Is there a canonical definition of "↗"?
I've seen it in a formulation of the Monotone convergence theorem as follows:
Let X and Xn be random variables for which Xn ↗ X ...
from which it would seem that "Xn ↗ X" means two things:
- The sequence Xn converges to X, and
- Xn+1 is greater or equal to Xn almost surely
although it is not clear which type of convergence is assumed.
This notation does indeed indicate convergence where $X_{n+1}\ge X_n$, but what kind of convergence needs to be specified, or it should be clear from the context. Likewise, whether the inequality is everywhere or just a.e. will depend on context, and should be specified if there is reasonable doubt. This notation can even be used if $X_n$ is a self-adjoint operator on a Hilbert space, with $\ge$ meaning that the difference is positive semidefinite. In short, it can be used in any context where there is both an ordering and a notion of convergence.