Cycle Representations of Markov Processes

206 Pages·1995·4.381 MB·English

by  Sophia L. Kalpazidou (auth.)

Applications of
Stochastic Mechanics
Mathematics
Random Media
Signal Processing Stochastic M odeling
and Image Synthesis and Applied Probability
M athematical Economics
28
Stochastic Optimization
Stochastic Control
Edited by I. Karatzas
M. Vor
Advisory Board P. Bremaud
E. Carlen
R. Do brushin
W. Fleming
D. Geman
G. Grimmett
G. Papanicolaou
1. Scheinkman
Applications of Mathematics
Fleming/Rishel, Deterministic and Stochastic Optimal Control (1975)
2 Marchuk, Methods of Numerical Mathematics, Second Ed. (1982)
3 Balakrishnan, Applied Functional Analysis, Second Ed. (1981)
4 Borovkov, Stochastic Processes in Queueing Theory (1976)
5 LiptserlShiryayev, Statistics of Random Processes I: General Theory (1977)
6 LiptserlShiryayev, Statistics of Random Processes 11: Applications (1978)
7 Vorob'ev, Game Theory: Lectures for Economists and Systems Scientists
(1977)
8 Shiryayev, Optimal Stopping Rules (1978)
9 Ibragimov/Rozanov, Gaussian Random Processes (1978)
10 Wonham, Linear Multivariable Control: A Geometrie Approach, Third Ed.
(1985)
11 Hida, Brownian Motion (1980)
12 Hestenes, Conjugate Direction Methods in Optimization (1980)
13 Kallianpur, Stochastic Filtering Theory (1980)
14 Krylov, Controlled Diffusion Processes (1980)
15 Prabhu, Stochastic Storage Processes: Queues, Insurance Risk, and Dams
(1980)
16 Ibragimov/Has'minskii, Statistical Estimation: Asymptotic Theory (1981)
17 Cesari, Optimization: Theory and Applications (1982)
18 Elliott, Stochastic Calculus and Applications (1982)
19 Marchuk/Shaidourov, Difference Methods and Their Extrapolations (1983)
20 Hijab, Stabilization of Control Systems (1986)
21 Protter, Stochastic Integration and Differential Equations (1990)
22 Benveniste/Metivier/Priouret, Adaptive Algorithms and Stochastic
Approximations (1990)
23 KloedeniPlaten, Numerical Solution of Stochastic Differential Equations
(1992)
24 Kushner/Dupuis, Numerical Methods for Stochastic Control Problems
in Continuous Time (1992)
25 FlemingiSoner, Controlled Markov Processes and Viscosity Solutions
(1993)
26 Baccelli/Bremaud, Elements of Queueing Theory (1994)
27 Winkler, Image Analysis, Random Fields, and Dynamic Monte Carlo
Methods: An Introduction to Mathematical Aspects (1994)
28 Kalpazidou, Cycle Representations of Markov Processes (1995)
29 Elliottl AggouniMoore, Hidden Markov Models: Estimation and Control
(1995)
Sophia L. Kalpazidou
Cycle Representations
of Markov Processes
With 14 Figures
Springer Science+Business Media, LLC
Sophia L. Kalpazidou
Department of Mathematics
Aristotle University of Thessaloniki
54006 Thessaloniki
Greece
M anaging Editors
I. Karatzas
Department of Statistics
Columbia University
New York, NY 10027, USA
M. Vor
CNRS, Laboratoire de Probabilites
Universite Pierre et Marie Curie
4 Place Jussieu, Tour 56
75252 Paris Cedex 05, France
Mathematics Subject C1assification (1991): 60B99; 60G05; 60J05; 60J25
Library of Congress Cataloging-in-Publication Data
Kalpazidou, Sophia L.
CycIe representations of Markov processes / Sophia L. Kalpazidou.
p. cm.-(Applications ofmathematics: vol. 28)
IncIudes bibliographical references and index.
1. Markov processes. 2. Algebraic cycIes. I. Title.
11. Series: Applications of mathematics : 28.
QA274.7.K354 1994
519.2'33-dc20 94-28223
Printed on acid-free paper.
ISBN 978-1-4757-3931-2 ISBN 978-1-4757-3929-9 (eBook)
DOI 10.1007/978-1-4757-3929-9
© 1995 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 1995.
