On the Second Borel-Cantelli Lemma for strongly mixing sequences of events
نویسنده
چکیده
holds. This is the assertion of the Second Borel-Cantelli Lemma. If the assumption of independence is dropped, (1) fails in general. However, (1) remains true if independence is replaced by pairwise independence (Durrett (1991), Theorem (6.6) ch. 1) or by uniform mixing (Iosifescu and Theodorescu (1969) Lemma 1.1.2’). Rieders (1993) showed by an example that strong mixing without any further assumption doesn’t guarantee the validity of (1). Hence, the fact that Yoshihara’s (1979, Theorem 1) Borel-Cantelli Lemma for strongly mixing events requires a condition on the size of the probabilities is not surprising. This article is intended to give refinements of Rieders’ and Yoshihara’s statements. Intuitively, one might expect that the less dependent the events are the better relation (1) will fit. Our results confirm this intuition in the following manner: For a fixed rate of convergence of the strong mixing coefficients of the events An, we determine how fast the rate of divergence of the series ∑
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