A Comparison of the Finite Sample Properties of Selection Rules of Factor Numbers in Large Datasets

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Preprint submitted on 20 Mar 2014

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A Comparison of the Finite Sample Properties of Selection Rules of Factor Numbers in Large Datasets

Liang Guo-Fitoussi, Olivier Darné

To cite this version:

Liang Guo-Fitoussi, Olivier Darné. A Comparison of the Finite Sample Properties of Selection Rules of Factor Numbers in Large Datasets. 2014. �hal-00962247�

EA 4272

A comparison of the finite sample properties of selection rules of factor

numbers in large datasets

Liang Guo-Fitoussi*

Olivier Darné**

2014/12

(*) RITM – Université Paris-Sud (**) LEMNA - Université de Nantes

Laboratoire d’Economie et de Management Nantes-Atlantique Université de Nantes

Chemin de la Censive du Tertre – BP 52231 44322 Nantes cedex 3 – France www.univ-nantes.fr/iemn-iae/recherche Tél. +33 (0)2 40 14 17 17 – Fax +33 (0)2 40 14 17 49

D o cu m en t d e T ra va il W o rk in g P ap er

❆ ❈♦♠♣❛&✐(♦♥ ♦❢ +❤❡ ❋✐♥✐+❡ ❙❛♠♣❧❡ 1&♦♣❡&+✐❡(

♦❢ ❙❡❧❡❝+✐♦♥ ❘✉❧❡( ♦❢ ❋❛❝+♦& ◆✉♠❜❡&( ✐♥ ▲❛&❣❡

❉❛+❛(❡+(

▲✐❛♥❣ ●✉♦✲❋✐*♦✉++✐^∗ ❖❧✐✈✐❡0 ❉❛0♥2^†

▼❛0❝❤ ✷✵✱ ✷✵✶✹

❆❜"#$❛❝#

■♥ "❤✐% ♣❛♣❡)✱ ✇❡ ❝♦♠♣❛)❡ "❤❡ ♣)♦♣❡)"✐❡% ♦❢ "❤❡ ♠❛✐♥ ❝)✐"❡)✐❛ ♣)♦✲

♣♦%❡❞ ❢♦) %❡❧❡❝"✐♥❣ "❤❡ ♥✉♠❜❡) ♦❢ ❢❛❝"♦)% ✐♥ ❞②♥❛♠✐❝ ❢❛❝"♦) ♠♦❞❡❧ ✐♥ ❛

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❛)❡ %"✉❞✐❡❞✳ ❙✐♠✉❧❛"✐♦♥% %❤♦✇ "❤❛" "❤❡ ●❘ )❛"✐♦ ♣)♦♣♦%❡❞ ❜② ❆❤♥ ❛♥❞

❍♦)❡♥%"❡✐♥ ✭✷✵✶✸✮ ❛♥❞ "❤❡ ❝)✐"❡)✐♦♥ ♣)♦♣♦%❡❞ ❜② ❖♥❛"%❦✐ ✭✷✵✶✵✮ ♦✉"♣❡)✲

❢♦)♠ "❤❡ ♦"❤❡)%✳ ❋✉)"❤❡)♠♦)❡✱ "❤❡ "✇♦ ❝)✐"❡)✐❛ ❝❛♥ %❡❧❡❝" ❛❝❝✉)❛"❡❧② "❤❡

♥✉♠❜❡) ♦❢ %"❛"✐❝ ❢❛❝"♦)% ✐♥ ❛ ❞②♥❛♠✐❝ ❢❛❝"♦)% ❞❡%✐❣♥✳ ❆❧%♦✱ "❤❡ ❝)✐"❡)✐❛

♣)♦♣♦%❡❞ ❜② ❍❛❧❧✐♥ ❛♥❞ ▲✐%❦❛ ✭✷✵✵✼✮ ❛♥❞ ❇)❡✐"✉♥❣ ❛♥❞ J✐❣♦)%❝❤ ✭✷✵✵✾✮

❝♦))❡❝"❧② %❡❧❡❝" "❤❡ ♥✉♠❜❡) ♦❢ ❞②♥❛♠✐❝ ❢❛❝"♦)% ✐♥ ♠♦%" ❝❛%❡%✳ ❍♦✇❡✈❡)✱

❡♠♣✐)✐❝❛❧ ❛♣♣❧✐❝❛"✐♦♥% %❤♦✇ ♠♦%" ❝)✐"❡)✐❛ %❡❧❡❝" ♦♥❧② ♦♥❡ ❢❛❝"♦) ✐♥ ♣)❡%❡♥❝❡

♦❢ ♦♥❡ %")♦♥❣ ❢❛❝"♦)✳

❑❡② ✇♦%❞'✿ ❞②♥❛♠✐❝ ❢❛❝/♦% ♠♦❞❡❧✱ ❢❛❝/♦% ♥✉♠❜❡%'✱ '♠❛❧❧ '❛♠♣❧❡ ♣%♦♣❡%/✐❡'

❏❊▲ ❈❧❛''✐✜❝❛/✐♦♥✿ ❈✶✸✱ ❈✺✷

✶ ■♥#$♦❞✉❝#✐♦♥

❚❤❡ ✐♠♣%♦✈❡♠❡♥/' ✐♥ ❝♦♠♣✉/❡% /❡❝❤♥♦❧♦❣②✱ ❛♥❞ ❝♦❧❧❡❝/✐♦♥ ❛♥❞ '/♦%❛❣❡ ♦❢ ❞❛/❛✱

❛♥❞ ❞❡✈❡❧♦♣♠❡♥/ ♦❢ ♣♦✇❡%❢✉❧ ♠❛/❤❡♠❛/✐❝❛❧ ❛♥❞ '/❛/✐'/✐❝❛❧ '♦❢/✇❛%❡ ✐' ❛❧❧♦✇✐♥❣

%❡'❡❛%❝❤❡%' ❛♥❞ ♣%♦❢❡''✐♦♥❛❧' ✐♥ ❡❝♦♥♦♠✐❝' ❛♥❞ ✜♥❛♥❝❡ /♦ ❜❡♥❡✜/ ❢%♦♠ ✐♥❝%❡❛'✲

✐♥❣❧② %✐❝❤ ❛♥❞ ✐♥❝%❡❛'✐♥❣❧② ❞✐'❛❣❣%❡❣❛/❡❞ ❞❛/❛✳ ■/ ✐' ✐♥ /❤✐' ❝♦♥/❡①/ /❤❛/ ❢❛❝/♦%

♠♦❞❡❧' ♦❢ ❧❛%❣❡ ❞✐♠❡♥'✐♦♥❛❧ ❞❛/❛'❡/ ❤❛✈❡ ❜❡❡♥ ♣%♦♣♦'❡❞ ❛♥❞ ❛❝❤✐❡✈❡❞ ♣♦♣✉✲

❧❛%✐/②✳ ❚❤❡ ❢❛❝/♦% ♠♦❞❡❧' ♦❢ ❧❛%❣❡ ❞❛/❛'❡/' ❛%❡ ✇✐❞❡❧② ❛♣♣❧✐❡❞ ❜❡❝❛✉'❡ /❤❡②

❝♦♥'/✐/✉/❡ ❛ ❣♦♦❞ ❝♦♠♣%♦♠✐'❡ ❜❡/✇❡❡♥ ❡①♣❧♦✐/✐♥❣ ❧❛%❣❡ ❛♠♦✉♥/' ♦❢ ✐♥❢♦%♠❛/✐♦♥

∗❯♥✐✈❡%&✐'② )❛%✐&✲❙✉❞✱ ❘■❚▼✱ ✺✹✱ ❇❉ ❉❊❙●❘❆◆●❊❙✱ ✾✷✸✸✶✱ ❙❝❡❛✉①✱ ❋❘❆◆❈❊✳

†❯♥✐✈❡%&✐'② ♦❢ ◆❛♥'❡&✱ ▲❊▼◆❆✱ ❈❤❡♠✐♥ ❧❛ ❝❡♥&✐✈❡ ❞✉ '❡%'%❡✱ ❇) ✺✷✷✸✶✱ ✹✹✸✷✷ ◆❛♥'❡&

❈❡❞❡① ✸✱ ❋❘❆◆❈❊✳

✶

❛♥❞ ♣❛$%✐♠♦♥✐♦✉% ♣❛$❛♠❡+❡$ ❡%+✐♠❛+✐♦♥✳ ❋♦$ $❡✈✐❡✇ ♦❢ $❡❝❡♥+ ❢❛❝+♦$ ♠♦❞❡❧ ❞❡✲

✈❡❧♦♣♠❡♥+%✱ %❡❡ ❘❡✐❝❤❧✐♥ ✭✷✵✵✸✮✱ ❇$❡✐+✉♥❣ ❛♥❞ ❊✐❝❦♠❡✐❡$ ✭✷✵✵✻✮✱ ❊✐❝❦♠❡✐❡$ ❛♥❞

❩✐❡❣❧❡$ ✭✷✵✵✽✮✱ ❇♦✐✈✐♥ ❛♥❞ ◆❣ ✭✷✵✵✺✮✱ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✽✮✱ ●✉♦ ✭✷✵✶✵✮✱ ❙+♦❝❦

