Stochastic Processes and Related Topics pp 253-273 | Cite as

# Cycle Representations of Markov Processes: An Application to Rotational Partitions

## Abstract

The cycle formula asserts that any finite-order recurrent stochastic matrix *P* is a linear combination of matrices *J* _{ c } = (*J* _{c}(*i, j*)) associated with the cycles *c* of the graph of *P*, and defined as follows: *J* _{ c }(*i, ii*) = 1 or 0, according to whether *i, j* are consecutive points of *c* or not.

In the present paper we investigate, by using the cycle formula, the asymptotic behavior of the sequence (*t*, ^{m} *S*), *t* ⩾ 0, of rotational representations associated to the powers *P* ^{m}, *m* = 0, 1, 2,…, of an irreducible stochastic matrix on a finite set *S* = {1,…, n}, n ⩾ 2. In particular, we give a criterion on the rotational partitions ^{m} *S*, *m* = 0, 1,…, for the sequence *P* ^{m} _{ m } to be convergent. A pair (*t*, *S*) is a rotational representation of *P* = (*p* _{ij}, i, j = 1,…, *n*) if *S* is a partition of [0, 1) into *n* sets *S* _{1},…, *S* _{n}, each of positive Lebesgue measure and consisting of a finite union of arcs, such that P_{ij} = . Here *f* _{ t } is the A-preserving transformation of [0, 1) onto itself defined by *f* _{ t }(*x*) = (*x* + *t*)(mod 1), and A denotes Lebesgue measure

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