## The Annals of Probability

### A new maximal inequality and invariance principle for stationary sequences

#### Abstract

We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713–724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.

#### Article information

Source
Ann. Probab. Volume 33, Number 2 (2005), 798-815.

Dates
First available in Project Euclid: 3 March 2005

http://projecteuclid.org/euclid.aop/1109868600

Digital Object Identifier
doi:10.1214/009117904000001035

Mathematical Reviews number (MathSciNet)
MR2123210

Zentralblatt MATH identifier
02164482

#### Citation

Peligrad, Magda; Utev, Sergey. A new maximal inequality and invariance principle for stationary sequences. Ann. Probab. 33 (2005), no. 2, 798--815. doi:10.1214/009117904000001035. http://projecteuclid.org/euclid.aop/1109868600.

#### References

• Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
• Bradley, R. C. (2002). Introduction to strong mixing conditions, 1. Technical report, Dept. Mathematics, Indiana Univ., Bloomington.
• Bradley, R. C. and Utev, S. (1994). On second order properties of mixing random sequences and random fields. In Probability Theory Mathematical Statistics (B. Grigelionis, J. Kubilius, H. Pragarauskas and V. Statulyavichus, eds.) 99--120. VSP/TEV, Vilnius.
• Chung, K. L. (1960). Markov Chains with Stationary Transition Probabilities. Springer, Berlin.
• Dedecker, J. and Rio, E. (2000). On the functional central limit theorem for stationary processes. Ann. Inst. H. Poincaré Probab. Statist. 36 1--34.
• Derriennic, Y. and Lin, M. (2003). The central limit theorem for Markov chains started at a point. Probab. Theory Related Fields 125 73--76.
• Esseen, C. G. and Janson, S. (1985). On moment conditions for normed sums of independent variables and martingale differences. Stochastic Process. Appl. 19 173--182.
• Garsia, A. M. (1965). A simple proof of E. Hopf's maximal ergodic theorem. J. Math. Mech. 14 381--382.
• Gordin, M. I. (1969). The central limit theorem for stationary processes. Soviet. Math. Dokl. 10 1174--1176.
• Halmos, P. R. (1956). Lectures on Ergodic Theory. The Mathematical Society of Japan, Tokyo.
• Ibragimov, I. A. (1975). A note on the central limit theorem for dependent variables. Theory Probab. Appl. 20 135--140.
• Isola, S. (1999). Renewal sequences and intermittency. J. Statist. Phys. 24 263--280.
• Liverani, C. (1996). Central limit theorem for deterministic systems. In International Conference on Dynamical Systems (F. Ledrappier, J. Lewowicz and S. Newhouse, eds.) 362 56--75. Longman, Harlow.
• Maxwell, M. and Woodroofe, M. (2000). Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 713--724.
• Peligrad, M. (1982). Invariance principle for mixing sequences of random variables. Ann. Probab. 10 968--981.
• Peligrad, M. (1999). Convergence of stopped sums of weakly dependent random variables. Electron. J. Probab. 4 1--13.
• Peligrad, M. and Utev, S. (1997). Central limit theorem for linear processes. Ann. Probab. 25 443--456.
• Rio, E. (2000). Theorie Asymptotique des Processus Aleatoires Faiblement Dependants. Springer, Berlin.
• Shao, Q. (1989). On the invariance principle for stationary $\rho$-mixing sequences of random variables. Chinese Ann. Math. 10B 427--433.
• Wu, B. W. and Woodroofe, M. (2002). Martingale approximation for sums of stationary processes. Ann. Probab. 32 1674--1690.