Consider a monotonically decreasing series of step sizes, { h$_{j}$ } ∞ j $_{=0}$, and note 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}$$_{j}$ ( x x, t ) } ∞ j $_{=0}$, and for which P ( H ≥ j ) > 0 for all j ≥ 0. Define where ω$_{j}$ = 1 ⧸ P ( H ≥ j ), and note that where $^{I$_{j}$}$≤ H is the indicator function 18 $^{I$_{H}$}$≥ j =    $^{1}$ if H ≥ j, 0 otherwise . E { u † ( x x, t ) } = E { H ∑ j =0 ω$_{j}$ ˜ Δ$_{j}$ } = E { ∞ ∑ j =0 $^{I$_{H}$}$≥ $_{j}$ω$_{j}$ ˜ Δ$_{j}$ } = ∞ ∑ j =0 E { $^{I$_{H}$}$≥ j } ω$_{j}$E { ˜ Δ$_{j}$ } = ∞ ∑ j =0 E { ˜ Δ$_{j}$ } = u ( x x, t ) , u † ( x x, t ) = H ∑ j =0 ω$_{j}$ ˜ Δ$_{j}$ , u ( x x, t ) = E { ˜ u$_{h}$$_{0}$ ( x x, t ) } + ∞ ∑ j =1 E {( ˜ u$_{h}$$_{j}$ ( x x, t ) − ˜ u$_{h}$$_{j}$$_{−}$$_{1}$ ( x x, t ) )} = ∞ ∑ j =0 E { ˜ Δ$_{j}$ } ,