Composition and mixture of Markov operators
Citation for published version (APA):Simons, F. H. (1976). Composition and mixture of Markov operators. (Memorandum COSOR; Vol. 7616). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1976
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EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics
PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP
Memorandum COSOR 76-16
composition and mixture of Markov operators
by
F.H. Simons
Eindhoven, October 1976
composition and mixture of Markov operators
by
F.H. Simons
Let (X,I,m) be a a-finite measure space and let Sand T be conservative Markov operators on (X,L,m). For the very definitions and notations we
refer to the book of Foguel [IJ.
One might wonder whether the Markov processes ST and
as
+ eT, 0 <a,3
<I,
a.+ e= 1 are again conservative Markov processes. This is the case for instance in the situation when S = T. However in general both statements are false, as we shall show in the next two examples.
EXAMPLE 1. Let X
=
{1,2,3},r
be the power set of X, and m be the counting measure. Consider 0 0 1 1 13"
3 3 S = 1 1 1 T 0 03"
33"
= 0 0 1 1 I 3 33"
I~
S S. I f we start instate 1, under ST we are at state 3 at time 1, and stay
there for ever. Hence the process ST cannot be recurrent, while S and T are irreducible processes on a finite state space, and therefore re-current. We can show this also in a more formal manner.
The process S has (I,3,1) as an equivalent finite invariant measure,
and therefore is conservative. The same reasoning holds for the process T, which has (1,1,2) as an equivalent finite invariant measure. The process
ST is given by the matrix
o
0 2"9
"9
2"9
52
-Then for every a e: Il, b e: :R we have
Bence 2 2 (a,a,b) (ST)
=
('9
a,'9
a, aDI
(1,1,1) (ST)n= (
1 2 ~O 1--9 14"'9
a + b) •Since (I, 1,1) e: £'1' i t follows that the conservative part of X with respect to
ST
is{3},
and the dissipative part is{1,2}.
In this example the two processes Sand T do not cOlllllute, Le. ST
:I
TS. The next example will show that even if ST=
TS and S and T are con-servative, the operators ST and as + BT, with 0 < a,S < 1, a +13= 1 can be dissipative.EXANPLE 2. Let X
=
%3,L
be the power set of X and m be the counting measure.Let the random walk S on X be given by
1 if (x,y, z) (a +
1, b, c)
4'
=(a - 1, b, c)
SI{ b }(x,y,z) = (a, b + 1, c)
a, ,c
(a, b - I , c)
o
otherwise.Then each set {z = c} is invariant under S, and S is a two-dimensional random walk on {z
=
c} with ~=
0,
and therefore, by[2],
8TI, the random walk S is recurrent onX.
Similarly we define the random walk T on X by
1
~
if
(x,y,z}=
(a, b,e
+ 1)( a , b , c - l ) TI{a, ,cb
3
-By an aualogous reasonina we have that T is recurrent on X. A straight-forward caaputation now shows that
I
in each of the points 8
STI
=
TSI=
(a ± I, b, c ± I){a,b,c} {a,b,c}
(a, b ± I, c ±
I),
0 otherwise
Bence ST is an irreducible raDdca walk on
z3
aDd therefore by [2] 8I1transient.
I f 0 < u,B < I aDd u + B
=
I, thenI I
in each of the points
-u+ -B
8
4
(uS + BT)I{ b } = (a ± I, b, c ± 1) a, ,c (a, b ± I, c ± I) 0 otherwise.Hence uS + BT is an irreducible random walk on
iJ
and therefore transient. In the last example the state space X of the process is infinite. This is rather essential in order to show that a convex combination of two conservative processes can be dissipative.Indeed, if the state space is finite, and S and T are recurrent, then P
=
as
+ BT is recurrent as well. This can easily be seen as follows. Put i!
j if under the process P the probability that, starting in the state i, the state j is reached in finitely many steps is positive. Then, if i!
j, there exists a finite sequence of states iO
=
i, i l, .•.,in=
j such that T i -+ i n-l n4 -T i -+ i -+ ••• n n-I T and therefore j ~ i •
Hence the state space X can be decomposed into finitely_many irreducible classes under P, each consisting of finitely many states. Then P is recurrent on each of this classes, and therefore P is recurrent on X.
References
[I] Foguel, S.R. : The ergodic theory of Markov processes. Van Nostrand
Mathematical Studies
H
21, New York 1969.[2] Spitzer, F. : Principles of random walk. Van Nostrand, Princeton
