lazyChain: Build a lazy version of a Markov chain

In markovchain: Easy Handling Discrete Time Markov Chains

Build a lazy version of a Markov chain

Description

Constructs the "lazy" chain associated with a markovchain object: at every step, stay put with probability alpha and otherwise take a step of the original chain.

Usage

lazyChain(object, alpha = 0.5)

## S4 method for signature 'markovchain'
lazyChain(object, alpha = 0.5)

Arguments

object	A markovchain object.
alpha	A single number in [0,1], the probability of staying in the current state at each step. The default, 0.5, matches the usual textbook "lazy random walk" construction.

Details

For a transition matrix P (in whichever storage convention object already uses) and a laziness parameter \alpha\in[0,1], the lazy chain's transition matrix is

L = \alpha I + (1-\alpha) P.

Laziness is a standard device for forcing aperiodicity without changing where the chain can go or its stationary distribution:

• L has the *same* stationary distribution as P (if \pi P=\pi then \pi L = \alpha\pi + (1-\alpha)\pi P = \pi), and the same communicating classes, since L_{ij}>0 \iff P_{ij}>0 for i\ne j.

• For 0<\alpha<1, L is aperiodic even if P is periodic, because L_{ii}=\alpha>0 for every state i rules out any period greater than 1. This is why mixingTime, which requires aperiodicity, is often applied to lazyChain(object) rather than to a periodic object directly (see mixingTime's own documentation for why it rejects periodic chains outright rather than lazifying them automatically).

• Every non-trivial eigenvalue of L is \alpha + (1-\alpha)\lambda for the corresponding eigenvalue \lambda of P: laziness shrinks the whole non-trivial spectrum towards \alpha, so slem and impliedTimescales generally get *worse* (mixing gets slower) as alpha increases towards 1.

\alpha=0 returns P unchanged; \alpha=1 returns the identity matrix (a chain that never moves).

Value

A new markovchain object with transition matrix L = \alpha I + (1-\alpha) P, the same states, and the same row/column-stochastic storage convention (byrow) as object.

References

Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times, 2nd edition. American Mathematical Society.

See Also

subchain, mixingTime, slem

Examples

# A 2-cycle is periodic (period 2); its lazy version is aperiodic.
statesNames <- c("a", "b")
cycle2 <- new("markovchain", states = statesNames,
  transitionMatrix = matrix(c(0, 1, 1, 0), byrow = TRUE, nrow = 2,
                            dimnames = list(statesNames, statesNames)))
period(cycle2)
lazyCycle2 <- lazyChain(cycle2, alpha = 0.5)
period(lazyCycle2)
steadyStates(cycle2)
steadyStates(lazyCycle2) # unchanged by laziness

markovchain documentation built on Oct. 10, 2026, 9:07 a.m.