arXiv:2610.29743v1 Announce Type: cross Abstract: We model insurance loss in each period as deterministic exposure multiplied by a nonnegative function of a long-memory Gaussian sequence. A weighted law of the iterated logarithm yields the exact almost-sure scale and sharp constant of centered cumulative loss. Loss per unit of cumulative exposure converges almost surely to its mean, with an exact error scale common to all admissible exposure profiles; the sharp constant depends on the exposure exponent. Lognormal and quadratic examples, of Hermite ranks one and two, and a simulation illustrate the results. A known spectral formula expresses the quadratic coefficient through a leading eigenvalue.