▁obtained ▁by ▁I ID ▁sampling , ▁by ▁the ▁K oks ma - H law ka ▁inequality . ▁Emp ir ically ▁it ▁allows ▁the ▁reduction ▁of ▁both ▁estimation ▁error ▁and ▁convergence ▁time ▁by ▁an ▁order ▁of ▁magnitude . ▁The ▁Array - R Q MC ▁method ▁comb ines ▁random ized ▁quasi - Mon te ▁Carlo ▁and ▁Mark ov ▁chain ▁simulation ▁by ▁sim ulating ▁ ▁ch ains ▁simultaneously ▁in ▁a ▁way ▁that ▁the ▁empir ical ▁distribution ▁of ▁the ▁ ▁states ▁at ▁any ▁given ▁step ▁is ▁a ▁better ▁approximation ▁of ▁the ▁true ▁distribution ▁of ▁the ▁chain ▁than ▁with ▁ordinary ▁MC MC . ▁In ▁empir ical ▁experiments , ▁the ▁variance ▁of ▁the ▁average ▁of ▁a ▁function ▁of ▁the ▁state ▁sometimes ▁converges ▁at ▁rate ▁ ▁or ▁even ▁faster , ▁instead ▁of ▁the ▁ ▁Monte ▁Carlo ▁rate . ▁ ▁Con ver gence ▁ ▁Usually ▁it ▁is ▁not ▁hard ▁to ▁construct ▁a ▁Mark ov ▁chain ▁with ▁the ▁desired ▁properties . ▁The ▁more ▁difficult ▁problem ▁is ▁to ▁determine ▁how ▁many ▁steps ▁are ▁needed ▁to ▁conver ge ▁to ▁the ▁station ary ▁distribution ▁within ▁an ▁acceptable ▁error . ▁A ▁good ▁chain ▁will ▁have ▁rapid ▁mixing : ▁the ▁station ary ▁distribution ▁is ▁reached ▁quickly ▁starting ▁from ▁an ▁arbitrary ▁position . ▁A ▁standard ▁empir ical ▁method ▁to ▁assess ▁convergence ▁is ▁to ▁run ▁several ▁independent ▁sim ulated ▁Mark ov ▁ch ains ▁and ▁check ▁that ▁the ▁ratio ▁of ▁inter - chain ▁to ▁int ra - chain ▁var ian ces ▁for ▁all ▁the ▁parameters ▁sample d ▁is ▁close ▁to ▁ 1 . ▁ ▁Typ ically , ▁Mark ov ▁chain