Asymptotic normality of recursive algorithms via martingale difference arrays

are such that if input data of size N produce random costs LN , then LN D = Ln + L̄N−n + RN for N ≥ n0 ≥ 2, where n follows a certain distribution PN on the integers {0, . . . ,N} and Lk D = L̄k for k ≥ 0. Ln, LN−n and RN are independent, conditional on n, and RN are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N− n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to ZN := LN−IE LN Var LN . Under certain compatibility assumptions on the sequence (PN)N≥0 we show that a pair of sufficient conditions (of Lyapunov type) for ZN D →N (0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (IE LN)N≥0. In the case that the PN are binomial distributions with the same parameter p, and for deterministic RN , we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (RN)N≥0 (and for the scale RN = Nα a characterization of those α) leading to asymptotic normality of ZN .