264 CHAPTER 10. MARKOV PROCESSES

where Ni satisfies Nrii = 0 for some ri > 0. It is clear that Ni (λiI) = (λiI)N and so

(Jri (λi))n=

n∑k=0

(n

k

)Nkλn−k

i =

r∑k=0

(n

k

)Nkλn−k

i

which converges to 0 due to the assumption that |λi| < 1. There are finitely many termsand a typical one is a matrix whose entries are no larger than an expression of the form

|λi|n−kCkn (n− 1) · · · (n− k + 1) ≤ Ck |λi|n−k

nk

which converges to 0 because, by the root test, the series∑∞

n=1 |λi|n−k

nk converges. Thusfor each i = 2, . . . , p,

limn→∞

(Jri (λi))n= 0.

By Condition 2, if anij denotes the ijth entry of An, then either

p∑i=1

anij = 1 or

p∑j=1

anij = 1, anij ≥ 0.

This follows from Lemma 10.1.2. It is obvious each anij ≥ 0, and so the entries of An mustbe bounded independent of n.

It follows easily from

n times︷ ︸︸ ︷P−1APP−1APP−1AP · · ·P−1AP = P−1AnP

thatP−1AnP = Jn (10.1)

Hence Jn must also have bounded entries as n → ∞. However, this requirement is incom-patible with an assumption that N ̸= 0.

If N ̸= 0, then Ns ̸= 0 but Ns+1 = 0 for some 1 ≤ s ≤ r. Then

(I +N)n= I +

s∑k=1

(n

k

)Nk

One of the entries of Ns is nonzero by the definition of s. Let this entry be nsij . Then this

implies that one of the entries of (I +N)nis of the form

(ns

)nsij . This entry dominates the

ijth entries of(nk

)Nk for all k < s because

limn→∞

(n

s

)/

(n

k

)= ∞

Therefore, the entries of (I +N)ncannot all be bounded. From block multiplication,

P−1AnP =

(I +N)

n

(Jr2 (λ2))n

. . .

(Jrm (λm))n

and this is a contradiction because entries are bounded on the left and unbounded on theright.