On large deviations mixing sequences

There is an extensive literature on the large deviation principle under various dependence structures. However, in comparison to other limit theorems, few papers deal with strong mixing dependence conditions. In this note we prove the large deviation principle for 4-mixing stationary random sequences with fast enough convergence rate and for $-mixing stationary random sequences. Our d-mixing results were motivated by Schonmann [17], where a weaker conclusion is proved under a much weaker mixing assumption. The large deviation principle under $-mixing was motivated by Orey and Pelikan [15] and Bryc [7]. Chiyonobu and Kusuoka [8] consider large deviations under a mixing condition, different from those ususally considered in the strong mixing approach. Yakimavichyus [19] (cf. also references there) states large deviation theorems under the assumption of uniform strong mixing, but considers one dimensional case only and his conclusions have a different form which makes it difficult to compare his results with ours. Let {XJnF~ be a stationary sequence. Define u-fields $a,h = a(& : a s k s b) and let