# Lemma 118 easily settles the question for the

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Unformatted text preview: hile the period of y is d &lt; c. Let k be such that pk (x, y ) &gt; 0 and let m be such that pm (y, x) &gt; 0. Since pk+m (x, x) pk (x, y )pm (y, x) &gt; 0 we have k + m 2 Ix . Since x has period c, k + m must be a multiple of c. Now let ` be any integer with p` (y, y ) &gt; 0. Since pk+`+m (x, x) pk (x, y )p` (y, y )pm (y, x) &gt; 0 k + ` + m 2 Ix , and k + ` + m must be a multiple of c. Since k + m is itself a multiple of c, this means that ` is a multiple of c. Since ` 2 Iy was arbitrary, we have shown that c is a divisor of every element of Iy , but d &lt; c is the greatest common divisor, so we have a contradiction. Lemma 1.18 easily settles the question for the inventory chain (Example 1.6) 012345 0 0 0 .1 .2 .4 .3 1 0 0 .1 .2 .4 .3 2 .3 .4 .3 0 0 0 3 .1 .2 .4 .3 0 0 4 0 .1 .2 .4 .3 0 5 0 0 .1 .2 .4 .3 Since p(x, x) &gt; 0 for x = 2, 3, 4, 5, Lemma 1.17 implies that these states are aperiodic. Since this chain is irreducible it follows from Lemma 1.18 that 0 and 1 are aperiodic. Con...
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