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principles of random walk
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Graduate
Texts
in Mathematics
34
Editorial Board
F.
W.
Gehring
P.R.
Halmos
Managing Editor
C.
C. Moore
Frank Spitzer
Principles of
Random
Walk
Second Edition
Springer Science+ Business Media,
LLC
1976
Frank
Spitzer
Cornell University
Department
of Mathematics
Ithaca,
New
York
14850
Editorial Board
P.R.
Halmos
Managing Editor
Indiana
University
Department
of
Mathematics
Swain
Hall
East
Bloomington,
Indiana
4740
I
AMS Subject
Classification
60]15
F.
W.
Gehring
University
of
Michigan
Department of Mathematics
Ann
Arbor, Michigan
48104
Library of Congress Cataloging in Publication
Data
Spitzer,
Frank Ludvig,
1926-
Principles
of random walk.
(Graduate texts in mathematics;
34)
C. C.
Moore
University
of
California
at
Berkeley
Department
of
Mathematics
Berkeley, California
94720
First edition published by D. Van Nostrand,
Princeton,
N.J.,
in series:
The
University series in higher mathematics,
edited by
M.A.
Stone, L. Nirenberg, and
S. S.
Chern.
Bibliography: p. 395
Includes index.
I. Random
walks
(Mathematics)
I.
Title.
II.
Series.
QA274.73.S65
1975 519.2'82 75-26883
All rights reserved.
No part of this
book
may
be
translated or reproduced
in any form without written permission from
Springer Science+ Business Media, LLC
©
1964
by Frank
Spitzer
Originally published
by
Springer
Verlag New York in
1964
Softcover reprint
of
the hardcover
2nd
edition
1964
ISBN 978-1-4757-4231-2
ISBN 978-1-4757-4229-9 (eBook)
DOI
10.1007/978-1-4757-4229-9
PREFACE
TO
THE
SECOND
EDITION
In
this edition a large number of errors have been corrected, an occasional
proof has been streamlined, and a number of references are made to recent pro-
gress.
These
references are to a supplementary bibliography, whose items are
referred to
as
[S1]
through
[S26].
A thorough revision was not attempted.
The
development of the subject
in
the last decade would have required a treatment in a much more general con-
text.
It
is
true that a number of interesting questions remain open in the concrete
setting of random walk on the integers.
(See [S
19]
for a recent survey).
On
the
other hand, much of the material of this book (foundations, fluctuation theory,
renewal theorems)
is
now available in standard texts, e.g. Feller
[S9],
Breiman
[S1],
Chung
[S4]
in the more general setting of random walk on the real line.
But
the major new development since the first edition occurred in 1969, when
D.
Ornstein
[S22]
and
C.
J.
Stone
[S26]
succeeded in extending the recurrent
potential theory
in·
Chapters
II
and
VII
from the integers to the reals. By now
there
is
an extensive and nearly complete potential theory of recurrent random
walk on locally compact groups, Abelian (
[S20],
[S25])
as
well
as
non-
Abelian (
[S17],
[S2]
).
Finally, for the non-specialist there exists now an
unsurpassed brief introduction to probabilistic potential theory, in the context of
simple random walk and Brownian motion,
by
Dynkin and Yushkevich
[S8].
In
view of the above mentioned developments it might seem that the intuitive
ideas of the subject have been left far behind and perhaps lost their vitality.
For-
tunately this
is
false.
New
types of random walk problems are now in the stage
of pioneering work, which were unheard of when the first edition appeared.
This
came about because the simple model of a single particle, performing a
random walk with given transition probabilities, may
be
regarded as a crude
approximation to more elaborate random walk models.
In
one of these a single
particle moves in a random environment, i.e. the transition probabilities are
themselves random variables.
In
other models one considers the simultaneous
random walk of a finite
or
even infinite system of particles, with certain types of
interaction between the particles. But this
is
an entirely different story.
PREFACE
TO
THE
FIRST
EDITION
This
book
is
devoted exclusively to a very special class
of
random
processes, namely to
random walk
on
the
lattice points
of
ordinary
Euclidean space. I considered this high degree
of
specialization worth
while, because
the
theory
of
such
random
walks
is
far more complete
than
that
of
any
larger class
of
Markov
chains.
Random
walk occupies
such a privileged position primarily because
of
a delicate
interplay
between methods from
harmonic analysis
on one
hand,
and
from
potential theory
on
the
other.
The
relevance
of
harmonic analysis to
random
walk
of
course stems from
the
invariance
of
the
transition
probabilities
under
translation
in
the
additive group which forms
the
state
space.
It
is
precisely for this reason
that,
until
recently,
the
subject
was
dominated
by
the
analysis
of
characteristic functions (Fourier
transforms
of
the
transition
probabilities).
But
if harmonic analysis
were the central theme
of
this book,
then
the
restriction to
random
walk on
the
integers
(rather
than
on
the
reals,
or
on o'ther Abelian
groups) would be
quite
unforgivable.
Indeed
it
was the need for a self-
contained
elementary
exposition
of
the
connection
of
harmonic analysis
with
the
much
more recent developments in
potential
theory
that
dictated
the
simplest possible setting.
The
potential theory associated with
Markov
processes
is
currently
being explored in the research
literature,
but
often
on
such
a high
plane
of
sophistication,
and
in such a general context
that
it
is
hard
for
the
novice to see
what
is
going on.
Poteatial
theory
is
basically con-
cerned with
the
probability laws governing the time
and
position
of
a
Markov
process when
it
first visits a specified subset
of
its
state
space.
These
probabilities satisfy equations entirely analogous to those in
classical
potential
theory,
and
there
is
one
Markov
process, namely
Brownian motion,
whose
potential
theory
is
exactly
the
classical one.
Whereas even for Brownian motion
the
study
of
absorption probabilities
involves delicate measure theory
and
topology, these difficulties
evap-
orate
in the case
of
random
walk.
For
arbitrary
subsets
of
the
space
of
lattice points the
time
and
place
of
absorption are
automatically
measurable,
and
the
differential equations encountered in the
study
of
Brownian motion reduce to difference equations for
random
walk.
In
this sense
the
study
of
random
walk leads one to
potential
theory in a
very simple setting.
VII
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