Abstract
We consider a random tree and introduce a metric in the space of trees to define the “mean tree” as the tree minimizing the average distance to the random tree. When the resulting metric space is compact we have laws of large numbers and central limit theorems for sequence of independent identically distributed random trees. As application we propose tests to check if two samples of random trees have the same law.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Billera LJ, Holmes SP, Vogtmann K (2001) Geometry of the space of phylogenetic trees. Adv Appl Math 27(4):733–767
Busch JR, Ferrari PA, Flesia AG, Fraiman R, Grynberg S, Leonardi F (2006) Testing statistical hypothesis on random trees. Preprint
Critchlow DE (1985) Metric methods for analysing partially ranked data. Lectures notes in statistics, vol. 34. Springer, Berlin
De Fitte PR (1997) Théorème ergodique ponctuel et lois fortes des grandes nombres pour des points aléatoires d’un espace métrique à courbure négative. Ann Probab 25(2):738–766
Es-Sahib A, Heinich H (1999) Barycentre canonique pour un espace métrique à courbure négative. In: Séminaire de probabilités, XXXIII. Lecture notes in math, vol 1709. Springer, Berlin, pp 355–370
Haldane JBS (1948) Note on the median of a multivariate distribution. Biometrika 35:414–415
Herer W (1992) Mathematical expectation and strong law of large numbers for random variables with values in a metric space of negative curvature. Probab Math Stat 13(1):59–70
Kingman JFC (1982) The coalescent. Stoch Proc Appl 13:235–248
Kurata K, Minami N (2004) The equivalence of two constructions of Galton–Watson processes. Preprint mp_arc 04-295
Ledoux M, Talagrand M (1991) Probability in Banach spaces. Isoperimetry and processes. Springer, Berlin
Liggett TM (1985) Interacting particle systems. Springer, New York
Milasevic P, Ducharme GR (1987) Uniqueness of the spatial median. Ann Stat 15:1332–1333
Neveu J (1986) Arbres et processus de Galton–Watson. Ann Inst Henri Poincaré, Probab Stat 22:199–207
Otter R (1949) The multiplicative process. Ann Math Stat 20:206–224
Sverdrup-Thygeson H (1981) Strong law of large numbers for measures of central tendency and dispersion of random variables in compact metric spaces. Ann Stat 9:141–145
Valiente G (2001) An efficient bottom-up distance between trees. In: Proc 8th int symposium on string processing and information retrieval, SPIRE 2001: IEEE Computer Science Press, pp 212–219
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Balding, D., Ferrari, P.A., Fraiman, R. et al. Limit theorems for sequences of random trees. TEST 18, 302–315 (2009). https://doi.org/10.1007/s11749-008-0092-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11749-008-0092-z