❛♥❞ ❲❛+%♦♥ ✭✷✵✶✵✮ ❛♥❞ ❇❛$❤♦✉♠✐ ❡+ ❛❧✳ ✭✷✵✶✸✮✳

■♥ ♠❛❝$♦❡❝♦♥♦♠✐❝%✱ ❢❛❝+♦$ ♠♦❞❡❧% ❛$❡ ✉%❡❞ ❢♦$ ♥♦✇❝❛%+✐♥❣ ✭❆❧+✐%%✐♠♦ ❡+ ❛❧✳✱

✷✵✵✻✮✱ ❢♦$❡❝❛%+✐♥❣ ✭❙+♦❝❦ ❛♥❞ ❲❛+%♦♥✱ ✶✾✾✽✮✱ ❝♦♥%+$✉❝+✐♦♥ ♦❢ ✐♥❞❡①❡%✱ %+$✉❝+✉$❛❧

❛♥❛❧②%✐% ✭%❡❡ ❡✳❣✳ ❙+♦❝❦ ❛♥❞ ❲❛+%♦♥✱ ✷✵✵✺❀ ❇❡$♥❛♥❦❡ ❛♥❞ ❛❧✳❀✷✵✵✺✮✱ ❛♥❞ ♠♦♥✲

❡+❛$② ♣♦❧✐❝② ✭%❡❡ ❡✳❣✳ ❇❡$♥❛❦❡ ❛♥❞ ❇♦✐✈✐♥✱ ✷✵✵✸✮✳ ■♥ ✜♥❛♥❝❡✱ +❤❡② ❛$❡ ✉%❡❞ +♦

%+✉❞② ❛$❜✐+$❛❣❡ ♣$✐❝✐♥❣ +❤❡♦$② ✭❆Q❚✮ ✭%❡❡✱ ❡✳❣✳ ❈❤❛♠❜❡$❧❛✐♥ ❛♥❞ ❘♦+❤%❝❤✐❧❞✱

✶✾✽✸✮✱ ♣❡$❢♦$♠❛♥❝❡ ❡✈❛❧✉❛+✐♦♥ ✭❈❤❛♣% ✺ ❛♥❞ ✻ ✐♥ ❈❛♠♣❡❧❧ ❡+ ❛❧✳✱ ✶✾✾✼✮✱ ❢❛❝+♦$%

✐♥ ✐♥+❡$❡%+ +❡$♠ %+$✉❝+✉$❡% ✭%❡❡ ❡✳❣✳ ❑♦♦♣♠❛♥ ❛♥❞ ❱❛♥ ❞❡$ ❲❡❧✱ ✷✵✶✸✮✱ ❛%%❡+

♠❛♥❛❣❡♠❡♥+ %+$❛+❡❣✐❡% %✉❝❤ ❛% ✧♠♦♠❡♥+✉♠ +$❛❞✐♥❣✧ ✭%❡❡ ❡✳❣✳ ❚♦♥❣✱ ✷✵✵✵✮✱

❛♥❞ ❝$❡❞✐+ ❞❡❢❛✉❧+ ❝♦$$❡❧❛+✐♦♥ ✭%❡❡ ❡✳❣✳ ❈✐♣♦❧❧✐♥✐ ❛♥❞ ▼✐%%❛❣❧✐❛✱ ✷✵✵✼❀ ●✉♦ ❛♥❞

❇$✉♥❡❛✉✱ ✷✵✶✵✮✳ ■♥ ♣$❛❝+✐❝❡✱ +❤❡ ❛♣♣$♦❛❝❤ ♦❢ ❙+♦❝❦ ❛♥❞ ❲❛+%♦♥ ✭✶✾✽✾✮ ❢♦$ ❝♦♥✲

%+$✉❝+✐♥❣ ❡❝♦♥♦♠✐❝ ✐♥❞✐❝❛+♦$% ✐% $❡❣✉❧❛$❧② ✉%❡❞ ❜② ◆❇❊❘ ❡❝♦♥♦♠✐%+% ❛♥❞ +❤❡

❋❡❞❡$❛❧ ❘❡%❡$✈❡ ❇❛♥❦ ♦❢ ❈❤✐❝❛❣♦✳ ■♥ ❊✉$♦♣❡✱ ❡❝♦♥♦♠✐❝ ❛♥❞ ✜♥❛♥❝✐❛❧ ✐♥%+✐+✉✲

+✐♦♥% ✉%❡ ❛ ❝♦✐♥❝✐❞❡♥+ ✐♥❞✐❝❛+♦$ ♦❢ ❡❝♦♥♦♠✐❝ ❝②❝❧❡% ✐♥ +❤❡ ❡✉$♦ ③♦♥❡ ✭❊✉$♦❈❖■◆✱

❋♦$♥✐ ❡+ ❛❧✳✱ ✷✵✵✵✮✱ ♣✉❜❧✐%❤❡❞ ♠♦♥+❤❧② ❜② +❤❡ ▲♦♥❞♦♥✲❇❛%❡❞ ❈❡♥+$❡ ❢♦$ ❊❝♦✲

♥♦♠✐❝ Q♦❧✐❝② ❘❡%❡❛$❝❤ ❛♥❞ ❇❛♥❝❛ ❞✬■+❛❧✐❛ ❢♦$ ❡❝♦♥♦♠✐❝ ❛❝+✐✈✐+② ❛♥❛❧②%✐% ❛♥❞

❢♦$❡❝❛%+✐♥❣✳

❆ ❝$✐+✐❝❛❧ %+❡♣ ✐♥ +❤❡ ❡%+✐♠❛+✐♦♥ ♦❢ ❢❛❝+♦$ ♠♦❞❡❧% ✐% %❡❧❡❝+✐♥❣ +❤❡ ♥✉♠❜❡$ ♦❢

❧❛+❡♥+ ❢❛❝+♦$%✳ ■♥ ❝❧❛%%✐❝❛❧ ❢❛❝+♦$ ♠♦❞❡❧%✱ ♦♥❡ ♦❢ +❤❡ ♠♦%+ ✇✐❞❡❧② ✉%❡❞ ♠❡+❤✲

♦❞% ✐% +❤❡ ❑❛✐%❡$✲●✉++♠❛♥ ❝$✐+❡$✐♦♥ ✭●✉++♠❛♥✱ ✶✾✺✹❀ ❑❛✐%❡$✱ ✶✾✻✵✮✱ ✐♥ ✇❤✐❝❤

♦♥❧② ❢❛❝+♦$% ✇✐+❤ ❡✐❣❡♥✈❛❧✉❡% ❣$❡❛+❡$ +❤❛♥ ✶ ❛$❡ $❡+❛✐♥❡❞✳ ❚❤❡ ✉♥❞❡$❧②✐♥❣ ✐❞❡❛

✐% +❤❛+ ✏❛ ❢❛❝+♦$ ♠✉%+ ❛❝❝♦✉♥+ ❢♦$ ❛+ ❧❡❛%+ ❛% ♠✉❝❤ ✈❛$✐❛♥❝❡ ❛% ❛♥ ✐♥❞✐✈✐❞✉❛❧

✈❛$✐❛❜❧❡✑ ✭◆✉♥♥❛❧❧② ❛♥❞ ❇❡$♥%+❡✐♥✱ ✶✾✾✹✮✳ ❆♥♦+❤❡$ ✐% +❤❡ ❙❝$❡❡ +❡%+✱ ❛ ❣$❛♣❤✐❝❛❧

+♦♦❧ ♣$♦♣♦%❡❞ ❜② ❈❛++❡❧❧ ✭✶✾✻✻✮✳ ❍♦✇❡✈❡$✱ +❤❡%❡ ✐♥❢♦$♠❛❧ ♠❡+❤♦❞% ❛$❡ %✉❜❥❡❝+

+♦ ❝$✐+✐❝✐%♠% ♦❢ ✈✉❧♥❡$❛❜✐❧✐+②✱ %✉❜❥❡❝+✐✈✐+②✱ ❛♥❞ ❧❛❝❦ ♦❢ %+❛+✐%+✐❝❛❧ +❤❡♦$② ✭❲✐%❧♦♥

❛♥❞ ❈♦♦♣❡$✱ ✷✵✵✽✮✳ ▼♦$❡♦✈❡$✱ ✐♥ ♣$❡%❡♥❝❡ ♦❢ ❝$♦%%✲%❡❝+✐♦♥❛❧ ❛♥❞ +❡♠♣♦$❛❧ ❞❡✲

♣❡♥❞❡♥❝❡ ♦❢ ❡$$♦$%✱ +②♣✐❝❛❧ ❢❡❛+✉$❡% ♦❢ ♠❛❝$♦❡❝♦♥♦♠✐❝ ❛♥❞ ✜♥❛♥❝✐❛❧ ❞❛+❛✱ +❤❡%❡

♠❡+❤♦❞% ❝❛♥♥♦+ ❝❧❡❛♥❧② $❡✈❡❛❧ +❤❡ +$✉❡ ♥✉♠❜❡$ ♦❢ ❢❛❝+♦$% ✭❆❤♥ ❛♥❞ ❍♦$❡♥✲

%+❡✐♥✱ ✷✵✶✸✮✳ ■♥ %♦♠❡ ❡❝♦♥♦♠✐❝ +❤❡♦$✐❡%✱ +❤❡ ♥✉♠❜❡$ ♦❢ ❢❛❝+♦$% ❛♥❞ +❤❡ ❢❛❝+♦$%

+❤❡♠%❡❧✈❡% ❛$❡ ✐♠♣♦%❡❞ $❛+❤❡$ +❤❛♥ ❜❡✐♥❣ %♣❡❝✐✜❡❞ ❜② +❤❡ ❞❛+❛✱ ❛ ✇❡❧❧✲❦♥♦✇♥

❡①❛♠♣❧❡ ✐% +❤❡ ❈❆Q▼✳ ❯♥❞❡$ +❤❡ ❛%%✉♠♣+✐♦♥ ♦❢ ❝$♦%%✲%❡❝+✐♦♥❛❧ ❛♥❞ +❡♠♣♦$❛❧ ❞❡✲

♣❡♥❞❡♥❝❡ ♦❢ ❡$$♦$%✱ ❈♦♥♥♦$ ❛♥❞ ❑♦$❛❥❝②❦ ✭✶✾✾✸✮✱ ❈❤❛♠❜❡$❧❛✐♥ ❛♥❞ ❘♦+❤%❝❤✐❧❞

✭✶✾✽✸✮✱ ❈$❛❣❣ ❛♥❞ ❉♦♥❛❧❞ ✭✶✾✾✼✮✱ ▲❡✇❜❡❧ ✭✶✾✾✶✮ ❛♥❞ ❉♦♥❛❧❞ ✭✶✾✾✼✮ ♣$♦♣♦%❡

❝$✐+❡$✐❛ ❢♦$ %❡❧❡❝+✐♥❣ +❤❡ ♥✉♠❜❡$ ♦❢ ❢❛❝+♦$%✳ ❍♦✇❡✈❡$✱ ❛❧❧ ♦❢ +❤❡%❡ ❝$✐+❡$✐❛ $❡d✉✐$❡

♦♥❡ ❞✐♠❡♥%✐♦♥ ✭◆ ♦$ ❚✮ ♦❢ ❞❛+❛%❡+ ✜①❡❞✳

❋♦$ ❢❛❝+♦$ ♠♦❞❡❧% ✇✐+❤ ❜♦+❤ ◆ ❛♥❞ ❚ ❛♣♣$♦❛❝❤✐♥❣ ✐♥✜♥✐+②✱ ❡❛$❧② ✇♦$❦ ♦♥ +❤❡

✐%%✉❡ ♦❢ %❡❧❡❝+✐♦♥ ♦❢ ♥✉♠❜❡$ ♦❢ ❢❛❝+♦$% ✐♥❝❧✉❞❡% ❙+♦❝❦ ❛♥❞ ❲❛+%♦♥ ✭✶✾✾✽✮✱ ❋♦$♥✐

❛♥❞ ❘❡✐❝❤❧✐♥ ✭✶✾✾✽✮ ❛♥❞ ❋♦$♥✐ ❡+ ❛❧✳ ✭✷✵✵✵✮✳ ❍♦✇❡✈❡$✱ +❤❡ ♣✐♦♥❡❡$✐♥❣ ❢♦$♠❛❧

%+❛+✐%+✐❝❛❧ ♣$♦❝❡❞✉$❡ ✐% +❤❡ ✐♥❢♦$♠❛+✐♦♥ ❝$✐+❡$✐❛ ❞❡✈❡❧♦♣❡❞ ❜② ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✳

❙✐♥❝❡ +❤❡♥✱ ❛ ❢❡✇ $❡%❡❛$❝❤❡$% ❤❛✈❡ ♣$♦♣♦%❡❞ ❛❧+❡$♥❛+✐✈❡ ❝♦♥%✐%+❡♥+ ❡%+✐♠❛+♦$%✳

❚❤❡%❡ ❡%+✐♠❛+♦$% ❝❛♥ ❜❡ ❝❧❛%%✐✜❡❞ ✐♥+♦ ❢♦✉$ +②♣❡%✳ ❚❤❡ ✜$%+ ✐% ✐♥❢♦$♠❛+✐♦♥ ❝$✐+❡✲

$✐❛✱ ❡✳❣✳✱ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✱ ❆♠❡♥❣✉❛❧ ❛♥❞ ❲❛+%♦♥ ✭✷✵✵✼✮✱ ❙+♦❝❦ ❛♥❞ ❲❛+%♦♥

✷

✭✷✵✵✺✮ ❛♥❞ ❆❧❡++✐ ❡- ❛❧✳ ✭✷✵✶✵✮✳ ❚❤❡ +❡❝♦♥❞ ✐+ ❜❛+❡❞ ♦♥ -❤❡ -❤❡♦5② ♦❢ 5❛♥❞♦♠

♠❛-5✐❝❡+ ❛♥❞ ❧✐♥❦❡❞ +♣❡❝✐✜❝❛❧❧② -♦ -❤❡ ♣5♦♣5✐❡-✐❡+ ♦❢ -❤❡ ❧❛5❣❡+- ❡✐❣❡♥✈❛❧✉❡+ ♦❢

-❤❡ ♠❛-5✐①✳ ❘❡♣5❡+❡♥-❛-✐✈❡ ✇♦5❦+ ❛5❡ ❍❛❧❧✐♥ ❛♥❞ ▲✐+❦❛ ✭✷✵✵✼✱ ✷✵✶✵✮✳ ❚❤❡ -❤✐5❞

-②♣❡ ✐+ ❜❛+❡❞ ♦♥ -❤❡ 5❛♥❦ ♦❢ ❛ ♠❛-5✐①✱ +✉❝❤ ❛+ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✼✮✳ ❚❤❡ ❢♦✉5-❤

