Sara Kropf

Variances and covariances in the Central Limit Theorem for the output of a transducer

We study the joint distribution of the input sum and the output sum of a deterministic transducer. Here, the input of this finite-state machine is a uniformly distributed random sequence. We give a simple combinatorial characterization of transducers for which the output sum has bounded variance, and we also provide algebraic and combinatorial characterizations of transducers for which the covariance of input and output sum is bounded, so that the two are asymptotically independent. Our results are illustrated by several examples, such as transducers that count specific blocks in the binary expansion, the transducer that computes the Gray code, or the transducer that computes the Hamming weight of the width-w non-adjacent form digit expansion. The latter two turn out to be examples of asymptotic independence.

Analysis of the Binary Asymmetric Joint Sparse Form

We consider redundant binary joint digital expansions of integer vectors. The redundancy is used to minimize the Hamming weight, i.e. , the number of non-zero digit vectors. This leads to efficient linear combination algorithms in abelian groups, which are used in elliptic curve cryptography, for instance. If the digit set is a set of contiguous integers containing zero, a special syntactical condition is known to minimize the weight. We analyse the optimal weight of all non-negative integer vectors with maximum entry less than N . The expectation and the variance are given with a main term and a periodic fluctuation in the second-order term. Finally, we prove asymptotic normality.

Higher dimensional quasi-power theorem and Berry–Esseen inequality

Hwang’s quasi-power theorem asserts that a sequence of random variables whose moment generating functions are approximately given by powers of some analytic function is asymptotically normally distributed. This theorem is generalised to higher dimensional random variables. To obtain this result, a higher dimensional analogue of the Berry–Esseen inequality is proved, generalising a two-dimensional version by Sadikova.

Fringe analysis of plane trees related to cutting and pruning

Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only the leftmost leaves or maximal paths. This generalizes the concept of pruning a tree. The results include exact expressions and asymptotic expansions for the expected value and the variance as well as central limit theorems.