Limit theorems under the Maxwell-Woodroofe condition in Banach spaces

2014/03/04 by Christophe Cuny, Cuny, Christophe
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR

paper · pdf · doi:10.48550/arxiv.1403.0772

openalex publication_date 2014/03/04 · arxiv created 2016/01/25 · arxiv updated 2016/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that, for (adapted) stationary processes, the so-called Maxwell-Wood-roofe condition is sufficient for the law of the iterated logarithm and that it is optimal in some sense. We obtain a similar conclusion concerning the Marcinkiewicz-zygmund strong law of large numbers. Those results actually hold in the context of Banach valued stationary processes, including the case of L^r-valued random variables, with 1≤ r<∞. In this setting we also prove the weak invariance principle, under a version of the Maxwell-Woodroofe condition, generalizing a result of Peligrad and Utev \citePU. Our results extend to non-adapted processes as well, and, partly to stationary processes arising from dynamical systems. The proofs make use of a new maximal inequality and of approximation by martingales, for which some of our results are also new.

Citations

• Strong invariance principles with rate for "reverse" martingales and applications

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