An almost sure invariance principle for hilbert space valued martingales

Transactions of the American Mathematical Society

Volumn 273, Issue 1, 1982, Pages 231-251

  Morrow, Gregory ^a   Philipp, Walter ^b  

^a WASHINGTON UNIVERSITY   (United States)

^b UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

Author keywords

Central limit theorem; Hilbert space; Invariance principles; Law of the iterated logarithm; Martingales

Indexed keywords

EID: 84968508604     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-1982-0664040-3     Document Type: Article

References (16)

1. Istvân Berkes and Walter Philipp, Approximation theorems for independent and weakly dependent random vectors, Ann. Probability 7 (1979), 29-54.
2. B. M. Brown, Martingale central limit theorems, Ann. Math. Statist. 42 (1971), 59-66.
3. D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probability 1 (1973), 19-42.
4. Herold Dehling and Walter Philipp, Almost sure invariance principles for weakly dependent vector-valued random variables, preprint, 1980.
5. X. Femique, Integrabilitë des vectours Gaussiens, C. R. Acad. Sei. Paris 270 (1970), 1698-1699.
6. Peter Gänssler and Winfried Stute, Wahrscheinlichkeitstheorie, Springer-Verlag, Berlin and New York, 1977.
7. V. Goodman, J. Kuelbs and J. Zinn, Some results on the LIL in Banach space with applications to the weighted empirical process y Ann. Probability 9 (1981), 713-752.
8. Naresh C. Jain, Kumar Jogdeo and William F. Stout, Upper and lower functions for martingales and mixing processes, Ann. Probability 3 (1975), 119-145.
9. N. C. Jain and S. J. Taylor, Local asymptotics for Brownian motion, Ann. Probability 1 (1973), 527-549.
10. J. Kuelbs and R. LePage, The law of the iterated logarithm for Brownian motion in a Banach space, Trans. Amer. Math. Soc. 185 (1973), 253-264.
11. J. Kuelbs and Walter Philipp, Almost sure invariance principles for partial sums of mixing B-valued random variables, Ann. Probability 8 (1980), 1003-1036.
12. S. Orey and W. E. Pruitt, Sample functions of the N-parameter Wiener process, Ann. Probability 1 (1973), 138-163.
13. Walter Philipp and William F. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. No. 161 (1975).
14. William F. Stout, Almost sure convergence, Academic Press, New York, 1974.
15. V. Strassen, An almost sure invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 3 (1964), 211-226.
16. V. Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Stat. Prob., Vol. 2, 1965, pp. 315-342.