▁unit ary ▁representation ▁U g . ▁As ▁above ▁we ▁have ▁f ▁given ▁by ▁some ▁vector ▁state ▁on ▁U g ▁▁▁▁ ▁therefore ▁positive - defin ite . ▁ ▁The ▁two ▁constru ctions ▁are ▁mut ual ▁in vers es . ▁ ▁Special ▁cases ▁▁ ▁Bo ch ner ' s ▁theorem ▁in ▁the ▁special ▁case ▁of ▁the ▁discrete ▁group ▁Z ▁is ▁often ▁referred ▁to ▁as ▁Her gl ot z ' s ▁theorem ▁( see ▁ ▁Her gl ot z ▁representation ▁theorem ) ▁and ▁says ▁that ▁a ▁function ▁f ▁on ▁Z ▁with ▁f ( 0 ) ▁= ▁ 1 ▁is ▁positive - defin ite ▁if ▁and ▁only ▁if ▁there ▁exists ▁a ▁probability ▁measure ▁ μ ▁on ▁the ▁circle ▁T ▁such ▁that ▁▁▁▁ ▁Similarly , ▁a ▁continuous ▁function ▁f ▁on ▁R ▁with ▁f ( 0 ) ▁= ▁ 1 ▁is ▁positive - defin ite ▁if ▁and ▁only ▁if ▁there ▁exists ▁a ▁probability ▁measure ▁ μ ▁on ▁R ▁such ▁that ▁ ▁App lications ▁ ▁In ▁statistics , ▁Bo ch ner ' s ▁theorem ▁can ▁be ▁used ▁to ▁describe ▁the ▁serial ▁correlation ▁of ▁certain ▁type ▁of ▁time ▁series . ▁A ▁sequence ▁of ▁random ▁variables ▁ ▁of ▁mean ▁ 0 ▁is ▁a ▁( wide - s ense ) ▁station ary ▁time ▁series ▁if ▁the ▁cov ar iance ▁▁▁▁ ▁only ▁depends ▁on ▁n   −   m . ▁The ▁function ▁▁▁▁ ▁is ▁called ▁the ▁aut oc ov ar iance ▁function ▁of ▁the ▁time ▁series . ▁By ▁the ▁mean ▁zero ▁assumption , ▁▁▁▁ ▁where ▁ ⟨ ⋅ , ▁ ⋅ ⟩ ▁denotes ▁the