HW3solns - March 8, 2007 HW3 Problem Solutions #2.6.4. What...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: March 8, 2007 HW3 Problem Solutions #2.6.4. What we must show (since the finite constant factor 1 / ( A ) 6 = 0 does not affect the validity of the assertion) is: i, j A : i q ij = j q ji The equation holds trivially when i = j , and it follows immediately from the corresponding equation for P when i 6 = j . #3.1.5. Begin by defining G ( N ) = N n =0 P n , and then observing, using first-step analysis, that G ( N ) ij = I [ i = j ] + X k S P ik G ( N- 1) kj from which you can take the monotone nondecreasing extended-real-valued lim- its to prove the equation G = I + P G . By definition of G = lim N G ( N ) , it is easy to see that P G = GP , where the entries on both sides may be + . Therefore G = I + GP . However, if all entries of G are finite and the number | S | of elements in the state-space is finite, then this equation is equivalent to G ( I- P ) = I , which leads to a contradiction since post-multiplying by the vector 1 containing all 1s gives the zero-vector on the left-hand side but 1 on the right. Therefore the finite-by-finite matrix G has at least some infinite entries G ij , which implies that the corresponding state j is recurrent, and in the irreducible case this means that all states are recurrent. #3.2.1. As mentioned in the correction/hint, we do this verification only for n 1: by definition, for i 6 = 0, o p oi ( n + 1) = P ( k : k 6 =0 { X n = k, X n +1 = i } [ X 1 , . . . , X n- 1 6 = 0]) which by the Markov property is equal to = X k : k 6 =0 p k ( n ) P ki ....
View Full Document

This note was uploaded on 09/09/2009 for the course STAT 650 taught by Professor Slud during the Spring '09 term at Maryland.

Page1 / 3

HW3solns - March 8, 2007 HW3 Problem Solutions #2.6.4. What...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online