Epilogue to Volume III: How Far Can a Rule Take Us?

A fair game initially offered only a promise about the next step. Following time onward exposed its limits: conditional means may remain unchanged at every finite time while the eventual random outcome loses part of the expectation. A player choosing when to leave is not merely replacing one clock reading with another. What was observed and what was risked before stopping are part of the probability relation.

Sometimes the present can summarize the past, but that present must be chosen. The latest observation may not suffice, while retaining every detail can obscure the useful structure. A Markov state does not make the world forget. It gathers the parts of the past that still matter to prediction into an object from which prediction can continue. Once such a state is found, a rule can be reused, but return is not guaranteed, nor is existence beyond every finite time.

Where observation takes place also changes the answer. Recording one cycle at each return differs from stopping the clock at an observation time, which gives longer intervals more opportunity. Recording one state per jump likewise differs from sampling its holding time. In a random family, size bias makes rare large families visible in proportion to their contribution to the mean population. These weights are not one formula to apply indiscriminately. They expose the need to specify how a supposedly typical outcome has been selected.

We also learned to take restarting seriously. Returning to the same position, completing a wait and having a residual wait that forgets age rely on different mathematical conditions. Exponential clocks make some restarts compatible; a change of state gives the next clock a different speed. Even when every step is valid, the sum of all waits may be finite. Local rules, path lifetime and continuation across a boundary have become three questions that can no longer be conflated.

These distinctions do not weaken judgments about the future; they make those judgments trustworthy. We know which average relations survive stopping, which states retain predictive information, which restarts truly supply an independent future, and which conclusions require further integrability, nonexplosion or boundary conditions. Determining how a path continues now requires more than writing down a distribution or a matrix.

Our records still consist of successive holds and jumps. If changes become smaller and denser, can they yield a continuous motion? Once all finite collections of observations have their joint laws, does continuity follow as well? Volume III ends here. What must next be constructed and compared is the entire random path itself.