Passeios aleatórios e grandes desvios em ambientes aleatórios

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

P❛ss❡✐♦s ❆❧❡❛tór✐♦s ❡ ●r❛♥❞❡s ❉❡s✈✐♦s

❡♠ ❆♠❜✐❡♥t❡s ❆❧❡❛tór✐♦s

❏❡❛♥♥❡ ❈❛r♠♦ ❆♠❛r❛❧

(2)

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❏❡❛♥♥❡ ❈❛r♠♦ ❆♠❛r❛❧

P❛ss❡✐♦s ❆❧❡❛tór✐♦s ❡ ●r❛♥❞❡s ❉❡s✈✐♦s

❡♠ ❆♠❜✐❡♥t❡s ❆❧❡❛tór✐♦s

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❝♦r♣♦ ❞♦❝❡♥t❡ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❞♦ ■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ▼✐♥❛s ●❡r❛✐s✱ ❝♦♠♦ ♣❛rt❡ ❞♦s r❡q✉✐s✐t♦s ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛✲ t❡♠át✐❝❛✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢❡ss♦r ❙❛❝❤❛ ❋r✐❡❞❧✐

(3)

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s✱ ♣♦r ♠✐♥❤❛ ✈✐❞❛ ❡ ♣❡❧❛s ❝♦✐s❛s ❜♦❛s q✉❡ ❝♦♥q✉✐st❡✐ ♥♦s ú❧t✐♠♦s ❞♦✐s ❛♥♦s✳ ❆♦ ❙❛❝❤❛✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✱ ❞❡❞✐❝❛çã♦ ❡ ❞✐s♣♦♥✐❜✐❧✐❞❛❞❡✳

❆♦s ♣r♦❢❡ss♦r❡s ❇❡r♥❛r❞♦✱ ▼❛r✐❛ ❊✉❧á❧✐❛ ❡ ❘❡♠② q✉❡ ❝♦♠♣✉s❡r❛♠ ❛ ❜❛♥❝❛✱ ♣❡❧❛ ❞✐s♣♦♥✐✲ ❜✐❧✐❞❛❞❡ ❡ ♣❡❧❛s s✉❣❡stõ❡s✳

❆♦s ❞❡♠❛✐s ♣r♦❢❡ss♦r❡s ❞♦ ❞❡♣❛rt❛♠❡♥t♦✱ ♣❡❧♦s ❡♥s✐♥❛♠❡♥t♦s✳ ❆♦ ❈♥♣q✱ ♣❡❧♦ ❛♣♦✐♦ ✜♥❛♥❝❡✐r♦✳

➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❝♦♠ ❛ q✉❛❧ ❞✐✈✐❞✐ ♠✉✐t♦s ♠♦♠❡♥t♦s ❞❡ ❛❧❡❣r✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ à ♠✐♥❤❛ ♠ã❡✱ q✉❡ ❛♠♦ t❛♥t♦ ❡ q✉❡ ❡stá s❡♠♣r❡ ♣r♦♥t❛ q✉❛♥❞♦ ❡✉ ♣r❡❝✐s♦✳

❆♦ ❏✉❧✐❛♥♦✱ ♣❡❧♦ ❝❛r✐♥❤♦ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦✳

(4)

❘❡s✉♠♦

❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦✳ ❈♦♠ ❜❛s❡ ♥♦ ❛rt✐❣♦ ✏❘❛♥❞♦♠ ❲❛❧❦ ✐♥ ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥t✑ ❡s❝r✐t♦ ♣♦r ❖✳ ❩❡✐t♦✉♥✐✱ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ❈r✐tér✐♦ ❞❡ ❘❡❝♦rrê♥❝✐❛ ❡ ❚r❛♥s✐ê♥❝✐❛ ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡ ❛ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ♣❛r❛ ♦ ♠❡s♠♦✱ ♦❜t✐❞♦s ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ❋✳ ❙♦❧♦♠♦♥✳ ❊♠ s❡❣✉✐❞❛✱ ♦ ❡st✉❞♦ ❞♦s ❣r❛♥❞❡s ❞❡s✈✐♦s ♣❛r❛ ❛ ✈❡❧♦❝✐❞❛❞❡ ❞♦ ♣❛ss❡✐♦✱ s❡rá ❜❛s❡❛❞♦ ♥♦ ❛r✲ t✐❣♦ ✏◗✉❡♥❝❤❡❞✱ ❆♥♥❡❛❧❡❞ ❛♥❞ ❋✉♥❝t✐♦♥❛❧ ▲❛r❣❡ ❉❡✈✐❛t✐♦♥s ❢♦r ❖♥❡✲❉✐♠❡♥s✐♦♥❛❧ ❘❛♥❞♦♠ ❲❛❧❦ ✐♥ ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥t✑ ❞❡ ❋✳ ❈♦♠❡ts✱ ◆✳ ●❛♥t❡rt ❡ ❖✳ ❩❡✐t♦✉♥✐✳ ❈♦♥s✐❞❡r❛r❡♠♦s ♦s ❣r❛♥❞❡s ❞❡s✈✐♦s ♣❛r❛ ♦ ♣❛ss❡✐♦ ❝♦♥❞✐❝✐♦♥❛❞♦ ❛♦ ❛♠❜✐❡♥t❡ ✭q✉❡♥❝❤❡❞✮ ❡ s♦❜r❡ ❛ ♠é❞✐❛ ❡♠ r❡❧❛çã♦ ❛♦ ❛♠❜✐❡♥t❡ ✭❛♥♥❡❛❧❡❞✮✳

