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The difference process $D_n$ is a Markov chain. If the columns are equal, the next block creates a height difference of one. If the difference is $k\ge 1$, landing in the taller column increases it to $k+1$, while landing in the shorter column makes it zero. Therefore its only nonzero transition probabilities areThese probabilities depend only on the present difference, not on the absolute heights or their history. This is the reset-or-increment Markov chain with $p=1/2$.

For its stationary distribution, the incoming-flow equations give ${\pi }_1={\pi }_0$ and ${\pi }_k={\pi }_{k-1}/2$ for $k\ge 2$. Normalization uses a geometric series:ThusThe remaining balance equation is also satisfied: the flow into zero is $(1-{\pi }_0)/2=1/3$. Every state can reach zero and every positive state can be reached from zero, so it is an irreducible Markov chain. Its invariant probability distribution makes it positive recurrent, and that distribution is unique.

The maximum-height process $H_n$ is not Markov. Starting empty, condition first on the positive-probability history $(H_1,H_2,H_3)=(1,1,2)$. After two blocks the columns were equal, so after three their difference is one. Hence $\mathbb{P}(H_4=3\mid H_1=1,H_2=1,H_3=2)=1/2$.

For the history $(1,2,2)$, the first two blocks built one column and the third caught the other column up, leaving equal heights. Hence $\mathbb{P}(H_4=3\mid H_1=1,H_2=2,H_3=2)=1$. Both histories have the same time and present height $H_3=2$, but different predictions. This violates the Markov property and illustrates why a function of a Markov chain need not be Markov.