Foundations of the theory of probability

§

6. ConditionalProbabilitiesasRandomVariables,

MarkovChains

13

P

m

(A?)=

P(Af)

q=\,2,...,r

2

.

Givenanydecompositions (experiments) 5l

(1)

,

5l

(2)

,

...

,

9l

(n)

,

we

we

shallrepresentby

2l

(1

>2l

(2)

..

.$

(»>

thedecomposition

ofsetEintotheproducts

Experiments3i

(1

\

2l

(2)

,...

,

%

(n)

are

mutuallyindependentwhen

andonlywhen

p

gB1

,

a

,»...p.

1

,(4»)

=P(4'),

kand

q

beingarbitrary

14

.

Definition: Thesequence

3l

(1)

,$

(2)

,

...

,5l

(n)

,

...forms

aMarkovchain ifforarbitraryn

and

q

P«»>«<«

...w-«>

W)

=Pa(n-D(4

n)

).

Thus, Markov chains form a natural generalization of se-

quencesofmutuallyindependentexperiments.Ifwe

set

pQmgn

(m,n)=

P

A

™

(A™)

m<n

,

thenthebasicformulaofthetheoryofMarkovchainswillassume

theform:

pQkqn

(k>n)==

*Zpqkqm

(k,m)

pgmqH

(m,n)

y

k<m<n.

(1)

Qm

Ifwedenotethematrix

\\pqmgn

(nt,n)\\

by

p(m,ri),

(1)

canbe

writtenas

15

:

p(k,n)

—

p(k,m)p(m,n) k<m<n.

(2)

14

ThenecessityoftheseconditionsfollowsfromTheoremII,

§ 5

;thatthey

arealso sufficient follows immediatelyfrom the Multiplication Theorem

(Formula

(7)

of

§4).

16

ForfurtherdevelopmentofthetheoryofMarkovchains,seeR.v.Mises

[1],§16,

andB.Hostinsky,Methodesgenerates

ducalculdes

probabilites,

"Mem.Sci.Math."V.52,Paris1931.