  A virtual Markov chain (VMC) is a sequence $\{X_N\}_{N=0}^{\infty}$ of Markov
chains (MCs) coupled together on the same probability space such that $X_N$ has
state space $\{0,1,\ldots, N\}$ and such that removing all instances of $N~+~1$
from the sample path of $X_{N+1}$ results in the sample path of $X_N$ almost
surely. In this paper, we prove an exact characterization of the triviality of
the $\sigma$-algebra $\bigcap_{N=0}^{\infty}\sigma(X_N,X_{N+1},\ldots)$. The
main tool for doing this is a decomposition theorem that the $\sigma$-algebra
generated by a VMC is equal to the $\sigma$-algebra generated by a certain
countably infinite collection of independent constituent MCs. These
constituents are so-called staircase MCs (SMCs), which are defined to be
inhomoheneous Markov chains on the non-negative integers which transition only
by holding or by jumping to a value equal to the current index. We also develop
some general aspects of the theory of SMCs, including a connection with some
classical but very much under-appreciated aspects of convex analysis.
