On the variance of a class of inductive valuations of data structures for digital search

An analytic approach to the asymptotic variance of trie statistics and related structures

“…where C > 0. ThenB(r) K (r) for a sufficiently large C > 0, and by standard Mellin argument [29] or by the proof used in [95]…”

“…where ∆ n,k := µ k + µ n−k − µ n + E(T n ). Different approaches have been proposed to the asymptotics of σ 2 n ; these include an elementary induction approach (see [7,52]), the second-moment approach (see [67,69,70,71]), Poissonized variance (by consideringf 2 (z) −f 2 1 (z)) approach (see [59,60,76,93,77,90]), (bivariate) characteristic function approach ( [57,77,60]), and Schachinger's differencing approach [95].…”

Limit behavior of maxima in geometric words representing set partitions

“…This follows with the same method of proof as Lemma 1, part (c) in [38]. Due to Lemma 5, we can apply the above proposition to ν n and obtain that ν n ≥ c 0 n − log 1/q ρ log n for n large enough with a suitable c 0 > 0.…”

“…The proof of the positiveness of σ 2 d for d ≥ 2 is now completed with the following proposition which was essentially proved by Schachinger in [38].…”

“…In order to show the above lower bound for the variance, we will directly work with recurrences and use some ideas of Schachinger from [38]. We first set…”