(5)

❆❜str❛❝t

❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ st✉❞② t❤❡ ♣r♦♣❡rt✐❡s ♦❢ r❛♥❞♦♠ ✇❛❧❦ ✐♥ r❛♥❞♦♠ ❡♥✲ ✈✐r♦♥♠❡♥t✳ ❇❛s❡❞ ♦♥ t❤❡ ❛rt✐❝❧❡ ✏ ❘❛♥❞♦♠ ❲❛❧❦ ✐♥ ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥t✑❜② ❖✳ ❩❡✐t♦✉♥✐✱ ✇❡ ♣r❡s❡♥t t❤❡ ❈r✐t❡r✐♦♥ ♦❢ ❘❡❝✉rr❡♥❝❡ ❛♥❞ ❚r❛♥s✐❡♥❝❡ ♦❢ r❛♥❞♦♠ ✇❛❧❦ ❛♥❞ t❤❡ ▲❛✇ ♦❢ ▲❛r❣❡ ◆✉♠❜❡rs ❢♦r t❤❡ s❛♠❡✱ ♦❜t❛✐♥❡❞ ✐♥✐t✐❛❧❧② ❜② ❋✳ ❙♦❧♦♠♦♥✳ ◆❡①t✱ t❤❡ st✉❞② ♦❢ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r t❤❡ ✇❛❧❦ ✇✐❧❧ ❜❡ ❜❛s❡❞ ♦♥ t❤❡ ❛rt✐❝❧❡ ✏ ◗✉❡♥❝❤❡❞✱ ❆♥♥❡❛❧❡❞ ❛♥❞ ❋✉♥❝t✐♦✲ ♥❛❧ ▲❛r❣❡ ❉❡✈✐❛t✐♦♥s ❢♦r ❖♥❡✲❉✐♠❡♥s✐♦♥❛❧ ❘❛♥❞♦♠ ❲❛❧❦ ✐♥ ❘❛♥❞♦♠ ❊♥✈✐r♦♥♠❡♥t✑❜② ❋✳ ❈♦♠❡ts✱ ◆✳ ●❛♥t❡rt ❛♥❞ ❖✳ ❩❡✐t♦✉♥✐✳ ❲❡ ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ❧❛r❣❡ ❞❡✈✐❛t✐♦♥s ❢♦r t❤❡ ✇❛❧❦ ❝♦♥❞✐t✐♦♥✐♥❣ t♦ ❡♥✈✐r♦♥♠❡♥t ✭q✉❡♥❝❤❡❞✮ ❛♥❞ ❛❜♦✉t t❤❡ ❛✈❡r❛❣❡ ✐♥ r❡❧❛t✐♦♥ t♦ ❡♥✈✐r♦♥♠❡♥t ✭❛♥♥❡❛❧❡❞✮✳

(6)

(7)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✶

✶✳ ❖ P❛ss❡✐♦ ❆❧❡❛tór✐♦ ❙✐♠♣❧❡s ✶

✷✳ ❖ ❆♠❜✐❡♥t❡ ❆❧❡❛tór✐♦ ✷

✸✳ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ❡ ●r❛♥❞❡s ❉❡s✈✐♦s ✸

❈❛♣ít✉❧♦ ✶✳ ❖ P❛ss❡✐♦ ❆❧❡❛tór✐♦ ❡♠ ❆♠❜✐❡♥t❡s ❆❧❡❛tór✐♦s ❡♠Z ✺

✶✳ ❖ ▼♦❞❡❧♦ Pr♦❜❛❜✐❧íst✐❝♦ ✺

✷✳ ❖ ❈r✐tér✐♦ ❞❡ ❘❡❝♦rrê♥❝✐❛✴❚r❛♥s✐ê♥❝✐❛ ✻

✸✳ ❆ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ✾

❈❛♣ít✉❧♦ ✷✳ ❖ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ◗✉❡♥❝❤❡❞ ✶✼

✶✳ Pr♦♣r✐❡❞❛❞❡s ❞❡ ϕ✱Λ ❡ Λ∗ _✶✽

✷✳ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ❋r❛❝♦ ♣❛r❛ Tn n ❡

T−n

n ✷✸

✸✳ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ♣❛r❛ Sn

n ✸✵

❈❛♣ít✉❧♦ ✸✳ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ❆♥♥❡❛❧❡❞ ✸✼

✶✳ ❈♦t❛ ❙✉♣❡r✐♦r ♣❛r❛ ❋❡❝❤❛❞♦s ✸✽

✷✳ ❈♦t❛ ■♥❢❡r✐♦r ♣❛r❛ ❆❜❡rt♦s ✹✸

❆♣ê♥❞✐❝❡ ❆✳ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ✹✼

❆♣ê♥❞✐❝❡ ❇✳ ❋✉♥çõ❡s ❍❛r♠ô♥✐❝❛s ✺✶

❆♣ê♥❞✐❝❡ ❈✳ ❯♠ ❚❡♦r❡♠❛ ❞❡ ❑❡st❡♥ ✺✸

❆♣ê♥❞✐❝❡✳ ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✼

(8)