❡♠♣❧♦②+ ❝❛♥♦♥✐❝❛❧ ❝♦55❡❧❛-✐♦♥ ❛♥❛❧②+✐+✳ ❚❤❡ 5❡♣5❡+❡♥-❛-✐✈❡ ♣❛♣❡5+ ❛5❡ ❏❛❝♦❜+

❛♥❞ ❖--❡5 ✭✷✵✵✽✮ ❛♥❞ ❇5❡✐-✉♥❣ ❛♥❞ K✐❣♦5+❝❤ ✭✷✵✵✾✮✳ ❍♦✇❡✈❡5✱ -❤❡+❡ ❡+-✐♠❛✲

-♦5+ ❛5❡ 5❡❧❛-❡❞ -♦ ❡❛❝❤ ♦-❤❡5✳ ❋♦5 ✐♥+-❛♥❝❡✱ ❖♥❛-+❦✐ ✭✷✵✵✼✮ +❤♦✇+ -❤❡ 5❡❧❛-✐♦♥

❜❡-✇❡❡♥ -❤❡ ✐♥❢♦5♠❛-✐♦♥ ❝5✐-❡5✐❛ ❡+-✐♠❛-♦5+ ❛♥❞ -❤❡ ❡✐❣❡♥✈❛❧✉❡ ❡+-✐♠❛-♦5+ ❜②

♣♦✐♥-✐♥❣ ♦✉- -❤❛- -❤❡ ✐♥❢♦5♠❛-✐♦♥ -②♣❡ ❡+-✐♠❛-♦5 ❡O✉❛❧+ -❤❡ ♥✉♠❜❡5 ♦❢ ❡✐❣❡♥✈❛❧✲

✉❡+ ❣5❡❛-❡5 -❤❛♥ ❛ -❤5❡+❤♦❧❞ ✈❛❧✉❡ +♣❡❝✐✜❡❞ ❜② ❛ ♣❡♥❛❧-② ❢✉♥❝-✐♦♥✳ ❚❤❡ ❝5✐-❡5✐❛

♣5♦♣♦+❡❞ ❜② ❖♥❛-+❦✐ ✭✷✵✵✼✮ ❛♥❞ ❆❤♥ ❛♥❞ ❍♦5❡♥+-❡✐♥ ✭✷✵✵✾✮ ❡①❛❝-❧② ❡①♣❧♦✐- -❤✐+

5❡❧❛-✐♦♥✳

❆❧❧ -❤❡+❡ +❡❧❡❝-✐♦♥ ❝5✐-❡5✐❛ ❞❡❧✐✈❡5 ❛ ❝♦♥+✐+-❡♥- ❡+-✐♠❛-♦5 ♦❢ -❤❡ ♥✉♠❜❡5 ♦❢

❢❛❝-♦5+❀ ❤♦✇❡✈❡5✱ ❡+-✐♠❛-❡❞ 5❡+✉❧-+ ✐♥ ✜♥✐-❡ +❛♠♣❧❡+ ♦❢-❡♥ ❞✐✈❡5❣❡✳ ❋✉5-❤❡5♠♦5❡✱

❛❧-❤♦✉❣❤ -❤❡ ❛++✉♠♣-✐♦♥+ ❛5❡ ♠♦5❡ ♦5 ❧❡++ 5❡+-5✐❝-✐✈❡ ❢♦5 ❞✐✛❡5❡♥- +❡❧❡❝-✐♦♥ 5✉❧❡+✱

♠♦+- ❛✉-❤♦5+ ❛5❣✉❡ -❤❛- -❤❡✐5 ❛♣♣5♦❛❝❤ ❝❛♥ ❜❡ ❡①-❡♥❞❡❞ -♦ -❤❡ ♠♦5❡ ❣❡♥❡5❛❧

❝❛+❡✳ ❚❤❡ ♣✉5♣♦+❡ ♦❢ -❤✐+ ♣❛♣❡5 ✐+ -♦ ❝♦♠♣❛5❡ -❤❡ ♣5♦♣❡5-✐❡+ ♦❢ -❤❡ ♠❛✐♥ ❝5✐-❡5✐❛

