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On the Variety of Shapes in Digital Trees

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Abstract

We study the joint distribution of the number of occurrences of members of a collection of nonoverlapping motifs in digital data. We deal with finite and countably infinite collections. For infinite collections, the setting requires that we be very explicit about the specification of the underlying measure-theoretic formulation. We show that (under appropriate normalization) for such a collection, any linear combination of the number of occurrences of each of the motifs in the data has a limiting normal distribution. In many instances, this can be interpreted in terms of the number of occurrences of individual motifs: They have a multivariate normal distribution. The methods of proof include combinatorics on words, integral transforms, and poissonization.

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Notes

  1. A set of probabilities \(p_1, \ldots , p_m\) is said to be periodic, when \(\log p_j / \log p_k\) is rational, for every \(1 \le j, k\le m\).

  2. In the aperiodic case, the o(n) estimate can be improved to \(O(n^{1-\varepsilon })\), for some \(0< \varepsilon < 1\).

  3. In our case, the variance \(\sigma ^{2}_\mathcal {C}(n)\) will always be strictly positive. For a more in-depth consideration of the variance for shape parameters in random tries, see Schachinger [23].

  4. We take an infinite-dimensional random vector to have a multivariate normal distribution, when every nonzero finite linear combination of its components has a univariate normal distribution.

  5. The same is not true in the fixed population model. That is, in case (i), \(I_{n, T_\kappa ,v}\) and \(I_{n, T_\nu ,w}\) can be dependent. So, we see the advantage of quickly switching to a Poisson model, rather than transforming recurrences in the fixed population model.

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Acknowledgments

The authors sincerely thank an anonymous referee for detailed and insightful comments about the entire paper. We acknowledge the referee for improving the paper in many ways. M. D. Ward’s research is supported by NSF Grant DMS-1246818, and by the NSF Science & Technology Center for Science of Information Grant CCF-0939370.

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Authors and Affiliations

  1. Mathematical Biosciences Institute, The Ohio State University, Jennings Hall 3rd Floor, 1735 Neil Ave., Columbus, OH, 43210, USA

    Jeffrey Gaither

  2. Department of Statistics, The George Washington University, 801 22nd Street, Washington, DC, 20052, USA

    Hosam Mahmoud

  3. Department of Statistics, Purdue University, 150 North University Street, West Lafayette, IN, 47907-2067, USA

    Mark Daniel Ward

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  1. Jeffrey Gaither
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  2. Hosam Mahmoud
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  3. Mark Daniel Ward
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Correspondence to Mark Daniel Ward.

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Gaither, J., Mahmoud, H. & Ward, M.D. On the Variety of Shapes in Digital Trees. J Theor Probab 30, 1225–1254 (2017). https://doi.org/10.1007/s10959-016-0700-x

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  • DOI: https://doi.org/10.1007/s10959-016-0700-x

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