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.