(9)

■♥tr♦❞✉çã♦

✶✳ ❖ P❛ss❡✐♦ ❆❧❡❛tór✐♦ ❙✐♠♣❧❡s

❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ❡♠Z❝♦♥s✐st❡ ❞❡ ✉♠❛ ♣❛rtí❝✉❧❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♥❛ ♦r✐❣❡♠✱ q✉❡ s❡ ♠♦✈❡ ♥♦s sít✐♦s ❞❡Z❡ ❛ ❝❛❞❛ ✐♥st❛♥t❡ ♣♦❞❡ ♣✉❧❛r ❞❡ ✉♠ ♣♦♥t♦x♣❛r❛ ✉♠ ❞❡ s❡✉s ♣ró①✐♠♦s ✈✐③✐♥❤♦s x+ 1 ♦✉ x−1✱ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p∈[0,1] ❞❡ ♣✉❧❛r ♣❛r❛ ♦ sít✐♦ à s✉❛ ❞✐r❡✐t❛ ❡ ♣r♦❜❛❜✐❧✐❞❛❞❡q= 1−p♣❛r❛ ♦ sít✐♦ à s✉❛ ❡sq✉❡r❞❛✳ ❖ ♣❛ss❡✐♦ é ❝❤❛♠❛❞♦ s✐♠étr✐❝♦ q✉❛♥❞♦ p = 1₂✳ ❉❡✜♥✐♠♦s S0 := 0 ❡ ♣❛r❛ n ≥ 1✱ Sn ❞❡♥♦t❛ ❛ ♣♦s✐çã♦ ❞❛ ♣❛rtí❝✉❧❛ ♥♦ ✐♥st❛♥t❡

n✳ ❊♥tã♦✱ (Sn)n≥0 ❞❡s❝r❡✈❡ ✉♠❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡

tr❛♥s✐çã♦✿

P(Sn+1 =x+ 1|Sn =x) = p, P(Sn+1 =x−1|Sn=x) = q= 1−p.

x+ 1 x−1 x

Z p

1−p

❋✐❣✉r❛ ✶✳ ❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s✳

❊st❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s sã♦ ❤♦♠♦❣ê♥❡❛s ♥♦ t❡♠♣♦✱ ♥♦ s❡♥t✐❞♦ q✉❡ p(❡ q) ♥ã♦ ❞❡♣❡♥❞❡♠ ❞❛ ♣♦s✐çã♦ x✳ ❉❡s❝r❡✈❡r❡♠♦s ❛ s❡❣✉✐r✱ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❜❡♠ ❝♦♥❤❡❝✐❞❛s ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ✭✈❡r ❬❋❡❧✼✶❪✮✿

− ❉❡✜♥✐♠♦s ♦ t❡♠♣♦ ❞❡ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ❡♠x✱

Tx := inf{n≥1;Sn=x},

❝♦♠ ❛ ❝♦♥✈❡♥çã♦ q✉❡ ♦ í♥✜♠♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦ é ✐❣✉❛❧ ❛∞✳ ❯♠ ♣❛ss❡✐♦

❛❧❡❛tór✐♦ é ❝❤❛♠❛❞♦ r❡❝♦rr❡♥t❡ s❡P(T0 <∞) = 1❡ tr❛♥s✐❡♥t❡ s❡P(T0 <∞)<1.

❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s é r❡❝♦rr❡♥t❡ s❡ ❡ s♦♠❡♥t❡ s❡p= 1₂✱ ❥á q✉❡

P(T0 <∞) = 1− |p−q|.

❆❧é♠ ❞✐ss♦✱ ♣❛r❛ ♦ ❝❛s♦ s✐♠étr✐❝♦✱ E[T0] ❡E[T1]sã♦ ❛♠❜❛s ✐♥✜♥✐t❛s✳

− ❆ ▲❡✐ ❋♦rt❡ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ✐♠♣❧✐❝❛ q✉❡ P q✉❛s❡ ❝❡rt❛♠❡♥t❡✱

Sn

n →2p−1≡vp.

