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Consider a monotonically decreasing series of step sizes, { h$_{j}$ } ∞ j $_{=0}$, and note 

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where we have denoted ˜ Δ$_{j}$ = ˜ u$_{h}$$_{j}$ ( x x, t ) − ˜ u$_{h}$$_{j}$$_{−}$$_{1}$ ( x x, t ), and defined ˜ u$_{h}$$_{−}$$_{1}$ ( x x, t ) = 0. It is possible to obtain an unbiased estimator of u ( x x, t ) by drawing a finite halting value, H , from some probability distribution P ( H ) that is indepen- dent of { ˜ u$_{h}$$_{j}$ ( x x, t ) } ∞ j $_{=0}$, and for which P ( H ≥ j ) > 0 for all j ≥ 0. Define 

where ω$_{j}$ = 1 ⧸ P ( H ≥ j ), and note that 

where $^{I$_{j}$}$≤ H is the indicator function 

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