Almost sure limit sets of random samples in Rd

If {Xj,https://static.cambridge.org/binary/version/id/urn:cambridge.org:id: binary:20180209072205981-0599:S0001867800018152:S0001867800018152_inline 1.gif?pub-status=live } is a sequence of i.i.d. random vectors in Rd, when do there exist scaling constants bn > 0 such that the sequence of random sets https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:bina ry:20180209072205981-0599:S0001867800018152:S0001867800018152_inline3.gi f?pub-status=live converges almost surely in the space of compact subsets of Rd to a limit set? A multivariate regular variation condition on a properly defined distribution tail guarantees the almost sure convergence but without certain regularity conditions surprises can occur. When a density exists, an exponential form of regular variation plus some regularity guarantees the convergence.