Analysis of carries in signed digit expansions
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- 作者：Clemens Heuberger ; Sara Kropf ; Helmut Prodinger
- 关键词：Carry ; Central limit theorem ; Transducer ; Probabilistic automaton ; Symmetric signed digit expansion ; Von Neumann’s addition
- 刊名：Monatshefte für Mathematik
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：182
- 期：2
- 页码：299-334
- 全文大小：1266KB
- 刊物主题：Mathematics, general;
- 出版者：Springer Vienna
- ISSN：1436-5081
- 卷排序：182
文摘
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.