Random processes and the differential equation method

Suppose the state coordinates Y_i(k) are of order n and one discrete update changes them by order one. Then the natural scaled state is Y_i/n and natural time is k/n. If the conditional expected increment is approximately a function f_i of scaled time and scaled state, an update changes the scaled state by approximately f_i/n. Dividing by the time increment 1/n suggests derivative f_i. A missing factor of n here creates a different ODE rather than a harmless notation change.

Consider independently placing balls, one at a time, into n uniformly chosen bins. Let U(k) count empty bins. Given the entire placement history, the next ball hits an empty bin with probability U(k)/n. Therefore U decreases by one with that probability and stays fixed otherwise. The conditional drift is exactly minus U/n. With u(0)=1 the proposed normalized equation is u′=−u, whose solution is exponential decay. The process remains random; the ODE describes its large-system trajectory.

Here is a restricted sufficient framework on a fixed finite time interval. Consider a fixed finite number of adapted coordinates bounded by a constant times n. Their initial normalized vector converges in probability to the specified deterministic ODE initial value y(0). Before exit from a fixed bounded domain, all coordinate jumps are bounded almost surely by a constant independent of n, and all conditional drifts differ from the corresponding f_i by an error tending uniformly to zero. The functions f_i are bounded and Lipschitz on a neighborhood of the candidate trajectory, including its time coordinate.

Assume the ODE solution exists through the fixed terminal time and stays a positive distance from the domain boundary. Under those hypotheses the maximum normalized tracking error over all discrete times through n times that terminal time tends to zero in probability. The martingale part has sublinear cumulative fluctuation by bounded-jump concentration, while the drift approximation and discretization errors accumulate sublinearly. A discrete Gronwall estimate prevents Lipschitz feedback from amplifying these errors to macroscopic size on a fixed interval. This is a bounded-jump specialization, not the most general form of Wormald’s theorem.

If the ODE approaches a singularity, a depleted resource, or the boundary where the drift formula changes, the previous uniform theorem cannot simply be extended by drawing a longer curve. Stop before that region and verify a new set of estimates if a later regime is needed. Likewise a bound allowing rare large jumps needs an explicit exceptional-event analysis; average small increments are not the same as almost-sure bounded increments. These are common points where a persuasive heuristic fails to become a theorem.

For empty bins the exact expectation is $n(1-1/n)^k$, because each specified bin is missed by all k independent balls. This expectation approaches $ne^{-k/n}$ on fixed scaled intervals, but an expectation calculation alone would not show that an entire realized trajectory stays close. The bounded-jump tracking argument supplies that additional probability statement. Neither result says the bin-empty indicators are independent: their dependence is compatible with a simple conditional drift and controlled fluctuations of the total count.