Discrete Mathematics & Theoretical Computer Science
| author: | Werner Schachinger |
|---|---|
| title: | Asymptotic normality of recursive algorithms via martingale difference arrays |
| keywords: | recursive algorithms, trie, martingales, asymptotic normality, central limit theorem |
| abstract: | We propose martingale central limit theorems as an tool to prove asymptotic normality of the costs of
certain recursive algorithms which are subjected to random input
data. The recursive algorithms that we have in mind are such that
if input data of size N produce random costs LN, then
LN=D Ln+ LN-n+RN for N ≥ n0≥2, where n follows
a certain distribution PN on the integers {0, .. ,N}
and Lk =D Lk 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 - E 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 → DN(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
(E LN)N≥0 and thus very likely a well studied object. 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.
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| reference: | Werner Schachinger (2001), Asymptotic normality of recursive algorithms via martingale difference arrays, Discrete Mathematics and Theoretical Computer Science 4, pp. 363-398 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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