♣5♦♣♦+❡❞ ✐♥ ❛ +♠❛❧❧ +❛♠♣❧❡ ❛♥❞ -❤✉+ ❤❡❧♣ -❤❡ ❝❤♦✐❝❡ ♦❢ ❝5✐-❡5✐❛ ✉+✐♥❣ ❞✐✛❡5❡♥-

❞❛-❛✳ ❚♦ -❤❡ ❜❡+- ♦❢ ♦✉5 ❦♥♦✇❧❡❞❣❡✱ -❤❡5❡ ✐+ ♥♦ ❡①✐+-✐♥❣ ❝♦♠♣❧❡-❡ ❝♦♠♣❛5✐+♦♥

♦❢ ❝5✐-❡5✐❛✱ ❛♣❛5- ❢5♦♠ -❤❡ ❛5-✐❝❧❡ ❜② ❇❛5❤♦✉♠✐ ❡- ❛❧✳ ✭✷✵✶✸✮✳ ❈♦♠♣❛5❡❞ -♦

❇❛5❤♦✉♠✐ ❡- ❛❧✳ ✭✷✵✶✸✮✱ ✇❤✐❝❤ ❢♦❝✉+❡+ ♦♥ ❢♦5❡❝❛+-✐♥❣ ♣❡5❢♦5♠❛♥❝❡✱ ♦✉5 ✇♦5❦

❢♦❝✉+❡+ ♦♥ ♣❡5❢♦5♠❛♥❝❡ ♦❢ ❝5✐-❡5✐❛ ✉♥❞❡5 ❞✐✛❡5❡♥- ❛++✉♠♣-✐♦♥+✳

❚❤❡ ♣❛♣❡5 ✐+ ♦5❣❛♥✐③❡❞ ❛+ ❢♦❧❧♦✇+✳ ❙❡❝-✐♦♥ ✷ +✉♠♠❛5✐③❡+ -❤❡ ❡①✐+-✐♥❣ ❢❛❝-♦5

♠♦❞❡❧+✳ ❙❡❝-✐♦♥ ✸ ♣5❡+❡♥-+ -❤❡ ❞✐✛❡5❡♥- -②♣❡+ ♦❢ ❡+-✐♠❛-♦5+✳ ❙❡❝-✐♦♥ ✹ 5❡♣♦5-+

▼♦♥-❡ ❈❛5❧♦ ❡①♣❡5✐♠❡♥-+✳ ❙❡❝-✐♦♥ ✺ ♣5♦✈✐❞❡+ -✇♦ ❡♠♣✐5✐❝❛❧ ❛♣♣❧✐❝❛-✐♦♥+✱ ✉+✐♥❣

5❡+♣❡❝-✐✈❡❧② ♠❛❝5♦❡❝♦♥♦♠✐❝ ❞❛-❛ ❛♥❞ +-♦❝❦ 5❡-✉5♥ ❞❛-❛✳ ❙❡❝-✐♦♥ ✻ ♣5❡+❡♥-+ -❤❡

❝♦♥❝❧✉+✐♦♥+✳

✷ ❋❛❝$♦& ▼♦❞❡❧+

❯+✉❛❧❧②✱ -❤❡ ❢❛❝-♦5 ♠♦❞❡❧ ✐+ ✇5✐--❡♥ ✐♥ ❛ ❣❡♥❡5❛❧ ❢♦5♠ ❛+ ❢♦❧❧♦✇+✿

x_t=ΛF_t+e_t ✭✶✮

x_t ❛5❡ ◆✲❞✐♠❡♥+✐♦♥❛❧ ♦❜+❡5✈❛❜❧❡ ✈❛5✐❛❜❧❡+✳ ❲❤❡♥ x_t ❛❞♠✐- ❛ ❢❛❝-♦5✐❛❧ 5❡♣✲

5❡+❡♥-❛-✐♦♥✱ -❤❡② ❝❛♥ ❜❡ ❞❡❝♦♠♣♦+❡❞ ✐♥-♦ ❛ +♠❛❧❧ ♥✉♠❜❡5 ♦❢ ❢❛❝-♦5+ ❛♥❞ N

✐❞✐♦+②♥❝5❛-✐❝ ❡55♦5+✳ F_t ✐+ ❛♥ r−❞✐♠❡♥+✐♦♥❛❧ ✈❡❝-♦5 ♦❢ ❝♦♠♠♦♥ ❢❛❝-♦5+✱ ✇❤❡5❡

r ❞❡♥♦-❡+ -❤❡ ♥✉♠❜❡5 ♦❢ ❢❛❝-♦5+✱ r ≪ N✳ Λ✐+ ❛♥ N ×r ❞✐♠❡♥+✐♦♥❛❧ ♠❛-5✐①

❝♦♥-❛✐♥✐♥❣ -❤❡ ❢❛❝-♦5 ❧♦❛❞✐♥❣+✳ ❲❡ ✉+❡ χt -♦ ❞❡♥♦-❡ -❤❡ ❝♦♠♠♦♥ ❝♦♠♣♦♥❡♥-✱

χt = ΛF_t✳ e_t ✐+ N×1 ❞✐♠❡♥+✐♦♥❛❧ ✐❞✐♦+②♥❝5❛-✐❝ ❡55♦5+✳ Λ✱ F_t ❛♥❞ e_t ❛5❡

✉♥♦❜+❡5✈❛❜❧❡✳

❙♣❡❝✐✜❝❛❧❧②✱ -❤❡ 5❡♣5❡+❡♥-❛-✐♦♥ ✭✶✮ ✐+ ❛ +-❛-✐❝ ❢❛❝-♦5 ♠♦❞❡❧ ❛♥❞Ft❛5❡ -❡5♠❡❞

+-❛-✐❝ ❢❛❝-♦5+ ❜❡❝❛✉+❡ -❤❡ 5❡❧❛-✐♦♥+❤✐♣ ❜❡-✇❡❡♥ ❢❛❝-♦5+ ❛♥❞ ❢❛❝-♦5 ❧♦❛❞✐♥❣+ ✐+

✸

!❛!✐❝✳ ◆♦♥❡!❤❡❧❡ ✱ ❡✈❡♥ ✐♥ ❛ !❛!✐❝ ♠♦❞❡❧✱ ❢❛❝!♦1 Ft ❝❛♥ ❜❡ ✧❞②♥❛♠✐❝✧ ✐♥ !❤❡

❡♥ ❡ !❤❛! !❤❡② ❝❛♥ ❡✈♦❧✈❡ ❢♦❧❧♦✇✐♥❣ ❛ ❞②♥❛♠✐❝ ♣1♦❝❡ ✉❝❤ ❛ ✱

Φ(L)Ft=B(L)υt ✭✷✮

❚❤❡ ✐❞✐♦ ②♥❝1❛!✐❝ ❡11♦1 ♠✐❣❤! ❛❧ ♦ ❜❡ ❛✉!♦❝♦11❡❧❛!❡❞✿

Ψ(L)eit=Di(L)ζit ✭✸✮

✇❤❡1❡ υt ❛♥❞ ζit ❛1❡ ✐✳✐✳❞✳ ✇❤✐!❡ ♥♦✐ ❡ ✇✐!❤ Ekυtk^4+δ < M < ∞ ❛♥❞

Ekζitk^4+δ < M <∞❢♦1 ♦♠❡δ >0✳ Φ(L)✱B(L)✱Ψ(L)❛♥❞Di(L)❛1❡ ❧❛❣❣❡❞

♣♦❧②♥♦♠✐❛❧ ✇✐!❤ 1♦♦! ✇❤✐❝❤ ❛❧❧ ❧✐❡ ♦✉! ✐❞❡ !❤❡ ✉♥✐! ❝✐1❝❧❡✳

❚❤❡ ❞②♥❛♠✐❝ ❢❛❝!♦1 ♠♦❞❡❧ ❝❛♥ ❜❡ ✇1✐!!❡♥ ❛ ❢♦❧❧♦✇ ✱ x_t=λ^′_i(L)f

t+e_t ✭✹✮

f_t❛1❡q✲❞✐♠❡♥ ✐♦♥❛❧ ❞②♥❛♠✐❝ ❢❛❝!♦1 ✱ ✇❤❡1❡q✐ !❤❡ ♥✉♠❜❡1 ♦❢ ❞②♥❛♠✐❝ ❢❛❝✲

!♦1 ✳ λ_i(L)❛1❡ ❧❛❣❣❡❞ ♣♦❧②♥♦♠✐❛❧ ✇✐!❤ 1♦♦! ♦✉! ✐❞❡ !❤❡ ✉♥✐! ❝✐1❝❧❡✳ ❋❛❝!♦1

❛♥❞ ✐❞✐♦ ②♥❝1❛!✐❝ ❡11♦1 ❢♦❧❧♦✇ ❞②♥❛♠✐❝ ♣1♦❝❡ ❡ ✐♠✐❧❛1 !♦ !❤♦ ❡ ✐♥ ❡B✉❛!✐♦♥

✭✷✮ ❛♥❞ ✭✸✮✳ ■♥ ✭✶✮ ❛♥❞ ✭✹✮✱ ❜♦!❤ ❞❡♣❡♥❞❡♥❝❡ ❛♥❞ ❤❡!❡1♦ ❦❡❞❛ !✐❝✐!② ♦❢ ✐❞✲

✐♦ ②♥❝1❛!✐❝ ❡11♦1 ❛♥❞ ❞❡♣❡♥❞❡♥❝❡ ❜❡!✇❡❡♥ ❢❛❝!♦1 ❛♥❞ ❡11♦1 ❛1❡ ❛❧❧♦✇❡❞✳ ❚❤❡

❛ ✉♠♣!✐♦♥ ♣1♦♣♦ ❡❞ ❜② ✈❛1✐♦✉ 1❡ ❡❛1❝❤❡1 ❞✐✛❡1 ♠❛✐♥❧② ✐♥ 1❡❧❛!✐♦♥ !♦ !❤❡

!1❛❞❡♦✛ ❜❡!✇❡❡♥ ♠♦♠❡♥! ❝♦♥ !1❛✐♥! ❛♥❞ ❞❡♣❡♥❞❡♥❝❡ ♣1♦♣❡1!✐❡ ♦❢ !❤❡ ❢❛❝!♦1

❛♥❞ ✐❞✐♦ ②♥❝1❛!✐❝ ❡11♦1 ✳ ❲❡ ❞♦ ♥♦! 1❡♣♦1! !❤❡ ❞❡!❛✐❧❡❞ !❡❝❤♥✐❝❛❧ ❛ ✉♠♣!✐♦♥

❤❡1❡^✶❀ ✇❡ ♣1♦✈✐❞❡ ❛ ❜1✐❡❢ ✉♠♠❛1② ✐♥ ❚❛❜❧❡ ✶ !♦ ❤♦✇ !❤❡ ❞✐✛❡1❡♥❝❡ ✳ ❲❡ ✇♦✉❧❞

♣♦✐♥! ♦✉! ❢♦1 ✐♠♣❧✐❝✐!②✱ !❛!✐♦♥❛1✐!② ✐ ❛ ✉♠❡❞✱ ❛❧!❤♦✉❣❤ ✐! ✐ ♥♦! ♥❡❝❡ ❛1② ❢♦1 ♦♠❡ ❝1✐!❡1✐❛^✷✳

✶❲❡ "❡❢❡" $❤❡ "❡❛❞❡" $♦ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮ ❋♦"♥✐ ❡$ ❛❧✳ ✭✷✵✵✵✮ ❛♥❞ ♦$❤❡"5 ❢♦" $❤❡ ❛55✉♠♣✲

$✐♦♥5✳

✷❋♦" ♥♦♥ 5$❛$✐♦♥❛"② ❞❛$❛✱ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✹✮ 5✉❣❣❡5$ $❤❛$ $❤❡ ♥✉♠❜❡" ♦❢ ❢❛❝$♦"5 ❝❛♥ ❜❡

❡5$✐♠❛$❡❞ ✇✐$❤ ❞✐✛❡"❡♥❝❡❞ ❞❛$❛✳

✹

❚❛❜❧❡ ✶✿ ❈♦♠♣❛+✐-♦♥- ♦❢ ❆--✉♠♣2✐♦♥- ♦❢ ❉✐✛❡+❡♥2 ❙♣❡❝✐✜❝❛2✐♦♥ ❈+✐2❡+✐❛

❲❤❡♥λi(L)❛+❡ ❧❛❣❣❡❞ ♣♦❧②♥♦♠✐❛❧- ♦❢ ❧✐♠✐2❡❞ ♦+❞❡+-✱ ✇❡ ❝❛❧❧ ✭✹✮ +❡-2+✐❝2❡❞

❞②♥❛♠✐❝ ❢❛❝2♦+ ♠♦❞❡❧✱ ✇❤✐❝❤ ✐- ✐♥ ❝♦♥2+❛-2 2♦ ❣❡♥❡+❛❧✐③❡❞ ❞②♥❛♠✐❝ ❢❛❝2♦+ ♠♦❞❡❧

✇✐2❤ λi(L)♦❢ ✐♥✜♥✐2❡ ♦+❞❡+-✳ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✼✮ -❤♦✇ 2❤❛2 2❤❡ +❡-2+✐❝2❡❞ ❞②✲

♥❛♠✐❝ ♠♦❞❡❧ ❛♥❞ 2❤❡ ❛♣♣+♦①✐♠❛2❡❞ -2❛2✐❝ ♠♦❞❡❧ ❝❛♥ ❜❡ ❞❡❞✉❝❡❞ ❜② ♠❛2❤❡♠❛2✲

✐❝❛❧ ✐❞❡♥2✐2✐❡-✳ ❍♦✇❡✈❡+✱ ♥♦2✐❝❡ 2❤❛2 ♦♥❧② 2❤❡ ❝♦♥2❡♠♣♦+❛♥❡♦✉- ❡✛❡❝2- ♦❢ 2❤❡

❢❛❝2♦+- ♦♥ 2❤❡ ✈❛+✐❛❜❧❡- ❛+❡ ❝♦♥-✐❞❡+❡❞ ✐♥ 2❤❡ -2❛2✐❝ ♠♦❞❡❧✱ ✇❤✐❧❡ ❧❛❣❣❡❞ ❞❡♣❡♥✲

❞❡♥❝✐❡- ❛+❡ ❛❧-♦ ❛❧❧♦✇❡❞ ✐♥ 2❤❡ ❞②♥❛♠✐❝ ♠♦❞❡❧✳ ■♥ ❛❞❞✐2✐♦♥✱ 2❤❡② ✐♠♣❧② ❞✐✛❡+❡♥2

❡-2✐♠❛2✐♦♥ ♠❡2❤♦❞-✳ ❆-②♠♣2♦2✐❝ ♣+✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥2- ❛♥❛❧②-✐- ✭❆N❈❆✮ ❝♦✉❧❞

❜❡ ❛♣♣❧✐❡❞ 2♦ 2❤❡ -❛♠♣❧❡ ❝♦✈❛+✐❛♥❝❡ ♠❛2+✐① ❢♦+ ❡-2✐♠❛2✐♥❣ ❢❛❝2♦+- ♦❢ -2❛2✐❝ ❢❛❝✲

2♦+ ♠♦❞❡❧- ✭-❡❡ ❙2♦❝❦ ❛♥❞ ❲❛2-♦♥ ✭✷✵✵✷✮ ❛♥❞ ♦2❤❡+-✳✮ ❍♦✇❡✈❡+✱ ♦♥❡ ❝♦✉❧❞ ✉-❡

❞②♥❛♠✐❝ ♣+✐♥❝✐♣❛❧ ❝♦♠♣♦♥❡♥2- ❛♥❛❧②-✐- ✭❉N❈❆✮ ✐♥ 2❤❡ ❢+❡P✉❡♥❝② ❞♦♠❛✐♥ ❢♦+ ❞②✲

♥❛♠✐❝ ❢❛❝2♦+ ♠♦❞❡❧- ✭❇+✐❧❧✐♥❣❡+✱ ✶✾✽✶❀ ❋♦+♥✐ ❡2 ❛❧✳✱ ✷✵✵✵✱ ✷✵✵✹✮✳ ❆❧2❡+♥❛2✐✈❡❧②✱

❉♦③ ❡2 ❛❧✳ ✭✷✵✵✻✮ ♣+♦♣♦-❡ ❛ P✉❛-✐ ♠❛①✐♠✉♠ ❧✐❦❡❧✐❤♦♦❞ ❛♣♣+♦❛❝❤✳ ❑❛♣❡2❛♥✐♦-

✺

❛♥❞ ▼❛$❝❡❧❧✐♥♦ ✭✷✵✵✹✮ ❛❧/♦ ♣$♦♣♦/❡❞ ❛ ♣❛$❛♠❡2$✐❝ ♠❡2❤♦❞ ❢♦$ ❡/2✐♠❛2✐♥❣ ❧❛$❣❡

❛♣♣$♦①✐♠❛2❡ ❢❛❝2♦$ ♠♦❞❡❧/✳ ❋♦$ $❡✈✐❡✇/ ❛♥❞ ❝♦♠♣❛$✐/♦♥/ ♦❢ 2❤❡/❡ ❡/2✐♠❛2✐♦♥

♠❡2❤♦❞/✱ /❡❡ ❙2♦❝❦ ❛♥❞ ❲❛2/♦♥ ✭✷✵✶✵✮✱ ❇♦✐✈✐♥ ❛♥❞ ◆❣ ✭✷✵✵✺✮✱ ▼❛$❝❡❧❧✐♥♦ ❡2 ❛❧✳

✭✷✵✵✺✮ ❛♥❞ ❉✬❆❣♦/2✐♥♦ ❛♥❞ ●✐❛♥♥♦♥❡ ✭✷✵✵✻✮✳

✸ ❈"✐$❡"✐❛ ♦❢ )❡❧❡❝$✐♦♥ ♦❢ ♥✉♠❜❡" ♦❢ ❢❛❝$♦")

■♥ 2❤✐/ /❡❝2✐♦♥✱ ✇❡ ❞✐/❝✉// ✈❛$✐♦✉/ ❝$✐2❡$✐❛✳ ❚❤❡② ❛$❡ ❝❧❛//✐✜❡❞ ✐♥ ❢♦✉$ ❣$♦✉♣/✿

✐♥❢♦$♠❛2✐♦♥ ❝$✐2❡$✐❛ 2②♣❡✱ ❝$✐2❡$✐❛ ❜❛/❡❞ ♦♥ ♣$♦♣❡$2✐❡/ ♦❢ ❡✐❣❡♥✈❛❧✉❡/ ♦$ /✐♥❣✉✲

❧❛$ ✈❛❧✉❡✱ ❝$✐2❡$✐❛ ❡①♣❧♦✐2✐♥❣ 2❤❡ $❛♥❦ ♦❢ ♠❛2$✐①✱ ❛♥❞ ❝$✐2❡$✐❛ ✉/✐♥❣ ❝❛♥♦♥✐❝❛❧

❝♦$$❡❧❛2✐♦♥ ❛♥❛❧②/✐/✳

✸✳✶ ■♥❢♦'♠❛*✐♦♥ ❝'✐*❡'✐❛ *②♣❡

❆/ ✐/ ✇❡❧❧✲❦♥♦✇♥✱ 2❤❡ ❣❡♥❡$❛❧ $✉❧❡ ❢♦$ ✐♥❢♦$♠❛2✐♦♥ ❝$✐2❡$✐❛ ✐/ /❡❧❡❝2✐♥❣ 2❤❡ ♥✉♠❜❡$

♦❢ ❢❛❝2♦$/ ✇❤✐❝❤ ♠✐♥✐♠✐③❡/ 2❤❡ ✈❛$✐❛♥❝❡ ❡①♣❧❛✐♥❡❞ ❜② 2❤❡ ✐❞✐♦/②♥❝$❛2✐❝ ❝♦♠♣♦✲

♥❡♥2✳ ❆ ♣❡♥❛❧2② ❢✉♥❝2✐♦♥ ✐/ ✐♥2$♦❞✉❝❡❞ ✐♥ ♦$❞❡$ 2♦ ❛✈♦✐❞ ♦✈❡$♣❛$❛♠❡2❡$✐③❛2✐♦♥✳

❚❤❡ ❝❤♦✐❝❡ ♦❢ ♣❡♥❛❧2② ❢✉♥❝2✐♦♥ ✐/ ♦❢2❡♥ $❡❧❛2❡❞ 2♦ 2❤❡ $❛2❡ ♦❢ ❝♦♥✈❡$❣❡♥❝❡ ♦❢

2❤❡ ❡/2✐♠❛2♦$/✳ ❙2❛♥❞❛$❞ ❝$✐2❡$✐❛ ❆■❈ ❛♥❞ ❇■❈ ❛$❡ ❣♦♦❞ ❡①❛♠♣❧❡/✳ ❍♦✇❡✈❡$✱

2❤❡/❡ ❝$✐2❡$✐❛ ❛$❡ ♥♦2 ❛♣♣❧✐❝❛❜❧❡ ✐♥ ❧❛$❣❡ ❢❛❝2♦$ ♠♦❞❡❧/ ❜❡❝❛✉/❡ 2❤❡ ❢❛❝2♦$/ ❛$❡

✉♥♦❜/❡$✈❛❜❧❡ ❛♥❞ ❞♦ ♥♦2 2❛❦❡ ❛❝❝♦✉♥2 ♦❢ 2❤❡ ❞♦✉❜❧❡ ❞✐♠❡♥/✐♦♥/ ✭❚ ❛♥❞ ♥✮✳

✸✳✶✳✶ ❊$%✐♠❛%✐♦♥ ♦❢ $%❛%✐❝ ❢❛❝%♦-$

❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮ ♠♦❞✐❢② ❆■❈ ❛♥❞ ❇■❈ ❜② 2❛❦✐♥❣

