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

Heuberger, Clemens; Kropf, Sara; Wagner, Stephan

doi:10.1016/j.ejc.2015.03.004

Cited by 5 publications

(12 citation statements)

References 25 publications

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“…A generalisation of the quasi-power theorem to dimension 2 has been provided in [13]. It has been used in [16], [17], [6], [15] and [19]. In [5,Thm.…”

Section: Introductionmentioning

confidence: 99%

Higher dimensional quasi-power theorem and Berry–Esseen inequality

Heuberger

Kropf

2018

Monatsh Math

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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.2010 Mathematics Subject Classification. 60F05; 60C05.

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“…A generalisation of the quasi-power theorem to dimension 2 has been provided in [13]. It has been used in [16], [17], [6], [15] and [19]. In [5,Thm.…”

Section: Introductionmentioning

confidence: 99%

Higher dimensional quasi-power theorem and Berry–Esseen inequality

Heuberger

Kropf

2018

Monatsh Math

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“…In this article, we consider the more general setting of Markov chains. The proofs are similar as those in [18], but the results are valid in a broader context and can be formulated more clearly. In contrast to [18], we allow the input sequence of the transducer to be generated by a Markov source.…”

Section: Introductionmentioning

confidence: 75%

“…In [18], the variance of the output of a transducer as well as the covariance between the input and the output were analyzed. In this article, we consider the more general setting of Markov chains.…”

Section: Introductionmentioning

confidence: 99%

Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

Kropf

2015

Preprint

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The partial sum of the states of a Markov chain or more generally a Markov source is asymptotically normally distributed under suitable conditions. One of these conditions is that the variance is unbounded. A simple combinatorial characterization of Markov sources which satisfy this condition is given in terms of cycles of the underlying graph of the Markov chain. Also Markov sources with higher dimensional alphabets are considered. Furthermore, the case of an unbounded covariance between two coordinates of the Markov source is combinatorically characterized. If the covariance is bounded, then the two coordinates are asymptotically independent.The results are illustrated by several examples, like the number of specific blocks in 0-1-sequences and the Hamming weight of the width-w non-adjacent form.

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“…Since the automaton is probabilistic and aperiodic, the unique dominant eigenvalue of A(1, 1) is 1. Thus the same arguments apply as in [5] after replacing "complete" by "probabilistic". We obtain the same formulas for the constants of the expectation, the variance and the covariance.…”

Section: Asymptotic Analysis Of the Standard Additionmentioning

confidence: 93%

“…After simplifying this construction as described in Lemma 5.4, the probabilistic automaton N SSDE has 12 states: (−1, 1)), (1, (1, −1))}, {(4, (0, 0)), (9, (0, 0))}, {(5, (0, 1)), (5 , (1, 0)), (10, (−1, 0)), (10, (0, −1))}, (11) {(2, (0, 1)), (2 , (1, 0)), (7, (−1, 0)), (7, (0, −1))}, {(5, (0, 0)), (10, (0, 0))}, (0, 0)), (7, (0, 0))}, (−1, 0)), (1, (0, −1)), (1, (0, 1)), (1, (1, 0))}, (0, 1)), (3 , (1, 0)), (8, (−1, 0)), (8, (0, −1))}, { (1, (0, 0))}, {(3, (0, 0)), (8, (0, 0))}, (1, 1)), (9, (−1, −1))}, {(4, (0, 1)), (4 , (1, 0)), (9, (−1, 0)), (9, (0, −1))}.…”

Section: Asymptotic Analysis Of Von Neumann's Additionmentioning

confidence: 99%

Analysis of carries in signed digit expansions

2016

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

Cited by 5 publications

References 25 publications

Higher dimensional quasi-power theorem and Berry–Esseen inequality

Higher dimensional quasi-power theorem and Berry–Esseen inequality

Variance and Covariance of Several Simultaneous Outputs of a Markov Chain

Analysis of carries in signed digit expansions

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