Let __\( M = \| m_{ij} \| \)__ be a __\( 4{\times}4 \)__ irreducible aperiodic Markov matrix such that __\( h_1 \neq h_2 \)__, __\( h_3 \neq h_4 \)__, where __\( h_i = m_{i1} + m_{i2} \)__. Let __\( x_1 \)__, __\( x_2, \dots \)__ be a stationary Markov process with transition matrix __\( M \)__, and let __\( y_n = 0 \)__ when __\( x_n = 1 \)__ or 2, __\( y_n = 1 \)__ when __\( x_n = 3 \)__ or 4. For any finite sequence
__\[ s = (\varepsilon_1____\( , \)__\varepsilon_2, \dots__\( , \)__\varepsilon_n) \]
of 0s and 1s, let
__\[ p(s) = \mathrm{Pr}\{y_1 = \varepsilon_1, \dots, y_n = \varepsilon_n\} .\]__
If
__\begin{equation*}\tag{1} p^2(00) \neq p(0)\,p(000) \quad\text{and}\quad p^2(01) \neq p(1)\,p(010), \end{equation*}__
the joint distribution of __\( y_1 \)__, __\( y_2, \dots \)__ is uniquely determined by the eight probabilities __\( p(0) \)__, __\( p(00) \)__, __\( p(000) \)__, __\( p(010) \)__, __\( p(0000) \)__, __\( p(0010) \)__, __\( p(0100) \)__, __\( p(0110) \)__, so that two matrices __\( M \)__ determine the same joint distribution of __\( y_1 \)__, __\( y_2, \dots \)__ whenever the eight probabilities listed agree, provided (1) is satisfied. The method consists in showing that the function __\( p \)__ satisfies the recurrence relation
__\[ p(s, \varepsilon, \delta, 0) = p(s, \varepsilon, 0)\,a(\varepsilon, \delta) + p(s, \varepsilon)\,b(\varepsilon, \delta) \]__
for all __\( s \)__ and __\( \varepsilon = 0 \)__ or 1, __\( \delta = 0 \)__ or 1, where __\( a(\varepsilon, \delta) \)__, __\( b(\varepsilon, \delta) \)__ are (easily computed) functions of __\( M \)__, and noting that, if (1) is satisfied, __\( a(\varepsilon, \delta) \)__ and __\( b(\varepsilon, \delta) \)__ are determined by the eight probabilities listed. The class of doubly stochastic matrices yielding the same joint distribution for __\( y_1 \)__, __\( y_2, \dots \)__ is described somewhat more explicitly, and the case of a larger number of states is considered briefly.