206 research outputs found
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
On the centroid of increasing trees
A centroid node in a tree is a node for which the sum of the distances to all
other nodes attains its minimum, or equivalently a node with the property that
none of its branches contains more than half of the other nodes. We generalise
some known results regarding the behaviour of centroid nodes in random
recursive trees (due to Moon) to the class of very simple increasing trees,
which also includes the families of plane-oriented and -ary increasing
trees. In particular, we derive limits of distributions and moments for the
depth and label of the centroid node nearest to the root, as well as for the
size of the subtree rooted at this node
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