带符号数字展开式中的进位分析
Analysis of Carries in Signed Digit Expansions
摘要：
对$(q, d)$ -系统和对称符号数展开式，分析了两个独立的给定长度的随机符号数展开式相加的正负进位数的渐近性.结果包括正负进位之间的期望、方差、协方差和一个中心极限定理。数字之间的重复性要求确定合适的转移概率，以在给定长度的所有展开上获得等分布。一个通用的程序来获得这样的转移概率任意正规语言。还分析了对称符号数位展开的冯·诺依曼并行加法中的迭代次数，再次包括期望、方差和收敛到双指数极限分布。这种分析是在生成函数序列的一般框架中进行的。
英文摘要：
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.
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