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A diffusion limit for a class of randomly-growing binary trees
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  • Published: November 1988

A diffusion limit for a class of randomly-growing binary trees

  • David Aldous1 &
  • Paul Shields2 

Probability Theory and Related Fields volume 79, pages 509–542 (1988)Cite this article

  • 118 Accesses

  • 47 Citations

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Summary

Binary trees are grown by adding one node at a time, an available node at height i being added with probability proportional to c -i, c>1. We establish both a “strong law of large numbers” and a “central limit theorem” for the vector X(t)=(X i(t)), where X i(t) is the proportion of nodes of height i that are available at time t. We show, in fact, that there is a deterministic process x i(t) such that

$$\sum {|X_i (t) - x_i (t)|} {\text{ converges to 0 a}}{\text{.s}}{\text{.,}}$$

and such that if c2\(\tfrac{1}{2}\),

$$Z_i^n (t) = 2^{n/2} \{ X_{n + 1} (tc^n ) - x_{n + 1} (tc^n )\} ,$$

and Z n(t)=(Z ni (t)), then Z n(t) converges weakly to a Gaussian diffusion Z(t). The results are applied to establish asymptotic normality in the unbiased coin-tossing case for an entropy estimation procedure due to J. Ziv, and to obtain results on the growth of the maximum height of the tree.

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References

  1. Aldous, D.: Stopping times and tightness, II. Tech. report No. 124, Dept. of Statistics, University of California, Berkeley, 1987

    Google Scholar 

  2. Athreya, K.B.: Discounted branching random walks. Adv. Appl. Probab. 17, 53–66 (1985)

    Google Scholar 

  3. Bradley, R.M., Strenski, P.N.: Directed diffusion-limited aggregation on the Bethe lattice: exact results. Phys. Rev. B30, 6788–6790 (1984)

    Google Scholar 

  4. Brennan, M.D., Durrett, R.: Splitting intervals. Ann. Probab. 14, 1023–1036 (1986)

    Google Scholar 

  5. Brennan, M.D., Durrett, R.: Splitting intervals II: Limit laws for lengths. Probab. Th. Rel. Fields 75, 109–127 (1987)

    Google Scholar 

  6. Durrett, R.: Oriented percolation in two dimensions. Ann. Probab. 12, 999–1040 (1984)

    Google Scholar 

  7. Elliott, R.: Stochastic calculus and its applications. New York: Springer 1982

    Google Scholar 

  8. Gihman, I.I., Skorohod, A.V.: The theory of stochastic processes, vol. 3. New York: Springer 1979

    Google Scholar 

  9. Grinblatt, L.S.: A limit theorem for measurable random processes and its applications. Proc. Amer. Soc. 61, 371–376 (1976)

    Google Scholar 

  10. Knuth, D.: The art of computer programming, vol. 3. Reading, Mass.: Addison-Wesley 1973

    Google Scholar 

  11. Loynes, R.M.: A criterion for tightness for a sequence of martingales. Ann. Probab. 4, 859–862 (1976)

    Google Scholar 

  12. Pittel, B.: On growing random binary trees. J. Math. Anal. Appl. 103, 461–480 (1984)

    Google Scholar 

  13. Pittel, B.: Asymptotic growth of a class of random trees. Ann. Probab. 13, 414–427 (1985)

    Google Scholar 

  14. Pittel, B.: Paths in a random digital tree: limiting distributions. Adv. Appl. Probab. 18, 139–155 (1986)

    Google Scholar 

  15. Ross, S.M.: Stochastic processes. New York: Wiley 1983

    Google Scholar 

  16. Wignarajah, G.: Complexity tests for statistical independence. M.S. thesis, University of Toledo, 1985

  17. Ziv, J.: Coding theorems for individual sequences. IEEE Trans. Inf. Theory 24, 405–412 (1978)

    Google Scholar 

  18. Ziv, J., Lempel, A.: A universal algorithm for sequential data compression. IEEE Trans. Inf. Theory 23, 337–343 (1977)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, University of California, 94720, Berkeley, CA

    David Aldous

  2. Department of Mathematics, University of Toled, 2801 W. Bancroft Street, 43606, Toledo, OH, USA

    Paul Shields

Authors
  1. David Aldous
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  2. Paul Shields
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Additional information

The work of the first author supported in part by NSF grant # MCS84-03239

The work of the second author supported in part by NSF grants # MCS83-03253 and # DMS85-07189 and in part by a Fulbright fellowship to visit the Mathematics Institute of the Hungarian Academy of Sciences, Budapest

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Aldous, D., Shields, P. A diffusion limit for a class of randomly-growing binary trees. Probab. Th. Rel. Fields 79, 509–542 (1988). https://doi.org/10.1007/BF00318784

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  • Received: 03 April 1987

  • Revised: 13 May 1988

  • Issue Date: November 1988

  • DOI: https://doi.org/10.1007/BF00318784

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Keywords

  • Entropy
  • Stochastic Process
  • Probability Theory
  • Statistical Theory
  • Central Limit
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