❛❝❝♦✉♥2 ♦❢ ❜♦2❤ ❞✐♠❡♥/✐♦♥/n❛♥❞T ♦❢ 2❤❡ ❞❛2❛/❡2 ❛♥❞ /✉❣❣❡/2 ❝$✐2❡$✐❛P Cp2♦

/♣❡❝✐❢② 2❤❡ ♥✉♠❜❡$ ♦❢ /2❛2✐❝ ❢❛❝2♦$/ r✿

P C(k) =V(k) +kp(n, T) ✭✺✮

✇❤❡$❡ V(k) = (nT)⁻¹Pⁿ

i=1

PT t=1

(Xit−λˆ^k_i^′Fˆ_t^k)²✱ λˆ^k_i ❛♥❞ Fˆ_t^k ❛$❡ 2❤❡ ❆S❈❆ ❡/✲

2✐♠❛2♦$/ ♦❢ 2❤❡ ❢❛❝2♦$ ❧♦❛❞✐♥❣/ ❛♥❞ ❢❛❝2♦$/✱ 2❤❡ /✉♣❡$/❝$✐♣2 k /✐❣♥✐✜❡/ k/2❛2✐❝

❢❛❝2♦$/ ❛$❡ ✉/❡❞✳

❚❤❡ /❡❧❡❝2❡❞ ♥✉♠❜❡$ ♦❢ ❢❛❝2♦$/ /❤♦✉❧❞ ♠✐♥✐♠✐③❡ P C(k)✱ ✐✳❡✳✱

bk= arg min_0≤k≤rmaxP C(k)✱ ✇❤❡$❡rmax✐/ ❛ ♣$❡❞❡2❡$♠✐♥❡❞ ❜♦✉♥❞❡❞ ✐♥2❡✲

❣❡$✳

❆/ ❢♦$ AIC ❡2 BIC✱ V(k) /❤♦✉❧❞ ❜❡ /♠❛❧❧ ✐❢ k > r✳ ❚♦ ❛✈♦✐❞ ✉♥❞❡$✲

❡/2✐♠❛2✐♦♥ ❛♥❞ ♦✈❡$❡/2✐♠❛2✐♦♥✱ 2❤❡ ♣❡♥❛❧2② ❢✉♥❝2✐♦♥ ♠✉/2 /❛2✐/❢② 2❤❡ ❝♦♥❞✐✲

2✐♦♥/ ✭✐✮ p(n, T) → 0 ❛♥❞ ✭✐✐✮ CN,T ·p(n, T) → ∞ ✇❤❡♥ n, T → ∞✱ ✇❤❡$❡

Cn,T = minh√n,√ Ti

✭❙❡❡ ❚❤❡♦$❡♠ ✷ ♦❢ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✮✳ ❚❤❡ ✐♥2✉✐2✐♦♥

❜❡❤✐♥❞ 2❤❡/❡ ❝♦♥❞✐2✐♦♥/ ✐/ 2❤❛2 2❤❡ ♣❡♥❛❧2② ❢✉♥❝2✐♦♥ p(n, T)❝♦♥✈❡$❣❡/ 2♦ ③❡$♦

✻

❜✉" ❧❡%% &✉✐❝❦❧② "❤❛♥ "❤❡ ❝♦♥✈❡0❣❡♥❝❡ 0❛"❡ ♦❢ ❡%"✐♠❛"♦0 ♦❢ ❢❛❝"♦0%✱ ✇❤✐❝❤ ✐% ♣0♦✈❡♥

"♦ ❜❡C_n,T⁻¹ ❜② ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✳ ❚❤❡0❡❢♦0❡✱ "❤❡ ♣❡♥❛❧"② ❢✉♥❝"✐♦♥ ❛♣♣0♦❛❝❤❡%

③❡0♦ ❜✉" ✐" ✏❞♦♠✐♥❛"❡% "❤❡ ❞✐✛❡0❡♥❝❡ ✐♥ "❤❡ %✉♠ ♦❢ %&✉❛0❡❞ 0❡%✐❞✉❛❧% ❜❡"✇❡❡♥

"❤❡ "0✉❡ ❛♥❞ "❤❡ ♦✈❡0♣❛0❛♠❡"❡0✐③❡❞ ♠♦❞❡❧✑ ✭❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✮✳ ❆♥♦"❤❡0 ❝❧❛%%

♦❢ ❝0✐"❡0✐❛ ❛❧❧♦✇✐♥❣ ❛ ❝♦♥%✐%"❡♥" ❡%"✐♠❛"♦0 ♦❢r✐% ♣0♦♣♦%❡❞ ❜② ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✱

❛♥❞ ✐% "❤❡ ❧♦❣❛0✐"❤♠✐❝ ✈❡0%✐♦♥ ♦❢P C(k)✳ ❋♦0 ❡❛❝❤ ❝❧❛%%❡ ♦❢ ❝0✐"❡0✐❛✱ ❇❛✐ ❛♥❞ ◆❣

✭✷✵✵✷✮ ♣0♦♣♦%❡ "❤0❡❡ %♣❡❝✐✜❝ ❢♦0♠✉❧❛"✐♦♥% ✳ ❙✐♥❝❡ ICp1❛♥❞P Cp1❛0❡ %❤♦✇♥ "♦

❜❡ ♠♦0❡ 0♦❜✉%" "❤❛♥ "❤❡ ♦"❤❡0% ❜② "❤❡ ▼♦♥"❡ ❈❛0❧♦ ❙✐♠✉❧❛"✐♦♥ ✐♥ ❇❛✐ ❛♥❞ ◆❣

✭✷✵✵✷✮^✸✱ ✇❡ ❝♦♥%✐❞❡0 ♦♥❧② "❤❡%❡ "✇♦ ❝0✐"❡0✐❛ ✐♥ "❤✐% ♣❛♣❡0✱

P C_p1^BN02(k) =V(k,Fc^k) +kbσ² n+T

nT

ln nT

n+T

✭✻✮

IC_p1^BN02(k) = ln(V(k,Fc^k)) +k n+T

nT

ln nT

n+T

✭✼✮

✇❤❡0❡ σb² ✐% ❛ ❝♦♥%✐%"❡♥" ❡%"✐♠❛"❡ ♦❢ (nT)⁻¹ Pn i=1

PT t=1

E(eit)²✳ ❇❛✐ ❛♥❞ ◆❣

✭✷✵✵✷✮ %✉❣❣❡%" "❤❛"σb²❝❛♥ ❜❡ 0❡♣❧❛❝❡❞ ❜②V(rmax✱Fd^rmax)✐♥ 0❡❛❧✐"②✳ ❍♦✇❡✈❡0✱

"❤✐% ✐♠♣❧✐❡% "❤❛" M❈ ❞❡♣❡♥❞ ❞✐0❡❝"❧② ♦♥ "❤❡ ❝❤♦✐❝❡ ♦❢ rmax✭❆❧❡%%✐ ❡" ❛❧✳✱ ✷✵✵✽❀

❋♦0♥✐ ❡" ❛❧✳✱ ✷✵✵✼✮✳

❚❤❡ ❝0✐"❡0✐❛ ♦❢ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮ ❤❛✈❡ ❛""0❛❝"❡❞ "✇♦ ❝0✐"✐❝✐%♠%✳ ❖♥❡ ✐%

"❤❛" "❤❡ ❡%"✐♠❛"♦0% ♥❡❡❞ "♦ ♣0❡✲%♣❡❝✐❢② ❛ ♠❛①✐♠✉♠ ♣♦%%✐❜❧❡ ♥✉♠❜❡0 ♦❢ ❢❛❝✲

"♦0%✱ rmax ✭❆❤♥ ❛♥❞ ❍♦0❡♥%"❡✐♥✱ ✷✵✶✸✮✳ ❆❧"❤♦✉❣❤ ❙❝❤✇❡0" ✭✶✾✽✾✮ %✉❣❣❡%"%

✉%✐♥❣ 8int

(T /100)^1/4

❛% ❛ 0✉❧❡ "♦ %❡"rmax❢♦0 "✐♠❡ %❡0✐❡% ❛♥❛❧②%✐%✱ ♥♦ ❣✉✐❞❡

✐% ❛✈❛✐❧❛❜❧❡ ❢♦0 ♣❛♥❡❧ ❛♥❛❧②%✐%✳ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮ %✉❣❣❡%" ❛♥ ❛0❜✐"0❛0② ❝❤♦✐❝❡✱

8int

(c²_n,T/100)^1/4

✱ ❢♦0 ❧❛0❣❡ ❞✐♠❡♥%✐♦♥❛❧ ❢❛❝"♦0 ♠♦❞❡❧% ✇✐"❤♦✉" ♣0♦♦❢%✳ ❆♥✲

♦"❤❡0 ♣0♦❜❧❡♠ ✐% "❤❛" "❤❡ "❤0❡%❤♦❧❞ ❝❛♥ ❜❡ ❛0❜✐"0❛0✐❧② %❝❛❧❡❞✳ ◆❛♠❡❧②✱ ✐❢ p(n, T)

❧❡❛❞% "♦ ❝♦♥%✐%"❡♥" ❡%"✐♠❛"✐♦♥ ♦❢r✱ %♦ ❞♦❡%αp(n, T)✱ ✇❤❡0❡α∈R⁺✳ ❆% ♣♦✐♥"❡❞

♦✉" ❜② ❍❛❧❧✐♥ ❛♥❞ ▲✐%❦❛ ✭✷✵✵✼✮ ❛♥❞ ❆❧❡%%✐ ❡" ❛❧✳ ✭✷✵✵✽✮✱ ❛❧"❤♦✉❣❤ ♠✉❧"✐♣❧②✐♥❣

"❤❡ ♣❡♥❛❧"② ❢✉♥❝"✐♦♥ ❜② ❛♥ ❛0❜✐"0❛0② ❝♦♥%"❛♥" ❤❛% ♥♦ ✐♥✢✉❡♥❝❡ ♦♥ "❤❡ ❛%②♠♣✲

"♦"✐❝ ♣❡0❢♦0♠❛♥❝❡ ♦❢ "❤❡ ❝0✐"❡0✐❛✱ "❤❡ 0❡%✉❧" ❝❛♥ ❜❡ ❛✛❡❝"❡❞ ✐♥ ❛ ✜♥✐"❡ %❛♠♣❧❡✳

❋✐♥❛❧❧②✱ ✐♥ "❤❡ ❛♣♣❧✐❝❛"✐♦♥% ✐♥ ❉✬❆❣♦%"✐♥♦ ❛♥❞ ●✐❛♥♥♦♥❡ ✭✷✵✶✸✮✱ ❆❤♥ ❛♥❞ ❍♦0❡♥✲

%"❡✐♥ ✭✷✵✶✸✮✱ ❋♦0♥✐ ❡" ❛❧ ✭✷✵✵✾✮ ❛♥❞ ❆❧❡%%✐ ❡" ❛❧✳ ✭✷✵✵✽✮✱ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✬%

