In this paper, we first establish a new fixed point theorem for a Meir-Keeler type condition. As an application, we derive a simultaneous generalization of Banach contraction principle, Kannan's fixed point theorem, Chatterjea's fixed point theorem and other fixed point theorems. Some new fixed point theorems are also obtained.

For a coordinate symmetric random vector (Y1, . . . ,Yn) = Y ∈ R, that is, one satisfying (Y1, . . . ,Yn) =d (e1Y1, . . . , enYn) for all (e1, . . . , en) ∈ {−1,1}, for which P(Yi = 0) = 0 for all i = 1,2, . . . ,n, the following Berry Esseen bound to the cumulative standard normal Φ for the standardized projection Wθ = Yθ/vθ of Y holds: sup x∈R |P(Wθ ≤ x)−Φ(x)| ≤ 2 n ∑ i=1 |θi |E|X i | + 8.4E(...