Abstract
We give rates of convergence in the strong invariance principle for stationary sequences satisfying some projective criteria. The conditions are expressed in terms of conditional expectations of partial sums of the initial sequence. Our results apply to a large variety of examples. We present some applications to a reversible Markov chain, to symmetric random walks on the circle, and to functions of dependent sequences.
Highlights
Introduction and notationsThe almost sure invariance principle is a powerful tool in both probability and statistics
It says that the partial sums of random variables can be approximated by those of independent Gaussian random variables, and that the approximation error between the trajectories of the two processes is negligible in a certain sense
We are interested in studying rates in the almost sure invariance principle for dependent sequences
Summary
The almost sure invariance principle is a powerful tool in both probability and statistics. In Theorem 2.3 and its corollaries, we give conditions expressed in terms of conditional expectations of the random variables Xi and XiXj with respect to the past σ-algebra F0, to obtain rates in the almost sure invariance principle The proofs of these results are postponed to the section 4. As we just mentioned before, the conditions involved in these results are well adapted to linear processes even generated by martingale differences sequences, and we would like to refer to Section 3 in Wu (2007) where it is shown that they are satisfied for a large variety of functions of iid sequences. Assume in addition that np n2(log n)(t−1)p/2
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