Strong Approximation for Mixing Sequences with Infinite Variance

Raluca Balan, University of Ottawa, Canada Ingrid-Mona Zamfirescu, City University of New York, USA

Abstract
In this paper we prove a strong approximation result for
a mixing sequence with infinite variance and logarithmic decay
rate of the mixing coefficient. The result is proved under the
assumption that the distribution is symmetric and lies in the
domain of attraction of the normal law. Moreover the truncated variance function
is supposed to be slowly varying
with log-log type remainder.

R.M. Balan and R. Kulik.
Self-normalized weak invariance principle for mixing
sequences. Tech. Rep. Series, Lab. Research Stat. Probab.,
Carleton University-University of Ottawa 417 (2005).
Math. Review number not available.

N.H. Bingham, C.M. Goldie and J.L. Teugels. Regular Variation. (1987) Cambridge
University Press, Cambridge.
Math. Review 91a:26003

R.C. Bradley. Basic properties of
strong mixing conditions. A survey and some open questions.
Probability Surveys2 (2005), 107-144.
Math. Review MR2178042

M. Csörgö, B. Szyszkowicz and Q. Wang. Donsker's theorem for
self-normalized partial sum processes. Ann. Probab.31 (2003),
1228-1240.
Math. Review 2004h:60031

W. Feller. An extension of the law of
the iterated logarithm to variables without variance. J.
Math. Mech.18 1968, 343-355.
Math. Review 0233399

W. Feller. An introduction to probability theory and its applications. Vol. II. Second Edition.
(1971), John Wiley, New York.
Math. Review 0270403

P. Hall and C.C.Heyde.
Martingale limit theory and its applications. (1980) Academic Press, New
York.
Math. Review 83a:60001

H. Kesten. Sums of independent random
variables - without moment conditions. Ann. Math. Stat.43 (1972), 701-732.
Math. Review MR0301786

H. Kesten and R.A. Maller. Some effects of trimming on the law of the iterated logarithm.
J. Appl. Probab.41A (2004), 253-271.
Math. Review 2005a:60038

M. Loève. Probability theory. Vol. II. Fourth Edition. (1978)
Springer, New York.
Math. Review MR0651018

J. Mijnheer. A strong approximation
of partial sums of i.i.d. random variables with infinite
variance. Z. Wahrsch. verw. Gebiete52 (1980), 1-7.
Math. Review 81e:60032

S. Y. Novak. On self-normalized sums
and Student's statistic. Theory Probab. Appl. 49 (2005),
336-344.
Math. Review 2005m:60038

W. Philipp and W.F. Stout. Almost sure invariance principles for partial sums of weakly
dependent random variables. Mem. AMS. Vol. 2161 (1975).
Math. Review MR0433597

Q.-M. Shao. Almost sure invariance principles for mixing sequences of random variables. Stoch.
Proc. Appl.48 (1993), 319-334.
Math. Review 95c:60030

Q.-M. Shao. An invariance principle for stationary rho-mixing sequences with infinite variance.
Chin. Ann. Math. Ser. B.14 (1993), 27-42.
Math. Review 94f:60051