The Annals of Probability

Law of the iterated logarithm for stationary processes

Ou Zhao and Michael Woodroofe

Full-text: Open access


There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes …, X−1, X0, X1, … whose partial sums Sn=X1+⋯+Xn are of the form Sn=Mn+Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(Rn2)=o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting ‖⋅‖ denote the norm in L2(P), a sufficient condition for the partial sums of a stationary process to have the form Sn=Mn+Rn is that n−3/2E(Sn|X0, X−1, …)‖ be summable. A sufficient condition for the LIL is only slightly stronger, requiring n−3/2log3/2(n)‖E(Sn|X0, X−1, …)‖ to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.

Article information

Ann. Probab. Volume 36, Number 1 (2008), 127-142.

First available: 28 November 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

Conditional central limit question ergodic theorem Fourier series martingales Markov chains operators on L^2


Zhao, Ou; Woodroofe, Michael. Law of the iterated logarithm for stationary processes. The Annals of Probability 36 (2008), no. 1, 127--142. doi:10.1214/009117907000000079.

Export citation


  • Arcones, M. A. (1999). The law of the iterated logarithm over a stationary Gaussian sequence of random vectors. J. Theoret. Probab. 12 615--641.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Bingham, N. H., Goldie, C. M. and Tuegels, J. L. (1987). Regular Variation. Cambridge Univ. Press.
  • Derriennic, Y. and Lin, M. (2001). Fractional Poisson equation and ergodic theorems for fractional coboundaries. Israel J. Math. 123 93--130.
  • Durrett, R. and Resnick, S. (1978). Functional limit theorems for dependent variables. Ann. Probab. 6 829--846.
  • Hartman, P. and Wintner, A. (1941). On the law of the iterated logarithm. Amer. J. Math. 63 169--176.
  • Heyde, C. C. and Scott, D. J. (1973). Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments. Ann. Probab. 1 428--436.
  • Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 1--19.
  • Krengel, U. (1985). Ergodic Theorems. de Gruyter, Berlin.
  • Lai, T. L. and Stout, W. F. (1980). Limit theorems for sums of dependent random variables. Z. Wahrsch. Verw. Gebiete 51 1--14.
  • Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713--724.
  • Merlevède, F., Peligrad, M. and Utev, S. (2006). Recent advances in invariance principles for stationary sequences. Probab. Surv. 3 1--36.
  • Peligrad, M. and Utev, S. (2005). A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 798--815.
  • Petersen, K. (1983). Ergodic Theory. Cambridge Univ. Press.
  • Rio, E. (1995). The functional law of the iterated logarithm for stationary strongly mixing sequences. Ann. Probab. 23 1188--1203.
  • Shao, Q. M. (1993). Almost sure invariance principles for mixing sequences of random variables. Stochastic Process. Appl. 48 319--334.
  • Stout, W. F. (1970). The Hartman--Wintner law of the iterated logarithm for martingales. Ann. Math. Statist. 41 2158--2160.
  • Yokoyama, R. (1995). On the central limit theorem and law of the iterated logarithm for stationary processes with applications to linear processes. Stochastic Process. Appl. 59 343--351.
  • Zygmund, A. (2002). Trigonometric Series. Paperback with a foreword by Robert A. Fefferman. Cambridge Univ. Press.