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  1. ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡ ❆❜♦✉❜❛❝❛r ❆♠✐r✐ ∗ ✱ ❈❤r✐st♦♣❤❡ ❈r❛♠❜❡s † ❛♥❞ ❇❛❜❛ ❚❤✐❛♠ ∗ ❏♦✉r♥é❡s ❞❡ ❙t❛t✐st✐q✉❡ ❋♦♥❝t✐♦♥♥❡❧❧❡✱ ▼♦♥t♣❡❧❧✐❡r ✷✽ ❏✉✐♥ ✷✵✶✷ ∗ ✳ ❯♥✐✈❡rs✐té ▲✐❧❧❡ ✸ † ✳ ❯♥✐✈❡rs✐té ▼♦♥t♣❡❧❧✐❡r ✷ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  2. ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ✷ ❆♣♣❧✐❝❛t✐♦♥ ✸ ❈♦♥❝❧✉s✐♦♥ ✹ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  3. ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ■♥tr♦❞✉❝t✐♦♥ ✶ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ✷ ❆♣♣❧✐❝❛t✐♦♥ ✸ ❈♦♥❝❧✉s✐♦♥ ✹ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  4. ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳ ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss ( X k , k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n � ❋♦r❡❝❛st X n +1 ✳ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  5. ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ ✭✶✮ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳ ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss ( X k , k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n � ❋♦r❡❝❛st X n +1 ✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ � ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  6. ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳ ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss ( X k , k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n � ❋♦r❡❝❛st X n +1 ✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ � ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ X n +1 = γ � � X n + (1 − γ ) X n +1 (0 < γ < 1) . ✭✶✮ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  7. ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳ ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss ( X k , k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n � ❋♦r❡❝❛st X n +1 ✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ � ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ X n +1 = γ � � X n + (1 − γ ) X n +1 (0 < γ < 1) . ✭✶✮ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  8. ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ▲❡t ✉s ❝♦♥s✐❞❡r ❛ ♣r♦❝❡ss ( X k , k ≥ 1) , ♦❜s❡r✈❡❞ ✉♥t✐❧ ❛ ❣✐✈❡♥ ✐♥st❛♥t n � ❋♦r❡❝❛st X n +1 ✳ ➨ ❚❤❡ s❛♠♣❧❡ s✐③❡ ✐s ♥♦t ✜①❡❞ ✐♥ ❛❞✈❛♥❝❡✱ ❜✉t t❤❡ ❞❛t❛❜❛s❡ ✐s ❝♦♥t✐♥✉♦✉s❧② ✉♣❞❛t❡❞ � ✉♣❞❛t❡ t❤❡ ❢♦r❡❝❛st ❛t ❡❛❝❤ t✐♠❡✳ ➨ ❘❡❝✉rs✐✈❡ ❛♣♣r♦❛❝❤ ✿ ❡✳❣✳ ❡①♣♦♥❡♥t✐❛❧ s♠♦♦t❤✐♥❣ t❡❝❤♥✐q✉❡s✳ ❚❤❡ ♣r❡❞✐❝t♦r ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ r❡❝✉rs✐✈❡❧② ❜② ✿ X n +1 = γ � � X n + (1 − γ ) X n +1 (0 < γ < 1) . ✭✶✮ ➨ ■t ❝❛♥ ❜❡ ✉♣❞❛t❡❞ ✇✐t❤ ❡❛❝❤ ♥❡✇ ♦❜s❡r✈❛t✐♦♥ ❛❞❞❡❞ t♦ t❤❡ ❞❛t❛❜❛s❡✳ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

  9. ❲❡ ❛r❡ ♥♦t r❡q✉✐r❡❞ t♦ st♦r❡ ❡①t❡♥s✐✈❡ ❞❛t❛ ✐♥ ♦r❞❡r t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ♣r❡❞✐❝t♦r✳ ❯♥❞❡r s♦♠❡ ❝♦♥❞✐t✐♦♥s✱ t❤❡ ♣r❡❞✐❝t✐♦♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❞✉❝❡❞ t♦ t❤❡ ❡st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ♦❢ ❛ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ♦♥ ❛♥ ❡①♣❧❛♥❛t♦r② ♦♥❡✳ ●♦❛❧ ✿ ❊st✐♠❛t✐♦♥ ♦❢ t❤❡ r❡❣r❡ss✐♦♥ ❢✉♥❝t✐♦♥ ❜② ♥♦♥♣❛r❛♠❡tr✐❝ ❦❡r♥❡❧ ❛♣♣r♦❛❝❤ ✇✐t❤ ❦❡❡♣✐♥❣ t❤❡ ❛❞✈❛♥t❛❣❡s ♦❢ ❛ r❡❝✉rs✐✈❡ r❡❧❛t✐♦♥ ❛s ✭✶✮✳ ■♥tr♦❞✉❝t✐♦♥ ❆s②♠♣t♦t✐❝ r❡s✉❧ts ❆♣♣❧✐❝❛t✐♦♥ ❈♦♥❝❧✉s✐♦♥ ▼♦t✐✈❛t✐♦♥s ➨ ❆ ❝♦♥s✐❞❡r❛❜❧❡ ❣❛✐♥ ♦❢ t✐♠❡ ♦❢ ❝♦♠♣✉t❛t✐♦♥✳ ❘❡❝✉rs✐✈❡ ❡st✐♠❛t✐♦♥ ♦❢ ♥♦♥♣❛r❛♠❡tr✐❝ r❡❣r❡ss✐♦♥ ✇✐t❤ ❢✉♥❝t✐♦♥❛❧ ❝♦✈❛r✐❛t❡

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