(10)

✷ ■◆❚❘❖❉❯➬➹❖

❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠♦s q✉❡ P lim

n→∞Sn =∞

= 1✱ q✉❛♥❞♦ p > 1₂ ❡ P lim

n→∞Sn =

∞= 1✱ q✉❛♥❞♦p < 1 2✳

✷✳ ❖ ❆♠❜✐❡♥t❡ ❆❧❡❛tór✐♦

❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ é ✉♠❛ ♠♦❞✐✜❝❛çã♦ ❡♠ q✉❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ♣♦❞❡♠ ❞❡♣❡♥❞❡r ❞❛ ♣♦s✐çã♦ x✿

P(Sn+1 =x+ 1|Sn=x) =px, P(Sn+1 =x−1|Sn =x) = qx = 1−px.

x+ 1 x−1 x

Z 1−px px

❋✐❣✉r❛ ✷✳ ❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦✳

❈♦♠♦ ❛ ❝♦❧❡çã♦ (px, qx)x∈Z s❡rá ❡♠ ❣❡r❛❧ ❛❧❡❛tór✐❛✱ ❡❧❛ s❡rá ❞❡♥♦t❛❞❛ ♣♦r ✉♠❛ ❞✉♣❧❛

s❡q✉ê♥❝✐❛ω= (ωx)x∈Z ❝❤❛♠❛❞❛ ❞❡ ❛♠❜✐❡♥t❡✱ ❞❡ ❢♦r♠❛ q✉❡ωx ∈[0,1]✱ωx ≡px ❡1−ωx ≡

qx. ◗✉❛♥❞♦ ♦ ❛♠❜✐❡♥t❡ é ✜①♦✱ ❛ ❞✐str✐❜✉✐çã♦ ❛❝✐♠❛ é ❞❡♥♦t❛❞❛ ♣♦r Pω ❡ é ❝❤❛♠❛❞❛ ❞❡

❞✐str✐❜✉✐çã♦ q✉❡♥❝❤❡❞✳

❉❡♥♦t❛r❡♠♦s ♣♦r ωp _{♦ ❛♠❜✐❡♥t❡ ❡♠ q✉❡} _ωp

x = p✱ ♣❛r❛ t♦❞♦ x ∈ Z✱ ♦✉ s❡❥❛✱ ωp ❞❡s❝r❡✈❡

❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ p ❞❡ tr❛♥s✐çã♦ ♣❛r❛ à ❞✐r❡✐t❛ ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r ωSRW _{♦ ❛♠❜✐❡♥t❡ ❡♠ q✉❡} _ωSRW

x = 12✱ ♣❛r❛ t♦❞♦x∈Z✱ ♦✉ s❡❥❛✱

ωSRW _{❞❡s❝r❡✈❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s s✐♠étr✐❝♦✳}

❙❡ A é ✉♠ ❡✈❡♥t♦ q✉❡ ❞❡♣❡♥❞❡ s♦♠❡♥t❡ ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ Sn✱ ❡♥tã♦ ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡

❛♥♥❡❛❧❡❞✶ _{❞♦ ❡✈❡♥t♦ é ❞❡✜♥✐❞❛ ❝♦♠♦}

P(A) =

Z

Ω

Pω(A)P(dω),

❡♠ q✉❡ P é ✉♠❛ ♠❡❞✐❞❛ s♦❜r❡ ♦ ❝♦♥❥✉♥t♦s ❞♦s ❛♠❜✐❡♥t❡s ω✱ ❞❡♥♦t❛❞♦ ♣♦r Ω✳

❉✐③❡♠♦s q✉❡ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ é r❡❝♦rr❡♥t❡ s♦❜ P ✭♦✉ s♦❜Pω) s❡

P(T0 <∞) = 1 ( ♦✉Pω(T0 <∞) = 1)❡ é tr❛♥s✐❡♥t❡ s♦❜ P✭♦✉ s♦❜ Pω)s❡P(T0 <∞)<1

( ♦✉Pω(T0 <∞)<1)✳

❆s q✉❡stõ❡s ♠❛✐s ♥❛t✉r❛✐s s♦❜r❡ ♦ ♣❛ss❡✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ sã♦✿ ◗✉❛✐s sã♦ ❛s ♣r♦♣r✐✲ ❡❞❛❞❡s tí♣✐❝❛s ❞♦ ♣❛ss❡✐♦ ♣❛r❛ ✉♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦❄ ❙♦❜ ❤✐♣ót❡s❡s ❛❞✐❝✐♦♥❛✐s✱ ♦ ♣❛ss❡✐♦ é ✉♠❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ✐rr❡❞✉tí✈❡❧ s♦❜Pω✱ ❧♦❣♦✱ ♦ ♣❛ss❡✐♦ ♦✉ é r❡❝♦rr❡♥t❡ ♦✉ é tr❛♥s✐❡♥t❡✳

◗✉❡♠ é ♦ ♣❛râ♠❡tr♦ ♥❛t✉r❛❧❄ ❊①✐st❡ ✉♠❛ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ♣❛r❛ Sn

n❄ ❚❡♠ ❞♦✐s

(11)

✸✳ ▲❊■ ❉❖❙ ●❘❆◆❉❊❙ ◆Ú▼❊❘❖❙ ❊ ●❘❆◆❉❊❙ ❉❊❙❱■❖❙ ✸

t✐♣♦s ❞❡ ❞❡❝r✐çõ❡s✿ q✉❡♥❝❤❡❞✱ r❡❧❛t✐✈♦ à ♠❡❞✐❞❛ Pω ♣❛r❛ ✉♠ ❛♠❜✐❡♥t❡ ✜①♦ ❡ ❛♥♥❡❛❧❡❞✱

r❡❧❛t✐✈♦ à ♠❡❞✐❞❛P✳

◆♦ ❈❛♣ít✉❧♦ ✷✱ ✈❡r❡♠♦s ♦ ❈r✐tér✐♦ ❞❡ ❘❡❝♦rrê♥❝✐❛✴❚r❛♥s✐ê♥❝✐❛ ♦❜t✐❞♦ ♣♦r ❙♦❧♦♠♦♥ ❬❙♦❧✼✺❪✱ q✉❡ ♥♦s ❢♦r♥❡❝❡ ♣❛r❛ P ❡r❣ó❞✐❝❛✱ ♦ ♣❛râ♠❡tr♦ EP[logρ0]✱ ❡♠ q✉❡ ρx := 1−_ωxωx✳ ❖ ❝r✐tér✐♦