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9 8 7 6 5 4 3 2 1
Preface
'A(!f.lOvi17 &.qJIY.V~r; qJIY.Vf.(!1ir; X(!ÜJ(J(JJV.
H(!rJ.XAW:Or;
Unrevealed harmony is superior to the visible one.
HERACLITOS
The purpose of the present book is to give a systematic and unified exposi
tion of stochastic processes of the Markovian type, homogeneous and with
either discrete or continuous parameter, which, under an additional assump
tion concerning the existence of invariant measures, can be defined by directed
cycles or circuits. These processes are called cycle (or circuit) processes, and the
corresponding collections of weighted cycles are called cycle representations.
The descriptions of the Markov transition law in terms of the cycles dis
close new and special properties which have given an impetus to very intensive
research concentrated on the connections between the geometric properties
of the trajectories and the algebraic characterization of Markov processes.
Let us start with a few heuristic motivations for this new topic. The simplest
ex amp ie leading to a cycle process arises when modeling the motion of a par
ticle on a closed curve. Observe the particle's motion through p (~ 1) points
of this curve at moments one unit of time apart. This amounts to a discretiza
ti on ofthe curve into an ordered sequence c = (c(n), c(n + 1), ... , c(n + p - 1),
c(n)), n = 0, ± 1, ± 2, ... , called a directed circuit with period p(c) = p. The
subsequence c = (c(n), c(n + 1), ... , c(n + p - 1)) will be called a directed
cycle (associated with the circuit c). Assign a positive number Wc to c. Then, a
normalized measure ofthe passage from state i = c(n) to j = c(n + 1) is given
by wjwc = 1. Therefore, if no influences occur, the passages from i to j can be
v
VI Preface
°cod ified by an infinite binary sequence Y(i,j) = (0, 1, 0, ... , 0, 1, ... ) where 1 or
means that at some moment n the particle passes through or does not pass
through (i,j).
The sequence Y(i,j) is understood as a "nonrandom" sequen°ce in the con
text of Kolmogorov's theory of complexities since both 1 and appear peri
odically after each P steps. This happens because of the sm all complexity of
the particle's trajectory wh ich consists of a circuit c alone. Then, when some
"chaos" arises, it necessarily presupposes some complexity in the form of the
particle's trajectory. So, let us consider a further two, or even more than two,
overlapping directed circuits CI' ... , Cr' r 2 2, each associated with some pos
itive number wc/, 1 = 1, ... , r. Imagine that the particle appears sometime at
the incident point i of certain circuits, say for simplicity, Cl' ... , Cl' 1 ~ r.
Then, the particle can continue its motion to another point j through which
some circuits cm" •.• , cms' S ~ I, m1, ... , ms E {1, ... , I}, pass. A natural mea
sure for the particle's transition when moving from i to j can be defined
as
(wc + Wc + ... + Wc )/(W c + Wc + ... + Wc ). (1)
ml "'12 ms 1 2 I
Accordingly, the binary sequence codifying as above the appearances
of the pair (i,j) along a trajectory is given by a "more chaotic" sequence
like Y(i,j) = (0,0,0, 1,0, 1,0,0, 1,0, ... ), where 1 means that at some moment
of time the particle passes through or does not pass through certain cir
cuits containing (i,j). Furthermore, since expression (1) provides transition
prob ability from i to j of a Markov chain ~ = (~n' n = 0, 1, ... ) we conclude
that:
there exist deterministic constructions to a M arkov chain ~ which rely on
collections Cf/ 01 directed circuits endowed with certain measures "If/ =
(wo C E Cf/). The pairs (Cf/, "If/) completely determine the process ~.
But the same conclusion can be conversely viewed as:
there are M arkov chains ~ which are defined by two distinct entities: a topo
logical entity given by a collection Cf/ 01 directed circuits, and an algebraic
entity given by a measure "If/ = (wo C E Cf/).
Plainly, both topological and algebraic components Cf/ and "If/ are not
uniquely determined, and this is motivated by the algebraic nature of our
construction. To assure the uniqueness, we should look for another approach
which can express the definite characteristic of the finite-dimensional distri
butions of ~.
A natural way to obtain a uniqueness criterion for (Cf/, "If/) can be given by
a behavioral approach. It is this approach that we shall furt her use.
