High-Precision Framework for Expected Hitting Times Analysis in the Dice-Sum Process
Provides a highly precise numerical method for a specific stochastic process, but the problem is niche and the result is incremental.
The authors develop a framework for computing expected hitting times in a dice-sum process, achieving the expected time for perfect squares as 7.079764... with 1,017 decimal places of provable accuracy.
We study the expected number of rolls required for the cumulative sum of a fair six-sided die to first enter a prescribed target set $H\subset\mathbb{Z}_{\ge0}$. A one-variable dynamic-programming formulation is introduced that removes dependence on the roll count. Within this framework, the infinite process is truncated at a large cutoff $N$ and corrected by an analytically derived overshoot term that accounts for the rare event of exceeding $N$ before entering $H$. Explicit bounds on this residual yield a strict two-sided estimate of the truncation error. The method is numerically efficient, requiring constant memory and linear time in the cutoff. For the perfect-square target set $H=\{n^2:n\in\mathbb{N}\}$, all quantities are evaluated explicitly, yielding \[ \mathbb{E}[T]=7.07976423755110510389555305690818489468\ldots, \] provably correct to 1,017 decimal places. This constitutes the most precise result known to date and establishes a general framework for high-accuracy computation of discrete hitting times.