❞✐③ q✉❡ ♦ ♣❛ss❡✐♦ é r❡❝♦rr❡♥t❡ s❡ ❡ s♦♠❡♥t❡ s❡ EP[logρ0] = 0 ❡ tr❛♥s✐❡♥t❡ s❡ ❡ s♦♠❡♥t❡

s❡ EP[logρ0] 6= 0✳ ❙✉❛ ❞❡♠♦♥str❛çã♦ é ❜❛s❡❛❞❛ ♥✉♠❛ té❝♥✐❝❛ ❡❧❡♠❡♥t❛r✿ ❛ r✉í♥❛ ❞♦

❛♣♦st❛❞♦r✳

✸✳ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ❡ ●r❛♥❞❡s ❉❡s✈✐♦s

◆❡st❡ tr❛❜❛❧❤♦✱ ❡st❛♠♦s ✐♥t❡r❡ss❛❞♦s ❡♠ ❡st✉❞❛r ♦ ❡❢❡✐t♦ ❞❛ ❛❧❡❛t♦r✐❡❞❛❞❡ ❞♦ ❛♠❜✐❡♥t❡ s♦❜r❡ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❣r❛♥❞❡s ❞❡s✈✐♦s ❞❡ Sn

n✳ ❱✐♠♦s q✉❡ ♣❛r❛ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦

s✐♠♣❧❡s✱

Sn

n →2p−1≡vp, P✲q✳❝✳ ❆s ✢✉t✉❛çõ❡s ❞❡ ♦r❞❡♠ n ❞❡ Sn

n ❡♠ t♦r♥♦ ❞❡ vp ♣♦ss✉❡♠ ✉♠❛ ❞❡s❝r✐çã♦ ❡♠ t❡r♠♦s ❞❡

✉♠ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s✷ _{✭P●❉✮✳ P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❈r❛♠ér✱ ✭❚❡♦r❡♠❛ ❆✳✷ ❞♦} ❆♣ê♥❞✐❝❡ ❆✮✿ ✭✉♠ s❡♥t✐❞♦ ♣r❡❝✐s♦ s❡rá ❞❛❞♦ à s❡❣✉✐♥t❡ ❡①♣r❡ssã♦✮

✭✶✮ P Sn

n ∈[a, b]

≃❡①♣ −n inf

v∈[a,b]Ip(v)

,

❡♠ q✉❡ Ip é ❛ ❢✉♥çã♦ t❛①❛✱ ❞❛❞❛ ♣❡❧❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡

Ip(v) := sup t∈R

{tv−Λ(t)},

❡Λé ❛ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♠♦♠❡♥t♦s ❞♦s ✐♥❝r❡♠❡♥t♦ ❞♦ ♣❛ss❡✐♦✱Xi :=Si−Si−1✱ ❞❡✜♥✐❞❛

♣♦r

Λ(t) := logE[etX1_]_. ❆tr❛✈és ❞❡ ✉♠❛ ❝♦♥t❛ ❡①♣❧í❝✐t❛✱ ❡♥❝♦♥tr❛♠♦s

Ip(v) =

1 +v 2 log

_{1 +}_v

2p

+ 1−v 2 log

₁₋_v

2(1−p)

.

❙✉♣♦♥❤❛ q✉❡ vp ∈/ [a, b]✳ ❊♥tã♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❋✐❣✉r❛ ✸✱ inf

v∈[a,b]Ip(v) > 0✳ ❈♦♠ ✐ss♦✱ ❛

❡①♣r❡ssã♦ ✭✶✮ ❛❝✐♠❛ ❞✐③ q✉❡ ❤á ✉♠❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡①♣♦♥❡♥❝✐❛❧♠❡♥t❡ ♣❡q✉❡♥❛ ❞❡ ♦❜s❡r✈❛r

Sn

n ♥♦ ✐♥t❡r✈❛❧♦ [a, b]. ■st♦ é✱ ♦ P●❉ ❛❝✐♠❛ ❞❡s❝r❡✈❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ✈❛❧♦r❡s ❛tí♣✐❝♦s

