Analysis of Carries in Signed Digit Expansions

TL;DR

This paper analyzes the statistical behavior of carries in signed digit addition, providing asymptotic results, distributional limits, and a general framework applicable to various digit systems and addition algorithms.

Contribution

It introduces a comprehensive analysis of carries in signed digit expansions, including new asymptotic formulas, distributional results, and a general method for transition probability determination.

Findings

Expected number of positive and negative carries derived

Central limit theorem established for carry distributions

Analysis of iteration counts in parallel addition methods

Abstract

The number of positive and negative carries in the addition of two independent random signed digit expansions of given length is analyzed asymptotically for the $(q, d)$-system and the symmetric signed digit expansion. The results include expectation, variance, covariance between the positive and negative carries and a central limit theorem. Dependencies between the digits require determining suitable transition probabilities to obtain equidistribution on all expansions of given length. A general procedure is described to obtain such transition probabilities for arbitrary regular languages. The number of iterations in von Neumann's parallel addition method for the symmetric signed digit expansion is also analyzed, again including expectation, variance and convergence to a double exponential limiting distribution. This analysis is carried out in a general framework for sequences of generating functions.