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A function of the current state of a Markov chain can discard information needed to predict its next value. A sufficient condition preserving the Markov property is strong lumpability: all original states with the same recorded value must give the same transition probabilities to every recorded class.

For the two-column model in growth rate of two-column sticky deposition, the pair of column heights is Markov, but its maximum alone is not. Starting empty, the maximum-height histories $(1,1,2)$ and $(1,2,2)$ both end at height two. The former leaves unequal columns, so the next height increases with probability $1/2$; the latter leaves equal columns, so it increases with probability one. Each history has positive probability. Their different predictions at the same time and current value violate the Markov property. By contrast, the absolute height difference retains exactly the information needed for its own transitions.

The precise conclusion isTo prove it, observe that the maximum rises exactly when the new height difference is positive:A block landing in a shorter column produces $D_m=0$ and no rise; every other case raises the maximum and leaves $D_m>0$. Starting from $H_0=0$, telescoping givesThe difference chain is irreducible and positive recurrent by part (i). Apply the countable-state Markov chain ergodic theorem to the bounded indicator function $g(k)=1_{\{k>0\}}$. Its stationary mean is ${\sum }_{k\ge 1}{\pi }_k=1-{\pi }_0=2/3$, giving the asserted almost sure convergence. No aperiodicity assumption is needed for this time average. This establishes the growth rate of two-column sticky deposition.