❞❡ Sn n✳

▼❛✐♦r ♣❛rt❡ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❞❡❞✐❝❛❞❛ ❛♦ ❡st✉❞♦ ❞❡ ✉♠ tr❛❜❛❧❤♦ ❞❡ ❈♦♠❡ts✱ ●❛♥t❡rt ❡ ❩❡✐t♦✉♥✐ ❬❈●❩✵✵❪ ❡♠ q✉❡ ✉♠ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ✭P●❉✮ ❞❛ ❢♦r♠❛ ✭✶✮ é ♦❜t✐❞♦ q✉❛♥❞♦ ♦ ❛♠❜✐❡♥t❡ é ❛❧❡❛tór✐♦✳ ❊st❡ P●❉ s❡rá ♦❜t✐❞♦ t❛♥t♦ ♣❛r❛ ❛ ♠❡❞✐❞❛ q✉❡♥❝❤❡❞ q✉❛♥t♦ ♣❛r❛ ❛ ❛♥♥❡❛❧❡❞✳ ◆♦ ❈❛♣ít✉❧♦ ✷ ✈❡r❡♠♦s q✉❡ ❡①✐st❡ ✉♠❛ ❢✉♥çã♦ t❛①❛Iq✉❡ ❞❡♣❡♥❞❡ ❞❡ P✱ q✉❡ ❢♦r♥❡❝❡ ♦ P●❉ s♦❜ Pω ♣❛r❛ P✲q✳t✳ω ❡ ♥♦ ❈❛♣ít✉❧♦ ✸✱ ♦❜t❡r❡♠♦s ♦ P●❉ s♦❜ P

❝♦♠ ❢✉♥çã♦ t❛①❛ I,q✉❡ t❛♠❜é♠ ❞❡♣❡♥❞❡ ❞❡ P✳ ❙❡ P é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✱ ❡♥tã♦✿ P●❉ ❛♥♥❡❛❧❡❞✿ P❛r❛ P✲q✉❛s❡ t♦❞♦ ❛♠❜✐❡♥t❡ ω✱

✭✷✮ Pω Sn_n ∈[a, b]

≃❡①♣ −n inf

v∈[a,b]I(v)

.

(12)

✹ ■◆❚❘❖❉❯➬➹❖

Ip(v)

v

−1 +1

✭❛✮

vp −1 +1

v

✭❜✮ Ip(v)

vp

❋✐❣✉r❛ ✸✳ ❊s❜♦ç♦ ❞❡ Ip q✉❛♥❞♦✿ ✭❛✮ p= 1₂ (vp = 0)✱ ✭❜✮ p= 3₄ (vp >0)✳

P●❉ q✉❡♥❝❤❡❞✿

✭✸✮ P Sn_n ∈[a, b]≃❡①♣ −n inf

v∈[a,b]I(v)

.

(13)

❈❆Pí❚❯▲❖ ✶

❖ P❛ss❡✐♦ ❆❧❡❛tór✐♦ ❡♠ ❆♠❜✐❡♥t❡s ❆❧❡❛tór✐♦s ❡♠

Z

❖ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ ❢♦✐ ❡st✉❞❛❞♦ ♣❡❧❛ ♣r✐♠❡✐r❛ ✈❡③ ♣♦r ❙♦❧♦♠♦♥ ❬❙♦❧✼✺❪✳ ◆❡st❡ ❝❛♣ít✉❧♦ s❡❣✉✐r❡♠♦s ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ❩❡✐t♦✉♥✐ ❬❩❡✐✵✷❪✳ ◆❛ ❙❡çã♦ ✶✳✶ ❞❡✜♥✐r❡♠♦s ♦ ♠♦❞❡❧♦✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ❛s ❞✉❛s ❞❡s❝r✐çõ❡s ♣r✐♥❝✐♣❛✐s ❞♦ ♣❛ss❡✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ ✭✐✳❡✳ q✉❡♥❝❤❡❞ ❡ ❛♥♥❡❛❧❡❞✮✳ ◆❛ ❙❡çã♦ ✶✳✷ ❞❛r❡♠♦s ♦ ❈r✐tér✐♦ s♦❜r❡ ❘❡❝♦rrê♥❝✐❛ ❡ ❚r❛♥s✐ê♥❝✐❛ ❡ ♥❛ ❙❡çã♦ ✶✳✸ ❞❛r❡♠♦s ❛ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s ♣❛r❛ Sn

n✳

✶✳ ❖ ▼♦❞❡❧♦ Pr♦❜❛❜✐❧íst✐❝♦

❆ ❞❡✜♥✐çã♦ ❞❡ ✉♠ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦✱ ❡♥✈♦❧✈❡ ❞♦✐s ❝♦♠♣♦♥❡♥t❡s✿ ♣r✐✲ ♠❡✐r♦✱ ♦ ❛♠❜✐❡♥t❡✱ q✉❡ é ❛❧❡❛t♦r✐❛♠❡♥t❡ ❡s❝♦❧❤✐❞♦ ♠❛s é ♠❛♥t✐❞♦ ✜①♦ ❞✉r❛♥t❡ ❛ ❡✈♦❧✉çã♦ ❞♦ t❡♠♣♦ ❡ s❡❣✉♥❞♦✱ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦✱ q✉❡ ❞❛❞♦ ♦ ❛♠❜✐❡♥t❡✱ é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❤♦♠♦❣ê♥❡❛ ❝✉❥❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞❡♣❡♥❞❡♠ ❞♦ ❛♠❜✐❡♥t❡✳ ❖ ❛♠❜✐❡♥t❡ é ❞❡✲ ✜♥✐❞♦ ❝♦♠♦ ✉♠❛ s❡q✉ê♥❝✐❛ ω := (ωx)x∈Z ❡♠ q✉❡ ωx ∈ [0,1] ♣❛r❛ t♦❞♦ x ∈ Z✳ ❙❡❥❛ Ω ♦

