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Seminar on Stochastics
December 27, 2010 15:30, St. Petersburg, PDMI, room 106 (nab. r. Fontanki, 27)
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Conditional limit theorems for random walks and convergence of random trees II
V. V. Vysotsky |
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Abstract:
We start with a short and clearly incomplete review of conditional
functional limit theorems for random walks. In particular, we consider
a Brownian excursion,
which appears as one of the limit processes, and give it a convenient
“non-conditional” characterization in terms of a standard Brownian
motion.
The second part of the talk is on random trees. A certain
straightforward bijection maps the set of all rooted trees with $n$
edges into so-called Dyck paths of length $2n$. Assuming that the
elements of these sets are equally likely, we observe that such random
trees could be coded by positive excursions of length $2n$ of a simple
random walk. The limit of the later is a Brownian excursion, and it is
natural to assume that it codes a certain continuous random tree
(CRT). We define this object called Aldous' CRT, and discuss, subject
to time constraints, in which sense it is the weak limit of discrete
random trees.
Series of reports
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