A fundamental class of results in probability theory are limit theorems, of which the Central Limit Theorem is the most familiar example. They are usually proved using Fourier transforms. An alternative approach using operator theory goes back to Feller. I will use this approach to prove limit theorems in ``monotonic probability theory''; at its heart this is simply a result about composition of analytic functions. No probabilistic background (monotonic or otherwise) will be assumed or required. This is joint work with John Williams.