▁inner ▁product ▁on ▁the ▁Hil bert ▁space ▁of ▁random ▁variables ▁with ▁finite ▁second ▁moments . ▁It ▁is ▁then ▁immediate ▁that ▁g ▁is ▁a ▁positive - defin ite ▁function ▁on ▁the ▁integers ▁ ℤ . ▁By ▁Bo ch ner ' s ▁theorem , ▁there ▁exists ▁a ▁unique ▁positive ▁measure ▁ μ ▁on ▁[ 0 ,   1 ] ▁such ▁that ▁▁▁▁▁ ▁This ▁measure ▁ μ ▁is ▁called ▁the ▁spectral ▁measure ▁of ▁the ▁time ▁series . ▁It ▁yields ▁information ▁about ▁the ▁" season al ▁tr ends " ▁of ▁the ▁series . ▁ ▁For ▁example , ▁let ▁z ▁be ▁an ▁m - th ▁root ▁of ▁unity ▁( with ▁the ▁current ▁identification , ▁this ▁is ▁ 1 / m ▁ ∈ ▁[ 0 ,   1 ]) ▁and ▁f ▁be ▁a ▁random ▁variable ▁of ▁mean ▁ 0 ▁and ▁variance ▁ 1 . ▁Consider ▁the ▁time ▁series ▁. ▁The ▁aut oc ov ar iance ▁function ▁is ▁▁▁▁ ▁Ev ident ly , ▁the ▁corresponding ▁spectral ▁measure ▁is ▁the ▁Di rac ▁point ▁mass ▁centered ▁at ▁z . ▁This ▁is ▁related ▁to ▁the ▁fact ▁that ▁the ▁time ▁series ▁repe ats ▁itself ▁every ▁m ▁periods . ▁ ▁When ▁g ▁has ▁sufficiently ▁fast ▁decay , ▁the ▁measure ▁ μ ▁is ▁absolutely ▁continuous ▁with ▁respect ▁to ▁the ▁Leb es gue ▁measure , ▁and ▁its ▁Rad on – N ik od ym ▁derivative ▁f ▁is ▁called ▁the ▁spectral ▁density ▁of ▁the ▁time ▁series . ▁When ▁g ▁lies ▁in ▁l 1 ( ℤ ), ▁f ▁is ▁the ▁Fourier ▁transform ▁of ▁g . ▁ ▁See ▁also ▁▁