❝♦♥❥✉♥t♦ ❞♦s ❛♠❜✐❡♥t❡s ❛❧❡❛tór✐♦s ω✱ ❡q✉✐♣❛❞♦ ❝♦♠ ❛ σ✲á❧❣❡❜r❛ ❞❡ ❇♦r❡❧ F✳

❊♠ZN✱ s❡❥❛G ❛ _σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❝✐❧✐♥❞r♦s✳ ❋✐①❛❞♦s_ω ∈Ω ❡z ∈Z✱ s❡❥❛ (Sn)n≥0 ❛

❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❡♠ Z t❛❧ q✉❡ Pz

ω(S0 =z) = 1 ❡ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞❡ tr❛♥s✐çã♦ ❞❛❞❛s

♣♦r

Pωz(Sn+1 =x+ 1|Sn =x) = ωx, Pωz(Sn+1 =x−1|Sn =x) = 1−ωx.

❊♥tã♦ (Sn)n≥0 ❞❡♥♦t❛ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦ ω ❡ Pωz ❞❡♥♦t❛ ❛ ❧❡✐

q✉❡♥❝❤❡❞ ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦✳ ◆♦ ❣❡r❛❧✱ s✉♣♦r❡♠♦s s❡♠♣r❡ q✉❡ z ≡ 0 ❡ ❞❡♥♦t❛r❡♠♦s P0

ω ≡ Pω✳ P❛r❛ ❡✈✐t❛r ❝❛s♦s tr✐✈✐❛✐s✱ t❛✐s ❝♦♠♦ ωx = 1✱ ♣❛r❛ t♦❞♦ x ∈ Z✱ s✉♣♦r❡♠♦s

s❡♠♣r❡ q✉❡ P é ❡❧í♣t✐❝❛✱ ✐st♦ é✱ q✉❡ ❡①✐st❡ǫ∈ 0,1₂ t❛❧ q✉❡ ǫ≤ωx ≤1−ǫ, ∀ x∈Z.

P❛r❛ t♦❞♦ G ∈ G✱ ❛ ❢✉♥çã♦ _ω 7→ Pω(G) é F✲♠❡♥s✉rá✈❡❧✳ ❉❡✜♥✐♠♦s ❡♥tã♦ ❛ ❧❡✐ ❛♥♥❡❛❧❡❞

❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ P✱ ❡♠ (Ω×ZN_,_F_×_G₎_♣♦r

P(F ×G) :=

Z

F

Pω(G)P(dω), F ∈F, G∈G.

❆ ❞✐✜❝✉❧❞❛❞❡ ❡♠ t❡♥t❛r ❞✐③❡r ❛❧❣✉♠❛ ❝♦✐s❛ s♦❜r❡ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s❡♠ s❛❜❡r q✉❡ ❛♠✲ ❜✐❡♥t❡ ♣❛rt✐❝✉❧❛r ω ❢♦✐ ❡s❝♦❧❤✐❞♦ é q✉❡✱ ❡♠ ❣❡r❛❧✱ Sn ♥ã♦ é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ s♦❜

P✳✶

✶_{P♦❞❡ s❡r ✈❡r✐✜❝❛❞♦ q✉❡ q✉❛♥❞♦} _P _{é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ♥ã♦ ❞❡❣❡♥❡r❛❞❛✱ ❡♥tã♦✱} _P₍_S₃ _{= 1}_|_S₂ ₌

0, S1= 1)6=P(S3= 1|S2= 0, S1=−1).

(14)

✻ ✶✳ ❖ P❆❙❙❊■❖ ❆▲❊❆❚Ó❘■❖ ❊▼ ❆▼❇■❊◆❚❊❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼Z

◆♦t❡ q✉❡ ✉♠ ❡✈❡♥t♦ r❡❧❛t✐✈♦ ❛♦ ♣❛ss❡✐♦ Sn

n q✉❡ ♦❝♦rr❡P✲q✳❝✳✱ t❛♠❜é♠ ♦❝♦rr❡ Pω✲q✳❝✳ ♣❛r❛

P q✉❛s❡ t♦❞♦ ❛♠❜✐❡♥t❡ ω✳ ❖✉ s❡❥❛✱ ♣❛r❛ t♦❞♦ A∈G✱

✭✹✮ P(Ω×A) = 1⇒Pω(A) = 1, ♣❛r❛ P✲q✳t✳ω.