❝0✐"❡0✐❛ ❧❡❛❞ "♦ ✉♥❞❡0❡%"✐♠❛"✐♦♥ ❛♥❞✴♦0 ♦✈❡0❡%"✐♠❛"✐♦♥ ♦❢ ♥✉♠❜❡0 ♦❢ ❢❛❝"♦0% ✐♥

♣0❛❝"✐❝❡✳

✸❇❛"✐❝❛❧❧②✱ (❤❡ ❞✐✛❡-❡♥❝❡ ❜❡(✇❡❡♥ICp1✱P Cp1 ❛♥❞ (❤❡ ♦(❤❡- ❝-✐(❡-✐❛ -❡"✐❞❡" ✐♥ ✉"❡ ♦❢ (❤❡

(❡-♠ ^n+T_nT ♦- ✉"❡ ♦❢ (❤❡ ❝♦♥✈❡-❣❡♥❝❡ -❛(❡C_n,T✳ ◆♦(❡ (❤❡ ❝♦♥✈❡-❣❡♥❝❡ -❛(❡ ❢❛✐❧" (♦ (❛❦❡

❛❝❝♦✉♥( ♦❢ ❜♦(❤ ❞✐♠❡♥"✐♦♥"✳ ❋♦- ❡①❛♠♣❧❡✱ ✇❡ ♦❜(❛✐♥ (❤❡ "❛♠❡Cn,T ❢♦-n= 50, T= 50❛♥❞

n= 200, T = 50✱ ✇❤✐❧❡ (❤❡ ❡"(✐♠❛(✐♦♥ ❡--♦- ✐" "♠❛❧❧❡- ✐♥ (❤❡ ❧❛((❡- ❝❛"❡✳ ❆❝❝♦-❞✐♥❣ (♦ ❇❛✐

❛♥❞ ◆❣ ✭✷✵✵✷✮✱ (❤❡ (❡-♠ ^n+T_nT ♣-♦✈✐❞❡" ❛ "♠❛❧❧ ❝♦--❡❝(✐♦♥ (♦ (❤❡ ❝♦♥✈❡-❣❡♥❝❡ -❛(❡ ❛♥❞ (❤❡

❛✉(❤♦-"✬ "✐♠✉❧❛(✐♦♥" "❤♦✇ (❤❛( ✐( ❤❛" ❛ ❞❡"✐-❛❜❧❡ ✉♣✇❛-❞" ♣❡♥❛❧(② ❛❞❥✉"(♠❡♥( ❡✛❡❝(✳

✼

❆❧❡##✐ ❡% ❛❧✳ ✭✷✵✶✵✮ ❖♥❡ ♦❢ %❤❡ ❝(✐%✐❝✐*♠* ♦❢ ❇❛✐ ❛♥❞ ◆❣✬* ✭✷✵✵✷✮ ❝(✐%❡(✐❛✱

(❡❧❛%❡❞ %♦ %❤❡ ❞❡❣(❡❡ ♦❢ ❢(❡❡❞♦♠ ✐♥ ♣❡♥❛❧%② ❢✉♥❝%✐♦♥✱ ✐* ❡①♣❧♦✐%❡❞ ❜② ❆❧❡**✐ ❡%

❛❧✳ ✭✷✵✶✵✮✱ ✇❤♦ ♣(♦♣♦*❡ ❛ (❡✜♥❡♠❡♥% ♦❢ %❤❡ ❝(✐%❡(✐❛ ✐♥ ❇❛✐ ❛♥❞ ◆❣ ✭✷✵✵✷✮✳ ❚❤❡

✐❞❡❛ ✇❛* ✐♥*♣✐(❡❞ ❜② ❍❛❧❧✐♥ ❛♥❞ ▲✐*❦❛ ✭✷✵✵✼✮✱ ✇❤♦ ♣(♦♣♦*❡❞ *❡❧❡❝%✐♦♥ ❝(✐%❡(✐❛ ❢♦(

❞②♥❛♠✐❝ ❢❛❝%♦(* ✭❝✳❢ *❡❝%✐♦♥ ✸✳✸✮✳ ■♥*%❡❛❞ ♦❢ ✉*✐♥❣ ♦♥❡ *♣❡❝✐✜❝ ♣❡♥❛❧%② ❢✉♥❝%✐♦♥✱

❆❧❡**✐ ❡% ❛❧✳ ✭✷✵✶✵✮ ❡✈❛❧✉❛%❡ ❛ ✇❤♦❧❡ ❢❛♠✐❧② ♦❢ ♣❡♥❛❧%② ❢✉♥❝%✐♦♥*✳ ■♥ ♣❛(%✐❝✉❧❛(✱

%❤❡② ♣(♦♣♦*❡ %❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❢♦(♠❛%✐♦♥ ❝(✐%❡(✐❛ ❜❛*❡❞ ♦♥ ICp1(k)♦❢ ❇❛✐ ❛♥❞ ◆❣

✭✷✵✵✷✮^✹✿

IC_a^ABC(k) = ln(V(k,Fc^k)) +αk n+T

nT

ln nT

n+T

✭✽✮

❚❤❡ ❛(❜✐%(❛(② ♣♦*✐%✐✈❡ (❡❛❧ ♥✉♠❜❡(α✐* ❝❛❧❧❡❞ ❛ %✉♥✐♥❣ ♣❛(❛♠❡%❡(✱ ❛♥❞ %✉♥❡*

%❤❡ ♣❡♥❛❧✐③✐♥❣ ♣♦✇❡( ♦❢ %❤❡ ❢✉♥❝%✐♦♥✳ ❚❤❡ ❡*%✐♠❛%❡❞ ♥✉♠❜❡( ♦❢ ❢❛❝%♦(* ✐* bkα= arg min0≤k≤kmaxIC_a^ABC(k)✱ ✇❤✐❝❤ ❞❡♣❡♥❞* ♦♥ %❤❡ ❝❤♦✐❝❡ ♦❢α✳ ❚❤❡ ❝❛❧✐❜(❛%✐♦♥

♦❢α✐* ❝❛((✐❡❞ ♦✉% ✐♥ %❤❡ ❢♦❧❧♦✇✐♥❣ *%❡♣*✿ ❋✐(*%✱ %❤❡ ❛✉%❤♦( *❡% ✉♥ ✉♣♣❡( ❜♦✉♥❞

❢♦( %❤❡ ❝♦♥*%❛♥% α✱ α ∈ [0, αmax]✳ ◆❡①%✱ J *✉❜*❛♠♣❧❡* ♦❢ *✐③❡ (nj, Tj) ❛(❡

❝♦♥*✐❞❡(❡❞✱ ✇✐%❤j= 0, . . . , J ✱0< n1< . . . < nJ =n❛♥❞0≤T1≤. . . < TJ = T✳ ❋♦( ❡❛❝❤ j✱ %❤❡ ♥✉♠❜❡( ♦❢ %❤❡ ❢❛❝%♦(*✱ ❞❡♥♦%❡❞ ❜② kˆ^Tα,n^j j✱ ✐* ❝♦♠♣✉%❡❞✳ ■❢

%❤❡(❡ ❡①✐*%* ❛♥ ✐♥%❡(✈❛❧[α, α]¯ ♦❢α✇❤✐❝❤ ❤❛* ❛ *%❛❜❧❡ ❜❡❤❛✈✐♦(✱ ✐✳❡✳✱ %❤❡ ♥✉♠❜❡(

♦❢ ❢❛❝%♦(*ˆkα,n^Tj j ✐* ❝♦♥*%❛♥% ❛❝(♦** *✉❜*❛♠♣❧❡* ♦❢ ❞✐✛❡(❡♥% *✐③❡*✱ %❤✐* ♠❡❛♥* %❤❛%

%❤❡ ❝❤♦✐❝❡ ♦❢ α ❤❛* ♥♦% ❜❡❡♥ ❛✛❡❝%❡❞ ❜② %❤❡ *✐③❡ ♦❢ %❤❡ *❛♠♣❧❡✳ ❚❤✐* ♥✉♠❜❡(

ˆk^Tα,n^j j ✐* %❤❡♥ %❤❡ ❡*%✐♠❛%❡❞ ♥✉♠❜❡( ♦❢ ❢❛❝%♦(*✳

❋♦❧❧♦✇✐♥❣ %❤❡ ♥♦%❛%✐♦♥ ✐♥ ❍❛❧❧✐♥ ❛♥❞ ▲✐*❦❛ ✭✷✵✵✼✮✱ %❤❡ *%❛❜✐❧✐%② ✐* ♠❡❛*✉(❡❞

❜② %❤❡ ❡♠♣✐(✐❝❛❧ ✈❛(✐❛♥❝❡ ♦❢ ˆkα,n^Tj j✿

Sα= 1 J

XJ j=1



kˆ^Tα,n^j j− 1 J

XJ j=1

ˆkα,n^Tj j



 ✭✾✮

❚❤✐* ♣(♦❝❡❞✉(❡ ✐* %❡(♠❡❞ %✉♥✐♥❣✲*%❛❜✐❧✐%② ❝❤❡❝❦✉♣ ♣(♦❝❡❞✉(❡ ✐♥ ❆❤♥ ❛♥❞

❍♦(❡♥*%❡✐♥ ✭✷✵✶✸✮✳ ❚❤❡ ❡*%✐♠❛%♦( ❤❛* %❤❡ *❛♠❡ ❛*②♠♣%♦%✐❝ ♣(♦♣❡(%✐❡* ❛* %❤❡

♦(✐❣✐♥❛❧ ❝(✐%❡(✐❛✱ ✇❤✐❧❡ ✐% ❝♦♥✈❡②* ❛ ♠♦(❡ (♦❜✉*% ❡*%✐♠❛%✐♦♥ ♦❢ %❤❡ ♥✉♠❜❡( ♦❢

❢❛❝%♦(* %❤❛♥ ✐% ✇♦✉❧❞ ✇❡(❡ %❤❡ ♣❡♥❛❧%② ✜①❡❞✳

✸✳✶✳✷ ❊#%✐♠❛%✐♦♥ ♦❢ ❞②♥❛♠✐❝ ❢❛❝%♦6#

❙%♦❝❦ ❛♥❞ ❲❛%#♦♥ ✭✷✵✵✺✮ ❛♥❞ ❆♠❡♥❣✉❛❧ ❛♥❞ ❲❛%#♦♥ ✭✷✵✵✼✮ ❚♦

❡*%✐♠❛%❡ %❤❡ ♥✉♠❜❡( ♦❢ ❞②♥❛♠✐❝ ❢❛❝%♦(* ✐♥ ❛ (❡*%(✐❝%❡❞ ❞②♥❛♠✐❝ ♠♦❞❡❧✱ ❙%♦❝❦

❛♥❞ ❲❛%*♦♥ ✭✷✵✵✺✮ ♣(♦♣♦*❡ ❛ ♠♦❞✐✜❝❛%✐♦♥ ♦❢ ❇❛✐ ❛♥❞ ◆❣✬* ✭✷✵✵✷✮ ❡*%✐♠❛%♦(✳

❚❤❡ ♣(♦♦❢ ♦❢ %❤❡ ❝♦♥*✐*%❡♥❝② ♣(♦♣❡(%✐❡* ♦❢ %❤❡ ❡*%✐♠❛%♦( ✐* ❣✐✈❡♥ ❜② ❆♠❡♥❣✉❛❧

❛♥❞ ❲❛%*♦♥ ✭✷✵✵✼✮✳ ❚❤❡ ♠♦❞✐✜❝❛%✐♦♥ ✐* *%(❛✐❣❤%❢♦(✇❛(❞✳ T(❡❝✐*❡❧②✱ %❤❡② ❛**✉♠❡

✹❆♥♦#❤❡& ❝&✐#❡&✐♦♥ ♣&♦♣♦*❡❞ ❜② ❆❧❡**✐ ❡# ❛❧✳ ✭✷✵✶✵✮ ✐* ❜❛*❡❞ ♦♥ICp2(k)♦❢ ❇❛✐ ❛♥❞ ◆❣

✭✷✵✵✷✮✱ ❢♦& #❤❡ &❡❛*♦♥ ❣✐✈❡♥ ✐♥ ❢♦♦#♥♦#❡ ✸✱ ✐# ✐* ♥♦# &❡♣♦&#❡❞ ❤❡&❡✳

✽

❤❛ Ft ✐$ ❛ ❱❆❘✭♣✮ ♣+♦❝❡$$✱ ✐✳❡✳ ✭✷✮ ❜❡❝♦♠❡$ Φ(L)Ft = υt✱ ✇✐ ❤ Φ(L) = I−A1L−· · ·−ApL^p✱ ❛♥❞ ❤❡ ✐♥♥♦✈❛ ✐♦♥$ ❝❛♥ ❜❡ +❡♣+❡$❡♥ ❡❞ ❛$υt=Gηt✱✇❤❡+❡

