Check whether a Markov chain is stochastically monotone

A Markov chain on ordered states is stochastically monotone if starting from a higher state never makes the next state stochastically smaller: for every pair of consecutive states \(i\), \(i + 1\) and every \(j\), \(\sum_{l \le j} P_{i+1, l} \le \sum_{l \le j} P_{i, l}\).

Usage

is.stochasticallyMonotone(object)

# S4 method for class 'markovchain'
is.stochasticallyMonotone(object)

# S4 method for class 'matrix'
is.stochasticallyMonotone(object)

# S4 method for class 'ANY'
is.stochasticallyMonotone(object)

Arguments

object

A markovchain object, or a row-stochastic transition matrix.

Value

TRUE if the chain is stochastically monotone, FALSE otherwise.

Details

The states are taken in the order of the transition matrix, and the inequalities are checked up to a tolerance of \(10^{-10}\).

References

Keilson, J. and Kester, A. (1977). Monotone matrices and monotone Markov processes. Stochastic Processes and their Applications 5(3), 231-241.

Examples

# a birth-death chain is monotone
P <- matrix(c(0.7, 0.3, 0,
              0.2, 0.5, 0.3,
              0, 0.4, 0.6), nrow = 3, byrow = TRUE,
            dimnames = list(c("low", "mid", "high"), c("low", "mid", "high")))
is.stochasticallyMonotone(new("markovchain", transitionMatrix = P))
#> [1] TRUE
# reversing the order of the rows breaks monotonicity
is.stochasticallyMonotone(P[3:1, ])
#> [1] FALSE