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Examples and further patterns

A4-1 sums of independent random variables

Suppose Y N is an independent, integrable sequence. Set X n = k = 0 n Y k n 0 .

If E [ Y n ] ( ) 0 n 1 , then X N is a (S)MG.

A4-2 products of nonnegative random variables

Suppose Y N Z N , Y n 0 a . s . n . Consider X N : X n = c k = 0 n Y k , c > 0 .

If E [ Y n + 1 | W n ] ( ) 1 a . s . n , then ( X N , Z N ) is a (S)MG

X n W n and X n + 1 = Y n + 1 X n . Hence, E [ X n + 1 | W n ] = X n E [ Z n + 1 | W n ] ( ) X n a . s . n

A4-3 discrete random walk

Consider Y 0 = 0 and { Y n : 1 n } iid. Set X n = k = 0 n Y k n 0 . Suppose P ( Y n = k ) = p k . Let

g Y ( s ) = E [ s Y n ] = k p k s k , s > 0

Now g Y ( 1 ) = 1 , g Y ' ( 1 ) = E [ Y n ] , g Y ' ' ( s ) = k k ( k - 1 ) p k s k - 2 > 0 for s > 0 . Hence, g Y ( s ) = 1 has at most two roots, one of which is s = 1 .

  1. s = 1 is a minimum point iff E [ Y n ] = 0 , in which case X N is a MG (see A4-1 )
  2. If g Y ( r ) = 1 for 0 < r < 1 , then E [ r Y n ] = 1 n 1 . Let Z 0 = 1 , Z n = r X n = k = 1 n r Y k . By A4-2, Z N is a MG

For the MG case in Theorem IXA3-6 , the Y n are centered at conditional expectation; that is

E [ Y n + 1 | W n ] = 0 a . s . The following is an extension of that pattern.

A4-4 more general sums

Consider integrable Y N Z N and bounded H N Z N . Let W n = a constant for n < 0 and H n = 1 for n < 0 . Set

X n = k = 0 n { Y k - E [ Y k | W k - 1 ] } H k - 1 n 0

Then ( X N , Z N ) is a MG.

X n W n ; n 0 and E [ X n + 1 | W n ] = X n + H n E { Y n + 1 - E [ Y n + 1 | W n ] | W n } = X n + 0 a . s .

IXA4-2

A4-5 sums of products

Suppose Y N is absolutely fair relative to Z N , with E [ | Y n | k ] < n , fixed k > 0 . For n k , set

X n = 0 i 1 < n Y i 1 Y i 2 Y i k G n

Then ( X N k , Z N k ) N k = { k , k + 1 , k + 2 , } is a MG,

X n + 1 = X n + K n + 1 , where

K n + 1 = Y n + 1 0 i 1 < n Y i 1 Y i 2 Y i k - 1 = Y n + 1 K n * K n * W n
E [ K n + 1 | W n ] = K n * E [ Y n + 1 | W n ] = 0 a . s . n k

We consider, next, some relationships with homogeneous Markov sequences .

Suppose ( X N , Z N ) is a homogeneous Markov sequence with finite state space E = { 1 , 2 , , M } and transition matrix P = [ p ( i , j ) ] . A function f on E is represented by a column matrix f = [ f ( 1 ) , f ( 2 ) , , f ( M ) ] T . Then f ( X n ) has value f ( k ) when X n = k . P f is an m × 1 column matrix and P f ( j ) is the j th element of that matrix. Consider E [ f ( X n + 1 ) | W n ] = E [ f ( X n + 1 ) | X n ] a . s . . Now

E [ f ( X n + 1 ) | X n = j ] = k E f ( k ) p ( j , k ) = P f ( j ) so that E [ f ( X n + 1 ) | W n ] = P f ( X n )

A nonnegative function f on E is called (super)harmonic for P iff P f ( ) f .

A4-6 positive supermartingales and superharmonic functions.

Suppose ( X N , Z N ) is a homogeneous Markov sequence with finite state space E = { 1 , 2 , , M } and transition matrix P = [ p ( i , j ) ] . For nonnegative f on E , let Y n = f ( X n ) n N . Then ( Y N , Z N ) is a positive (super)martingale P(SR)MG iff f is (super)harmonic for P .

As noted above E [ f ( X n + 1 ) | W n ] = P f ( X n ) .

  1. If f is (super)harmonic P f ( X n ) ( ) f ( X n ) = Y n , so that
    E [ Y n + 1 | W n ] ( ) Y n a . s .
  2. If ( Y N , Z N ) is a P(SR)MG, then
    Y n = f ( X n ) ( ) E [ f ( X n + 1 ) | W n ] = P f ( X n ) a . s . , so that f is (super)harmonic

IX A4-3

An eigenfunction f and associated eigenvalue λ for P satisfy P f = λ f (i.e., ( λ I - P ) f = 0 ). In most cases, | λ | < 1 . For real λ , 0 < λ < 1 , the eigenfunctions are superharmonic functions. We may use the construction of Theorem IXA3-12 to obtain the associated MG.

A4-7 martingales induced by eigenfunctions for homogeneous markov sequences

Let ( Y N , Z N ) be a homogenous Markov sequence, and f be an eigenfunction with eigenvalue λ . Put X n = λ - n f ( Y n ) . Then, by Theorem IAXA3-12 , ( X N , Z N ) is a MG.

A4-8 a dynamic programming example.

We consider a horizon of N stages and a finite state space E = { 1 , 2 , , M } .

  • Observe the system at prescribed instants
  • Take action on the basis of previous states and actions.

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Source:  OpenStax, Topics in applied probability. OpenStax CNX. Sep 04, 2009 Download for free at http://cnx.org/content/col10964/1.2
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