● ✐$r×q❞✐♠❡♥$✐♦♥❛❧ ❢✉❧❧ ❝♦❧✉♠♥ +❛♥❦ ♠❛ +✐① ❛♥❞ ηt✐$ ✐✳✐✳❞✳ $❤♦❝❦$✳ ■ ❢♦❧❧♦✇$

❤❛ ❤❡ ♥✉♠❜❡+ ♦❢ ❝♦♠♠♦♥ $❤♦❝❦$ ✐$ ✐❞❡♥ ✐❝❛❧ ♦ ❤❡ ♥✉♠❜❡+ ♦❢ ❞②♥❛♠✐❝ ❢❛❝ ♦+$

q✳ ❚♦ ❡$ ✐♠❛ ❡q✱ ❛ ✇♦✲$ ❡♣ ♣+♦❝❡❞✉+❡ ✐$ ♣+♦♣♦$❡❞✳ ■♥ ❤❡ ✜+$ $ ❡♣✱ ❤❡ $ ❛ ✐❝

❢❛❝ ♦+$ ❛+❡ ❡$ ✐♠❛ ❡❞ ❢+♦♠ xt ✉$✐♥❣ ❤❡ ❆D❈❆ ❡$ ✐♠❛ ♦+ ❛♥❞ ❤❡ ♥✉♠❜❡+ ♦❢

$ ❛ ✐❝ ❢❛❝ ♦+$ ✐$ ❞❡ ❡+♠✐♥❡❞ ❜② ❛♣♣❧②✐♥❣ ❇❛✐ ❛♥❞ ◆❣✬$ ✭✷✵✵✷✮ ✐♥❢♦+♠❛ ✐♦♥ ❝+✐ ❡+✐❛✳

■♥ ❤❡ $❡❝♦♥❞ $ ❡♣✱ ❤❡ ♥✉♠❜❡+ ♦❢ ❞②♥❛♠✐❝ ❢❛❝ ♦+$ ✐$ ❡$ ✐♠❛ ❡❞ ❜② ❛♣♣❧②✐♥❣

❛❣❛✐♥ ❇❛✐ ❛♥❞ ◆❣✬$ ✭✷✵✵✷✮ ✐♥❢♦+♠❛ ✐♦♥ ❝+✐ ❡+✐❛ ♦ ❤❡ $❛♠♣❧❡ ❝♦✈❛+✐❛♥❝❡ ♠❛ +✐①

♦❢ ❡$ ✐♠❛ ❡❞ ✐♥♥♦✈❛ ✐♦♥$✱ ✇❤✐❝❤ ✐$ ♦❜ ❛✐♥❡❞ ❛$ ❤❡ +❡$✐❞✉❛❧ ♦❢ ❛ +❡❣+❡$$✐♦♥ ♦❢ xt

♦♥ ❧❛❣$ ♦❢xt ❛♥❞Fˆt✳

✸✳✷ ❆♣♣❧✐❝❛)✐♦♥ ♦❢ )❤❡♦/② ♦❢ /❛♥❞♦♠ ♠❛)/✐① ❛♥❞ ❡✐❣❡♥✲

✈❛❧✉❡ ♣/♦♣❡/)✐❡8

❚❤❡ $❡❝♦♥❞ ②♣❡ ♦❢ $❡❧❡❝ ✐♦♥ +✉❧❡$ ✐$ ❜❛$❡❞ ♦♥ $♦♠❡ +❡$✉❧ $ ❞❡✈❡❧♦♣❡❞ ❛❝❝♦+❞✐♥❣

♦ ❤❡ ❤❡♦+② ♦❢ +❛♥❞♦♠ ♠❛ +✐① ❛♥❞ ❡$♣❡❝✐❛❧❧② ❤❡ ❡✐❣❡♥✈❛❧✉❡$✬ ♣+♦♣❡+ ✐❡$✳ ❚❤❡

❜❛$✐❝ ✐❞❡❛ ✐$ ❤❛ ✐❢ ❤❡ ✈❛+✐❛❜❧❡$ ❛❞♠✐ ❛♥r❢❛❝ ♦+ $ +✉❝ ✉+❡✱ ❤❡ + ❧❛+❣❡$ ❡✐❣❡♥✲

✈❛❧✉❡$ ✐♥ ❤❡ $❛♠♣❧❡ ❝♦✈❛+✐❛♥❝❡ ♠❛ +✐① $❤♦✉❧❞ ❡①♣❧♦❞❡✱ ✇❤✐❧❡ ❤❡ +❡$ $❤♦✉❧❞

❡♥❞ ♦ ✵✳ ❚❤✉$✱ ❤❡ ♥✉♠❜❡+ ♦❢ ❡✐❣❡♥✈❛❧✉❡$ ❞✐✈❡+❣✐♥❣ ❛$ N✱ T ❞✐✈❡+❣❡ ✐$ ❡J✉❛❧

♦ ❤❡ ♥✉♠❜❡+ ♦❢ ❢❛❝ ♦+$✳ ❚❤❡ ✜+$ ❡①♣❧♦+❛ ✐♦♥ ♦❢ ♣+♦♣❡+ ✐❡$ ♦❢ ❡✐❣❡♥✈❛❧✉❡$ ❣♦❡$

❜❛❝❦ ♦ ❤❡ ❙❝+❡❡ ❡$ ✐♥ +♦❞✉❝❡❞ ❜② ❈❛ ❡❧❧ ✭✶✾✻✻✮ ✐♥ ♣$②❝❤♦❧♦❣②✳ ❈❛ ❡❧❧ ✭✶✾✻✻✮

$ ❛ ❡$ ❤❛ ✐❢ ♦♥❡ ♣❧♦ $ ❤❡ ❞❡❝+❡❛$✐♥❣ ❡✐❣❡♥✈❛❧✉❡$ ✐♥ ❤❡ $❛♠♣❧❡ ❝♦✈❛+✐❛♥❝❡ ♠❛✲

+✐① ♦❢ ❤❡ ❞❛ ❛ ❛❣❛✐♥$ ❤❡✐+ +❡$♣❡❝ ✐✈❡ ♦+❞❡+ ♥✉♠❜❡+$✱ ❤❡ ♣❧♦ $❤♦✇$ ❛ $❤❛+♣

❜+❡❛❦ ✇❤❡♥ ❤❡ +✉❡ ♥✉♠❜❡+ ♦❢ ❢❛❝ ♦+$ ❡♥❞$✱ ✇❤✐❝❤ ✐$ ❤❡ $♦✲❝❛❧❧❡❞ ✏$❝+❡❡✑ ❝♦+✲

+❡$♣♦♥❞✐♥❣ ♦ ❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ ✐❞✐♦$②♥❝+❛ ✐❝ ❡✛❡❝ $✳ ❍♦✇❡✈❡+✱ ❤❡ ❙❝+❡❡ ❡$

+❡♠❛✐♥$ ❛ ✈✐$✉❛❧ ✐♥$♣❡❝ ✐♦♥✳ ❆♥♦ ❤❡+ ❤❡✉+✐$ ✐❝ ❡②❡✲✐♥$♣❡❝ ✐♦♥ +✉❧❡ ❜❛$❡❞ ♦♥ ❤❡

+❡❧❛ ✐✈❡ $✐③❡ ♦❢ ❤❡ ❡✐❣❡♥✈❛❧✉❡$ ✐$ ♣+♦♣♦$❡❞ ❜② ❋♦+♥✐ ❡ ❛❧✳ ✭✷✵✵✵✮ ✐♥ ❢+❡J✉❡♥❝②

❞♦♠❛✐♥✳ ▼♦+❡ ❢♦+♠❛❧ ❡$ $ ✇❡+❡ ❞❡✈❡❧♦♣❡❞ ❜② ❖♥❛ $❦✐ ✭✷✵✵✾✱ ✷✵✶✵✮✳

✸✳✷✳✶ ❊%&✐♠❛&✐♦♥ ♦❢ %&❛&✐❝ ❢❛❝&♦.%

❖♥❛&%❦✐ ✭✷✵✵✾❛✮ ❖♥❛ $❦✐ ✭✷✵✵✾✮ ❞❡✈❡❧♦♣$ ❛ $❡J✉❡♥ ✐❛❧ ♣+♦❝❡❞✉+❡ ❜② ❛♣♣❧②✐♥❣

❤❡ ❛$②♠♣ ♦ ✐❝ ❞✐$ +✐❜✉ ✐♦♥ ♦❢ ❤❡ ❡✐❣❡♥✈❛❧✉❡$✱ ♥❛♠❡❧②✱ ❛ ❢❡✇ $❝❛❧❡❞ ❛♥❞ ❝❡♥ ❡+❡❞

❧❛+❣❡$ ❡✐❣❡♥✈❛❧✉❡$ ♦❢ ❤❡ ❝♦✈❛+✐❛♥❝❡ ♠❛ +✐① ♦❢ ❛ ♣❛+ ✐❝✉❧❛+ ❍❡+♠✐ ✐❛♥ +❛♥❞♦♠

♠❛ +✐①✱ ✇❤✐❝❤ ❛$②♠♣ ♦ ✐❝❛❧❧② ❞✐$ +✐❜✉ ❡ ❛$ ❛ ❚+❛❝②✲❲✐❞♦♠ ♦❢ ②♣❡ ✷ ✭T W2✱

❚+❛❝② ❛♥❞ ❲✐❞♦♠ ✭✶✾✾✹✮✮ ❛$ ❚ ❣+♦✇$ ♥♦ ✐❝❡❛❜❧② ❢❛$ ❡+ ❤❛♥ ♥ ^✺✳ ▼♦+❡♦✈❡+✱

❖♥❛ $❦✐ ✭✷✵✵✾✮ ❝♦♥$ +✉❝ $ ❛ $ ❛ ✐$ ✐❝ ❜② ❛❦✐♥❣ ❤❡ +❛ ✐♦ ♦❢ ❤❡ ❞✐✛❡+❡♥❝❡ ✐♥

❛❞❥❛❝❡♥ ❡✐❣❡♥✈❛❧✉❡$✱ ✇❤✐❝❤ ❣❡ $ +✐❞ ♦❢ ❜♦ ❤ ❤❡ ❝❡♥ ❡+✐♥❣ ❛♥❞ $❝❛❧✐♥❣ ♣❛+❛♠❡ ❡+$

♦❢ ❤❡ ❡✐❣❡♥✈❛❧✉❡$✳ ❚❤❡ $❡❧❡❝ ✐♦♥ +✉❧❡ ✐♥ ❖♥❛ $❦✐ ✭✷✵✵✾✮ ✐$ ❞❡✈❡❧♦♣❡❞ ❢♦+ ❛

❣❡♥❡+❛❧✐③❡❞ ❞②♥❛♠✐❝ ❢❛❝ ♦+ ♠♦❞❡❧✱ ✇❤✐❧❡ ✐ ✐$ ❛❧$♦ ❛♣♣❧✐❝❛❜❧❡ ❢♦+ ❛♣♣+♦①✐♠❛ ❡

❢❛❝ ♦+$✳ ■♥ ❤❡ ❝❛$❡ ♦❢ ❛♣♣+♦①✐♠❛ ❡ ❢❛❝ ♦+$✱ ❤❡ $❡❧❡❝ ✐♦♥ ♣+♦❝❡❞✉+❡ ❝♦♥$✐$ $ ♦❢✿

✺❚❤❡ ❛$$✉♠♣(✐♦♥ (❤❛( ❚ ❣-♦✇$ ❢❛$(❡- (❤❛♥ ♥ ✐$ ♦❜✈✐♦✉$❧② ♥♦( -❡❛❧✐$(✐❝ ✐♥ (❤❡ ♠❛❝-♦❡❝♦♥♦♠✐❝

❛♣♣❧✐❝❛(✐♦♥✳ ❲❤✐❧❡ (❤❡ ▼♦♥(❡ ❈❛-❧♦ $✐♠✉❧❛(✐♦♥$ ✐♥ ❖♥❛($❦✐ ✭✷✵✵✾✮ $❤♦✇ (❤❡ (❡$( ❞❡✈❡❧♦♣❡❞