Let S be a finite set, and let ~ = (~n)n;o,O be a homogeneous and irreducible
S-state Markov chain whose transition matrix is P = (Pij' i,j ES). Denote
by n = (ni' i E S) the invariant probability distribution of P, that is, ni > 0,
Lini = 1, and
L niPij = L njPji, (2)
j j
Preface vii
It turns out that system (2) of the "balance equations" can be equivalently
written as follows:
i,j E S, (3)
where c ranges over a collection rri of directed cycles (or circuits) in S, We are posi
tive numbers, and Je(i, j) = 1 or 0 according to whether or not (i, j) is an edge of c.
The equivalence of the systems (2) and (3) pressupposes the existence of
an invertible transform of the "global coordinates" expressed by the cycle
weights w" C E rri, into the "local coordinates" given by the edge-weights niPij,
i, j E S. That is, geometry (topology) enters into equations (2) and (3). The
inverse transform of the edge-coordinates niPij, i, jE S, into the cycles ones
w" C E rri, is given by equations of the form
(4)
where c = (il' i2, •.• , i., i1), s > 1, with il =1= ik, I, k = 1, ... , S, 1 =1= k, and t/I is a
function depending on i1, .•• , is' P, and n. The wc's have frequently physical
counterparts in what are called "through-variables."
To conclude, any irreducible (in general, recurrent) Markov chain ~ admits
two equivalent definitions. A first definition is given in terms of a stochastic
matrix P = (Pij) which in turn provides the edge-coordinates (niPi), and a
second definition is given in terms of the cycle-coordinates (we, C E rri).
To see how the edges and cycles describe the random law, we shall examine
the definitions of the niPij and We in the context of Kolmogorov's theory of
complexities as exposed in his last work with V.A. Uspensky "Algorithms
and Randomness" (see also A.N. Kolmogorov (1963, 1968, 1969, 1983a, b)
and V.A. Uspensky and A.L. Semenov (1993)).
Kolmogorov defined the entropy of a binary sequence using the concept of
complexity as follows: Given a mode (method) M of description, the com
plexity KM(Yn) of any finite string Yn = (0(0' 0(1' •.• , O(n-l), n;::: 1, under the
mode M is defined to be the minimal (space) length of adescription of Yn in
this mode (since there can be several descriptions with respect to M). (We
have to note here that there are two types oflengths: the time length and the
space length; see A.N. Uspensky and A.L. Semenov (1993), pp. 52-53). Then,
considering the class Jt of all computable modes of description of a set Y of
objects (to which Yn belongs), Kolmogorov proved that there is an optimal
mode 0 of description, not necessarily unique, which provides the shortest
possible descriptions, that is, Ko(Yn) ~ KM(Yn) + constant, for all ME Jt.
The complexity Ko(Yn) is called the entropy of Yn'
Now, turning back to our question of how the edges and cycles provide
descriptions of the probability distribution Prob(~k = i, ~k+l = j), i,j ES, we
shall examine the binary sequences assigned to this distribution. To this end
let us fix a pair (i, j) of states. Then for any k the probability
Prob(~k = i, ~k+l = j)
1
= lim - card {m :s; n - 1: ~m(O)) = i, ~m+1 (0)) = j} a.s. (5)
n-oo n
Preface
Vlll
can be assigned to an infinite binary sequence Y(i,j) = Y(i,j)(w) ==
(y(O), y(l), ... , y(m), ... ) whose coordinates are defined as
y(m) = {I, if the di.r ected pair (i,j) occurs on W at the time m; (6)
0, otherwlse;
where W is suitably chosen from the convergence set of (5). A directed pair
(i,j) occurs on trajectory W at moment m if ~m-l(W) = i and ~m(w) = j.
On the other hand, it turns out that the recurrent behavior of ~ deter
mines the appearances of directed circuits c = (i1, i2, ... , is' i1), S ~ 1, with
distinct points i1, i2, ... , is when s > 1, along the sampie paths, whose weights
are given by
Wc
Wc = lim ~ card {m ::;; n - 1: the cycle c appears, modulo the cyclic
n~CXJ n permutations, along w}, (7)
c.
almost surely, where m counts the appearances of the cycle
Equations (5) and (7) are connected by the following relation:
1 . . 1 .. en(w)
- card{m ::;; n - 1: ~m(w) = I, ~m+l (w) = j} = " --Wc n(W)Jc(/,j) + --, (8)
n "1n ' n
c
where ranges over the set ~n(w) containing all the directed cycles occurring
c
until n along w, wc,n(w°) denotes the number of the appearances of up to n
along w, and Bn(W) = or 1 according to whether or not the last step from i
to j corresponds or does not correspond to an edge of a circuit appearing up
to n. Then, we may assign the Y(i,j)(w) above to another description, say,
(0,0,0,0, 1,0,0, 1,0, ... ), where 1 codifies the appearances of a circuit pass
ing through (i,j) along (~k(w)h at certain moments.
