The definition of stochastic monotonicity and convexity is given by "Stochastic Orders and Their Applications" by Moshe Shaked and George Shanthikumar (1994) as:
Let $P = \{p_{i,j} \}$ be a stochastic matrix on $\mathbb{N}_+$. Then P is said to be
1) Stochastically monotone if $\sum_{j=k}^\infty p_{i,j} \leq \sum_{j=k}^\infty p_{(i+1),j}$ for any $i$ and $k$ in $\mathbb{N}_+$;
2) Stochastically monotone convex $\sum_{j=k}^\infty p_{i,j} \leq \sum_{j=k+1}^\infty p_{(i+1),j}$ for any $i$ and $k$ in $\mathbb{N}_+$;
That is, the row vectors of the matrix are stochastically increasing. Many results follow from this definition. For example, one can show stochastic ordering between two transition matrices of discrete-time Markov chains. This in turn, implies that their respective steady-state probability distributions are ordered. I want to show that similar properties hold for the steady-state probabilities in my system. However, my problem is defined slightly differently.
Let $Q = \{q_{i,j} \}$ be my stochastic matrix on $\mathbb{N}_+$. In my system, the following relations hold:
1) $\sum_{j=k}^\infty q_{i,j} \geq \sum_{j=k}^\infty q_{(i+1),j}$ for any $i$ and $k$ in $\mathbb{N}_+$;
2) $\sum_{j=k}^\infty q_{i,j} \geq \sum_{j=k+1}^\infty q_{(i+1),j}$ for any $i$ and $k$ in $\mathbb{N}_+$;
Note that the $\leq$ sign in the original definition is replaced by a $\geq$ sign in my system. I have tried redefining the underlying state space, using a coupling argument, and tried to find a bounding matrix that satisfies stochastic monotonicity and convexity but also has the same steady-state distribution as the one induced by Q, all to no avail. If anybody has any ideas and/or could provide me with some help in this matter it would be much appreciated.
If some $Q$ satisfies your property (1) then $Q^2$ is stochastically monotone.