✇♦-❦$ ✇❡❧❧ ❡✈❡♥ ✇❤❡♥ ♥ ✐$ ♠✉❝❤ ❧❛-❣❡- (❤❛♥ ❚✳

✾

✶✳ ❉✐✈✐❞❡ '❤❡ )❛♠♣❧❡ '♦ '✇♦ )✉❜)❛♠♣❧❡) ♦❢ ❡3✉❛❧ ❧❡♥❣'❤✱ ♠✉❧'✐♣❧②✐♥❣ '❤❡

)❡❝♦♥❞ ❤❛❧❢ ❜② ✐♠❛❣✐♥❛9② ✉♥✐'i✱Xˆj=Xj+iX_j+^T

2 ❝♦♠♣✉'❡ '❤❡ ❞✐)❝9❡'❡ ❋♦✉9✐❡9 '9❛♥)❢♦9♠❛'✐♦♥Xˆj=PT

t=1Xt·e^−iwjt ♦❢ '❤❡ ❞❛'❛ ❛' ❢9❡3✉❡♥❝✐❡)ωj = ^2πs_T^j✳

✷✳ ❈♦♠♣✉'❡ i✲'❤ ❧❛9❣❡)' ❡✐❣❡♥✈❛❧✉❡ ♦❢ '❤❡ )♠♦♦'❤❡❞ ♣❡9✐♦❞♦❣9❛♠ ❡)'✐♠❛'❡

2 T

PT /2

j=1XˆjXˆ_j^′✻✱µi✱ ❛♥❞ ❝♦♥)'9✉❝' '❤❡ )'❛'✐)'✐❝✳

R^O09≡maxk0<i<ki

µi−µi+1

µi+1−µi+2 ✭✶✵✮

❯♥❞❡9 '❤❡ ♥✉❧❧✱ )'❛'✐)'✐❝R^O09❝♦♥✈❡9❣❡) ✐♥ '❤❡ ❞✐)'9✐❜✉'✐♦♥ '♦maxk0<i<ki

λi−λi+1

λi+1−λi+2✱

✇❤❡9❡λi ❛9❡ 9❛♥❞♦♠ ✈❛9✐❛❜❧❡) ✇✐'❤ ❥♦✐♥' ♠✉❧'✐✈❛9✐❛'❡ T W2 ❞✐)'9✐❜✉'✐♦♥^✼✳ ❯♥✲

❞❡9 '❤❡ ❛❧'❡9♥❛'✐✈❡✱ R^O09 ❡①♣❧♦❞❡) )✐♥❝❡ µk ❡①♣❧♦❞❡) ✇❤✐❧❡ µi+1 ❛♥❞ µi+2 ❛9❡

❜♦✉♥❞❡❞✳ ❆ '❛❜❧❡ ♦❢ ❝9✐'✐❝❛❧ ✈❛❧✉❡) ♦❢ '❡)' )'❛'✐)'✐❝ ✐) ❣✐✈❡♥ ✐♥ ❖♥❛')❦✐ ✭✷✵✵✾✮✳

❚❤❡ ♥✉❧❧ ✐) 9❡❥❡❝'❡❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❘ ✐) ❧❛9❣❡9 '❤❛♥ ♦9 ❡3✉❛❧ '♦ '❤❡ ❝9✐'✐❝❛❧ ✈❛❧✉❡✳

❖♥❛#$❦✐ ✭✷✵✶✵✮ ❆♥♦'❤❡9 )❡❧❡❝'✐♦♥ ♣9♦❝❡❞✉9❡ ✐) ❞❡✈❡❧♦♣❡❞ ❜② ❖♥❛')❦✐ ✭✷✵✶✵✮✱

❜❛)❡❞ ♦♥ '❤❡ )'9✉❝'✉9❡ ♦❢ '❤❡ ✐❞✐♦)②♥❝9❛'✐❝ ❝♦♠♣♦♥❡♥' ✐♥ '❤❡ ❞❛'❛✳ ❍❡ ✐♠♣♦)❡) ❛ )'9✉❝'✉9❡ ♦♥ '❤❡ ✐❞✐♦)②♥❝9❛'✐❝ ❝♦♠♣♦♥❡♥') ✐♥ '❤❡ ❞❛'❛✿ e=AεB✱ ✇❤❡9❡ ❆ ❛♥❞ ❇

❛9❡ '✇♦ ✉♥9❡)'9✐❝'❡❞ ❞❡'❡9♠✐♥✐)'✐❝ ♠❛'9✐❝❡)✱ ❛♥❞ ε✐) ❛♥N×T ♠❛'9✐① ✇✐'❤ ✐✳✐✳❞✳

❣❛✉))✐❛♥ ❡♥'9✐❡)^✽✳ ❚❤✉)✱ ❜♦'❤ '❤❡ ❝9♦))✲)❡❝'✐♦♥❛❧ ❛♥❞ '❡♠♣♦9❛❧ ❝♦99❡❧❛'✐♦♥ ♦❢

'❤❡ ✐❞✐♦)②♥❝9❛'✐❝ ❝♦♠♣♦♥❡♥') ❛9❡ ❛❧❧♦✇❡❞✳ ❇❡)✐❞❡)✱ ❝♦♠♣❛9✐♥❣ '❤❡ ❛))✉♠♣'✐♦♥)

❛❜♦✉' ♣9♦♣♦9'✐♦♥❛❧ ❣9♦✇'❤ '♦ ♥ ♦❢ '❤❡ ❝✉♠✉❧❛'✐✈❡ ❡✛❡❝' ♦❢ ❢❛❝'♦9) ♦❢ ❇❛✐ ❛♥❞

◆❣ ✭✷✵✵✷✱ ✷✵✵✼✮✱ ❖♥❛')❦✐ ✭✷✵✶✵✮ ❛))✉♠❡) ♦♥❧② '❤❡ ❝✉♠✉❧❛'✐✈❡ ❡✛❡❝' ♦❢ '❤❡ ✏❧❡❛)'

✐♥✢✉❡♥'✐❛❧ ❢❛❝'♦9)✑ ❞✐✈❡9❣❡) '♦ ✐♥✜♥✐'② ✐♥ ♣9♦❜❛❜✐❧✐'② ❛)n→ ∞✳ ❚❤✐) ❛))✉♠♣'✐♦♥

❛❧❧♦✇) '❤❡ ❡①✐)'❡♥❝❡ ♦❢ )♦♠❡ ✏✇❡❛❦✑ ❢❛❝'♦9) ✇❤♦)❡ ❡①♣❧❛♥❛'♦9② ♣♦✇❡9 ❞♦❡) ♥♦'

♣9♦♣♦9'✐♦♥❛❧❧② ✐♥❝9❡❛)❡ ✇✐'❤ ◆✳ ❍♦✇❡✈❡9✱ ✐♥)'❡❛❞ ♦❢ ❛ ❝❧♦)❡❞ ❢♦9♠ ❡①♣9❡))✐♦♥

♦❢ '❤❡ ✉♣♣❡9 ❜♦✉♥❞ ♦♥ '❤❡ ✐❞✐♦)②♥❝9❛'✐❝ ❡✐❣❡♥✈❛❧✉❡)✱ ❖♥❛')❦✐ ✭✷✵✶✵✮ ❞❡9✐✈❡)

❛♥ ✐♠♣❧✐❝✐' ❢✉♥❝'✐♦♥ ❢♦9 '❤❡ ✉♣♣❡9 ❜♦✉♥❞✳ ❆) ♣9♦✈❡❞ ❜② ❩❤❛♥❣ ✭✷✵✵✻✮✱ ✇❤❡♥

'❤❡ ✐❞✐♦)②♥❝9❛'✐❝ ❝♦♠♣♦♥❡♥') ❛9❡ ♥♦♥✲'9✐✈✐❛❧❧② ❝♦99❡❧❛'❡❞ ❜♦'❤ ❝9♦))✲)❡❝'✐♦♥❛❧❧②

❛♥❞ '❡♠♣♦9❛❧❧②✱ '❤❡ ❡✐❣❡♥✈❛❧✉❡ ❞✐)'9✐❜✉'✐♦♥ ♦❢ ee^′/T(n)❝♦♥✈❡9❣❡) ❛✳)✳ '♦ ♥♦♥

9❛♥❞♦♠ ❝❞❢F^κ,A,B✭❛ )❛♠♣❧❡ )✐③❡ ♦❢n❛♥❞T(n)✐) ❛))✉♠❡❞ ✇✐'❤n/T(n)→κ >0

❛) n → ∞✮ ✭❩❤❛♥❣ ✷✵✵✻✱ ❚❤❡♦9❡♠ ✶✳✷✳✶✮✳ ❍♦✇❡✈❡9✱ F^κ,A,B ✐) ❛ ❝♦♠♣❧✐❝❛'❡❞

❢✉♥❝'✐♦♥ ✇✐'❤♦✉' ❡①♣❧✐❝✐' ❢♦9♠✳ ❖♥❛')❦✐ ✭✷✵✶✵✮ )❤♦✇) '❤❛' ❛♥② ✜♥✐'❡ ♥✉♠❜❡9 ♦❢

'❤❡ ❧❛9❣❡)' ♦❢ '❤❡ ❜♦✉♥❞❡❞ ❡✐❣❡♥✈❛❧✉❡) ✐♥ '❤❡ )❛♠♣❧❡ ❝♦✈❛9✐❛♥❝❡ ♠❛'9✐① ❝❧✉)'❡9

❛9♦✉♥❞ ❛ )✐♥❣❧❡ ♣♦✐♥'✱ u(F^κ,A,B)✱ ✇❤❡9❡ u(·) ❞❡♥♦'❡) '❤❡ ✉♣♣❡9 ❜♦✉♥❞ ♦❢ '❤❡

)✉♣♣♦9' ♦❢ '❤❡ ❞✐)'9✐❜✉'✐♦♥ F^κ,A,B✳ ❚❤✉)✱ ❢♦9 ❛♥② k > r✱ '❤❡ ❞✐✛❡9❡♥❝❡ ❜❡'✇❡❡♥

'❤❡ '✇♦ ❛❞❥❛❝❡♥' ❡✐❣❡♥✈❛❧✉❡)µk−µk+1❝♦♥✈❡9❣❡) '♦ ③❡9♦✱ ✇❤✐❧❡µr−µr+1❞✐✈❡9❣❡) '♦ ✐♥✜♥✐'②✳ ❖♥❛')❦✐ ✭✷✵✶✵✮ ❞❡✜♥❡) ❛ ❢❛♠✐❧② ♦❢ ❡)'✐♠❛'♦9)✿

✻■♥ "❤❡ ❝❛'❡ ♦❢ "❤❡ ❡'"✐♠❛"✐♦♥ ✐♥ ❢,❡-✉❡♥❝② ❞♦♠❛✐♥✱ "❤❡ ✏♣,✐♠❡✑ ❞❡♥♦"❡' "❤❡ ❝♦♥❥✉❣❛"❡✲

❝♦♠♣❧❡① ",❛♥'♣♦'❡ ♦❢ "❤❡ ♠❛",✐①✳

✼λi✐' "❤❡i✲"❤ ❧❛,❣❡'" ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❛ ❝♦♠♣❧❡① ❲✐'❤❛,"W_n^C(m, S^e_n(ω0)♦❢ ❞✐♠❡♥'✐♦♥n❛♥❞

❞❡❣,❡❡' ♦❢ ❢,❡❡❞♦♠m✱S_n^e(ω0)✐' "❤❡ '♣❡❝",❛❧ ❞❡♥'✐"② ♠❛",✐❝❡' ♦❢et(n)❛" ❢,❡-✉❡♥❝②ω0.

✽❋♦, ♥♦♥✲●❛✉''✐❛♥ε✱ ❡✐"❤❡, ❆ ♦, ❇ ✐' ,❡-✉✐,❡❞ "♦ ❜❡ ❞✐❛❣♦♥❛❧ "❤❡ ♦"❤❡, ,❡♠❛✐♥✐♥❣ ✉♥,❡✲

'",✐❝"❡❞✳

✶✵