631 research outputs found
On martingale tail sums for the path length in random trees
For a martingale converging almost surely to a random variable ,
the sequence is called martingale tail sum. Recently, Neininger
[Random Structures Algorithms, 46 (2015), 346-361] proved a central limit
theorem for the martingale tail sum of R{\'e}gnier's martingale for the path
length in random binary search trees. Gr{\"u}bel and Kabluchko [to appear in
Annals of Applied Probability, (2016), arXiv 1410.0469] gave an alternative
proof also conjecturing a corresponding law of the iterated logarithm. We prove
the central limit theorem with convergence of higher moments and the law of the
iterated logarithm for a family of trees containing binary search trees,
recursive trees and plane-oriented recursive trees.Comment: Results generalized to broader tree model; convergence of moments in
the CL
On martingale tail sums in affine two-color urn models with multiple drawings
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and
arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn
schemes with multiple drawings. We show that, in large-index urns (urn index
between and ) and triangular urns, the martingale tail sum for the
number of balls of a given color admits both a Gaussian central limit theorem
as well as a law of the iterated logarithm. The laws of the iterated logarithm
are new even in the standard model when only one ball is drawn from the urn in
each step (except for the classical Polya urn model). Finally, we prove that
the martingale limits exhibit densities (bounded under suitable assumptions)
and exponentially decaying tails. Applications are given in the context of node
degrees in random linear recursive trees and random circuits.Comment: 17 page
Random recursive trees: A boundary theory approach
We show that an algorithmic construction of sequences of recursive trees
leads to a direct proof of the convergence of random recursive trees in an
associated Doob-Martin compactification; it also gives a representation of the
limit in terms of the input sequence of the algorithm. We further show that
this approach can be used to obtain strong limit theorems for various tree
functionals, such as path length or the Wiener index
Galton-Watson trees with vanishing martingale limit
We show that an infinite Galton-Watson tree, conditioned on its martingale
limit being smaller than \eps, agrees up to generation with a regular
-ary tree, where is the essential minimum of the offspring
distribution and the random variable is strongly concentrated near an
explicit deterministic function growing like a multiple of \log(1/\eps). More
precisely, we show that if then with high probability as \eps
\downarrow 0, takes exactly one or two values. This shows in particular
that the conditioned trees converge to the regular -ary tree, providing an
example of entropic repulsion where the limit has vanishing entropy.Comment: This supersedes an earlier paper, arXiv:1006.2315, written by a
subset of the authors. Compared with the earlier version, the main result
(the two-point concentration of the level at which the Galton-Watson tree
ceases to be minimal) is much stronger and requires significantly more
delicate analysi
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Persisting randomness in randomly growing discrete structures: graphs and search trees
The successive discrete structures generated by a sequential algorithm from
random input constitute a Markov chain that may exhibit long term dependence on
its first few input values. Using examples from random graph theory and search
algorithms we show how such persistence of randomness can be detected and
quantified with techniques from discrete potential theory. We also show that
this approach can be used to obtain strong limit theorems in cases where
previously only distributional convergence was known.Comment: Official journal fil
Searching for a trail of evidence in a maze
Consider a graph with a set of vertices and oriented edges connecting pairs
of vertices. Each vertex is associated with a random variable and these are
assumed to be independent. In this setting, suppose we wish to solve the
following hypothesis testing problem: under the null, the random variables have
common distribution N(0,1) while under the alternative, there is an unknown
path along which random variables have distribution , , and
distribution N(0,1) away from it. For which values of the mean shift can
one reliably detect and for which values is this impossible? Consider, for
example, the usual regular lattice with vertices of the form and oriented edges , where . We show that for paths of length starting at
the origin, the hypotheses become distinguishable (in a minimax sense) if
, while they are not if . We derive
equivalent results in a Bayesian setting where one assumes that all paths are
equally likely; there, the asymptotic threshold is . We
obtain corresponding results for trees (where the threshold is of order 1 and
independent of the size of the tree), for distributions other than the Gaussian
and for other graphs. The concept of the predictability profile, first
introduced by Benjamini, Pemantle and Peres, plays a crucial role in our
analysis.Comment: Published in at http://dx.doi.org/10.1214/07-AOS526 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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