Foraprobabilityspace (ΩBox,E,PBox),withΩBox Rd,theGaussian-boxprocessisgeneratedas µi ΩBox, σi Rd+, r Rd+ Ci N(µi,σi), Xi, =Ci+ri, Xi, =Ci ri, Box(Xi) = dY

All coordinates will be modeled by independent Gumbel distributions, and thus it is enough to calculate the expected side-length of a box as the expected volume will simply be the product of the expected side-lengths. To properly restrict the Gumbel distributions to[0,1], we can either formcensoredortruncated distributions. Thetruncateddistribution,ontheotherhand,multipliesthe densities with the indicator function for[0,1]and renormalizes them to integrate to 1. The higher the temperature of the boxes, the more the true integral will tend to provide larger conditional probabilities. Monte Carlo experiments support this conclusion.