Now we shall appeal to Kolmogorov's theory of complexities which, as we
have already seen, uses an object-description relation. Accordingly, the object
to be considered here will be the binary sequence Yn = (y(O), y(1), ... , y(n - 1))
whose coordinates are defined by (6), while the corresponding descriptions
will be expressed in terms of two modes of description as folIows.
One mode of description for the Yn will use the edges and will be denoted
by E. The corresponding description in the mode E for each finite string
Yn = (y(O), y(1), ... , y(n - 1)) is given by the binary sequence x = (x(O),
x(1), ... , x(n - 1)) whose coordinates are defined as
x(m) = y(m), m = 0, 1, ... , n - 1.
The second mode of description, denoted by C, is based on the directed
cycles, and the corresponding description of Yn above in the mode Cis given
by the sequence z = (z(O), ... , z(n - 1)) where
I, if a cycle passing through (i,j) occurs along w at moment m;
z(m) = { .
0, otherwlse;
for all m = 0, 1, ... , n - 1.
Preface ix
Nevertheless, it seems that another mode of description would be given by
the k-cells, k = 0, 1, 2, were we to extend the graph of ~ to the next higher
topological structure which is the corresponding 2-complex. But in this case
a serious drawback would arise: the descriptions in terms of the k-cells would
comprise surface elements (the 2-cells) so that no reasonable algorithmic
device would be considered. This motivates the choice of the mode C of
description in preference to that provided by the k-cells, k = 0, 1, 2, and
in this direction we find another two strengthening arguments. First, the
replacement of the 2-cells by their bounding circuits leaves invariant the
orthogonality equation of the boundary operators which act on the k-cells,
k = 0, 1,2; that is, the boundary operators connecting the homology sequence
circuits-edges-points will still satisfy the orthogonality equation. Then the
use of the 2-cells instead of the circuits becomes superfluous.
Second, the circuit-weights Wc given by (7) enjoy a probabilistic interpreta
tion: is the rnean nurnber of occurrences of c along almost all the sampie
Wc
paths of ~. Furthermore, the circuits (cycles) used in mode C can be deter
mined by suitable equations called cycle generating equations.
To conclude, the cycles and edges provide two methods of description
connected by equation (8). Under this light, cycle representation theory of
Markov processes is devoted to the study of the interconnections between the
edge-coordinates and cycle-coordinates along with the corresponding impli
cations for the study of the stochastic properties of the processes. Only after
the definition of the cycle representations for continuous parameter Markov
processes can the idea of separating the geometric (topological) ingredients
from their algebraic envelope become clear and lead to the investigations of
fine stochastic properties such as Levy's theorem concerning the positiveness
of the transition probabilities.
A systematic development of the fundamentals of the cycle theory, in the
spirit of Kolmogorov's algorithmic approach to chaos, started in the 1980s
at Thessaloniki from the idea of interconnecting the principles of algebraic
topology (network theory), algebra, convex analysis, theory of algorithms,
and stochastic processes. For instance, the resulting cycle-decomposition
formula pro vi des the homological dimension of Betti, the algebraic dimen
sion of Caratheodory, and the rotational dimension as new revelations of the
Markov processes.
Another school, which developed independently the cycle representations,
is that of Qians in Pe king (Qian Gong, Qian Minping, Qian Min, Qian
Cheng, Gong Guang, Guang-Lu Gong, and others). The Chinese school, us
ing mainly a behavioral approach, defined and explored with exceptional
completeness the probabilistic analogues of certain basic concepts wh ich rule
nonequilibrium statistical physics such as Hill's cycle flux, Schnakenberg's
entropy production, the detailed balance, etc. For instance, conceived as a
function on cycles, the entropy production can be regarded as a measure for
characterizing how far a process is from being reversible.