This is a chatty discussion of my older research, intended to be understandable to a Ph. D. student in theoretical or applied probability. Numbers like [22] refer to the bibliography and are accompanied by Math Reviews links. Instead of summarizing the results (for which see Math Reviews or the actual papers) I focus on context: where did the problem come from, and what subsequent work has been done?

This idiosyncratic 1989 book
*Probability Approximations via the Poisson Clumping Heuristic*
MR 90k:60004
developes a method for writing down first-order approximations in
a wide range of "generalized extrema" settings, and illustrates
with 100 examples from areas like
Markov chain hitting times,
extrema of stationary processes,
combinatorial maxima,
stochastic geometry coverage problems,
multidimensional diffusions escaping from potential wells,
maxima of Gaussian random fields.
Here is a
two page account
from Encyclopedia of Statistical Science.
Here is a
1992 update
keyed to the book.

Feingold
MR 94j:60138
has a nice application to Markov chain hitting times
arising in genetics.
A recent preprint by Adler
*On Excursion Sets, Tube Formulae, and Maxima of Random Fields*
relates the "maxima of random fields" chapter to subsequent
developments of Worsley
MR 96b:60132
and others.

Consider an infinite array $(X_{ij}, 1 \leq i,j < \infty)$ of random variables of the form

(*) $X_{ij} = f(A_i,B_j,C_{ij})$

for i.i.d. $(A_i)$, $(B_j)$ and $(C_{ij})$.
The array $X = (X_{ij})$ has the *row and column exchangeability*
(RCE) property. It is natural (e.g. as a general Bayesian analog of
classical statistical analysis of variance) to seek a converse:
is every RCE array essentially of the form (*)?
This is proved in [11]
MR 82m:60022.
A different proof based on techniques from logic was given independently
by Hoover
MR 84b:60016.
Here is Hoover's original technical report.
At the same time Diaconis and Freedman
MR 83e:92071
were thinking about the same question in connection with human visual
pattern recognition.
My monograph-length survey [22]
MR 88d:60107
of probabilistic aspects of exchangeability gave the state of the art in 1983.
Characterization theorems were subsequently developed in several
directions in a sequence of papers by Kallenberg: see e.g.
MR 96f:60063.

**Update (2007).**
Kallenberg's 2005 monograph
*Probabilistic Symmetries and Invariance Principles*
elegantly covers the field.
The theory of RCE arrays has been rediscovered at last twice since 2000.
By Lovasz and co-workers in the context of limits of dense graphs:
see e.g. their
Limits of dense graph sequences
and also see
Diaconis-Janson Graph Limits and Exchangeable Random Graphs
for the precise connection.
And by Vershik
Random Metric Spaces and Universality
in the context of isometry classes of metric spaces with probability measures.

Around 1979 I circulated drafts of a monograph "Weak Convergence and the General Theory of Processes" but this was never completed.

Since 1989 I (and now Jim Fill) have been working on a monograph Reversible Markov Chains and Random Walks on Graphs which may be completed next millennium!

Call a countable rooted tree with a single path from the root
to infinity a *tree with one end* or a *sin*-tree.
The simplest occurrence of random sin-trees is as
critical Galton-Watson processes conditioned to be infinite,
e.g. Kesten
MR 88b:60232.
My 1991 "general abstract nonsense" paper [52]
MR 92j:60009
considers families of $n$-vertex rooted trees and supposes that
as $n \to \infty$ the subtrees seen from random vertices
converge to some limit random tree.
Such a limit has a kind of "one-sided stationarity" property,
and therefore (by analogy with stationary sequences)
specifies a "two-sided" structure, which is a random
sin-tree with a certain stationarity property.
[52] gives many examples plus structure theory.
The particular case of the (random graph) mean-field
analog of the minimal spanning tree was studied in [49]
MR 93a:05111.
Later Penrose
MR 97i:60014
showed the same tree arises as the high-dimensional limit of
Euclidean minimal spanning trees.

Random sin-trees have subsequently appeared in many other contexts, e.g. as components of uniform spanning forests on countable graphs (see preprint by Benjamini et al). Two of my own recent papers involve sin-trees: Paper [79] discusses pruning processes on conditioned Galton-Watson trees, and the limit of the Erdos-Renyi random graph process as seen from a specified vertex; Paper [89] discusses the percolation process on a tree where infinite clusters are forbidden to grow further.