([xi]n)) to (f[phi]1([xi]0),..., f[phi]m([xi]0)). Necessary and sufficient conditions for weak tightness are stated and applications of the results to the case of LEp-valued stochastic processes are given, ln particular it is shown that the usual tightness condition for weak convergence of such processes can be considerably weakened.

Suggested Citation

  • Cremers, Heinz & Kadelka, Dieter, 1986. "On weak convergence of integral functionals of stochastic processes with applications to processes taking paths in LEP," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 305-317, February.
    • Handle: RePEc:eee:spapps:v:21:y:1986:i:2:p:305-317
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    Cited by:

    1. Höpfner Reinhard & Kutoyants Yury A., 2009. "On LAN for parametrized continuous periodic signals in a time inhomogeneous diffusion," Statistics & Risk Modeling, De Gruyter, vol. 27(4), pages 309-326, December.
    2. Pilipauskaitė, Vytautė & Surgailis, Donatas, 2015. "Joint aggregation of random-coefficient AR(1) processes with common innovations," Statistics & Probability Letters, Elsevier, vol. 101(C), pages 73-82.
    3. repec:hal:journl:peer-00834425 is not listed on IDEAS
    4. Surgailis, Donatas & Teyssière, Gilles & Vaiciulis, Marijus, 2008. "The increment ratio statistic," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 510-541, March.
    5. Abadir, Karim M. & Distaso, Walter & Giraitis, Liudas, 2011. "An I(d) model with trend and cycles," Journal of Econometrics, Elsevier, vol. 163(2), pages 186-199, August.
    6. Alfredas Račkauskas & Charles Suquet, 2023. "Asymptotic Normality in Banach Spaces via Lindeberg Method," Journal of Theoretical Probability, Springer, vol. 36(1), pages 409-455, March.
    7. Höpfner, R. & Löcherbach, E., 1999. "On local asymptotic normality for birth and death on a flow," Stochastic Processes and their Applications, Elsevier, vol. 83(1), pages 61-77, September.
    8. Düker, Marie-Christine, 2018. "Limit theorems for Hilbert space-valued linear processes under long range dependence," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1439-1465.
    9. Jun, Sung Jae & Pinkse, Joris & Wan, Yuanyuan, 2015. "Classical Laplace estimation for n3-consistent estimators: Improved convergence rates and rate-adaptive inference," Journal of Econometrics, Elsevier, vol. 187(1), pages 201-216.
    10. James Cameron & Pramita Bagchi, 2022. "A test for heteroscedasticity in functional linear models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(2), pages 519-542, June.
    11. Karim M. Abadir & Walter Distaso & Liudas Giraitis, 2011. "An I() model with trend and cycles," Post-Print hal-00834425, HAL.
    12. Aue, Alexander & Van Delft, Anne, 2017. "Testing for stationarity of functional time series in the frequency domain," LIDAM Discussion Papers ISBA 2017001, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    13. Mathias Mørck Ljungdahl & Mark Podolskij, 2020. "A minimal contrast estimator for the linear fractional stable motion," Statistical Inference for Stochastic Processes, Springer, vol. 23(2), pages 381-413, July.
    14. Oliveira, P. E. & Suquet, Ch., 1998. "Weak convergence in Lp(0,1) of the uniform empirical process under dependence," Statistics & Probability Letters, Elsevier, vol. 39(4), pages 363-370, August.
    15. Characiejus, Vaidotas & Račkauskas, Alfredas, 2014. "Operator self-similar processes and functional central limit theorems," Stochastic Processes and their Applications, Elsevier, vol. 124(8), pages 2605-2627.

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