■ss♦ ♣♦rq✉❡P(Ω×Ac_{) =} R

ΩPω(A

c₎_P₍_dω_{) = 0} _{✐♠♣❧✐❝❛}_P

ω(Ac) = 0, ♣❛r❛ P✲q✳t✳ω✳

❉❡♥♦t❛r❡♠♦s ♣♦rθ♦ s❤✐❢t ❞♦ ❛♠❜✐❡♥t❡✱ ♦✉ s❡❥❛✱ ♣❛r❛i∈Z❡x∈Z✱θi_ω_{❞❡♥♦t❛ ♦ ❛♠❜✐❡♥t❡}

❡♠ q✉❡(θi_ω₎

x =ωx+i✳ P❛r❛F ∈F✱ s❡❥❛ θiF :={ω :θiω ∈F}✳ ❯♠❛ ♠❡❞✐❞❛ P ❡♠ (Ω,F)

é ❞✐t❛ ❡st❛❝✐♦♥ár✐❛ s❡

P(F) =P(θiF), ∀F ∈F_{, i}∈Z.

❯♠ ❝♦♥❥✉♥t♦ F ∈F é ✐♥✈❛r✐❛♥t❡ s❡_θ−1_F ₌_F_{✳ ❯♠❛ ♠❡❞✐❞❛ ❡st❛❝✐♦♥ár✐❛} _P _❡♠ _(Ω_,_F₎ _é

❡r❣ó❞✐❝❛ s❡

P(F)∈ {0,1}, ∀ F ∈F ✐♥✈❛r✐❛♥t❡✳

❉✐r❡♠♦s q✉❡ ✉♠❛ ♠❡❞✐❞❛ P ❡♠ (Ω,F) é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✱ s❡ ❛ s❡q✉ê♥❝✐❛ (ωx)x∈Z ❢♦r

✐♥❞❡♣❡♥❞❡♥t❡ ❡ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐str✐❜✉í❞❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ P é ✉♠❛ ♣r♦❞✉t♦✱ ❡♥tã♦P é ❡st❛❝✐♦♥ár✐❛ ❡ ❡r❣ó❞✐❝❛✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ❛❧❣✉♥s r❡s✉❧t❛❞♦s ♣❛r❛ ❛s ♠❡❞✐❞❛s P ❡st❛❝✐♦♥ár✐❛s ❡ ❡r❣ó❞✐❝❛s✳ ◆♦s ❝❛♣ít✉❧♦s s❡❣✉✐♥t❡s ♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s s❡rã♦ ♣❛r❛P ♣r♦❞✉t♦✳

❊s❝r❡✈❡r❡♠♦s Eω ♣❛r❛ ❡s♣❡r❛♥ç❛s ❝♦♠ r❡❧❛çã♦ à Pω✱ E ♣❛r❛ ❡s♣❡r❛♥ç❛s ❝♦♠ r❡❧❛çã♦

à P ❡ E ♣❛r❛ ❡s♣❡r❛♥ç❛s ❝♦♠ r❡❧❛çã♦ à P✳ P❛r❛ P ❡r❣ó❞✐❝❛✱ ✉t✐❧✐③❛r❡♠♦s ❡♠ ♠✉✐t❛s ❞❡♠♦♥str❛çõ❡s ♦ ❚❡♦r❡♠❛ ❊r❣ó❞✐❝♦ ❞❡ ❇✐r❦❤♦✛ ✭✈❡r ❬❙❤✐✾✻❪✱ ♣✳✹✵✾✮✱ q✉❡ ❢♦r♥❡❝❡ ♣❛r❛ ❛s ❢✉♥çõ❡s f : Ω→R✱ ♦ ❧✐♠✐t❡ ❞❛ sér✐❡

1 n

n−1

X

i=0

f(θi_ω₎_→_E_[_f_]_, _♣❛r❛ _P_✲q✳t✳_ω.

✷✳ ❖ ❈r✐tér✐♦ ❞❡ ❘❡❝♦rrê♥❝✐❛✴❚r❛♥s✐ê♥❝✐❛

❖ ♣r✐♠❡✐r♦ r❡s✉❧t❛❞♦ q✉❡ ❛♣r❡s❡♥t❛r❡♠♦s ♥♦s ❞✐③ q✉❡ ❛ r❡❝♦rrê♥❝✐❛ ♦✉ tr❛♥s✐ê♥❝✐❛ ❞♦ ♣❛ss❡✐♦ é ❞❡t❡r♠✐♥❛❞♦ ♣❡❧♦ ✈❛❧♦r E[logρ0], ❡♠ q✉❡ ρx := 1−_ωxωx✳ P❛r❛ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦

s✐♠♣❧❡s✱ ❡♠ q✉❡P =δ⊗Z

p ✱ ❝♦♠ p∈[ǫ,1−ǫ] t❡♠♦s

(a)E[logρ0]<0⇔p > 1₂ ⇔ lim

n→∞Sn =∞, P✲q✳❝✳

(b)E[logρ0]>0⇔p < 1₂ ⇔ lim

n→∞Sn =−∞, P✲q✳❝✳

(c)E[logρ0] = 0⇔p= 1₂ ⇔ −∞= lim inf

n→∞ Sn ≤lim sup_n_→∞ Sn =∞, P✲q✳❝✳

❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦✿

❚❡♦r❡♠❛ ✶✳✶✳ ❙❡❥❛ (Sn) ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ ❡♠ ❛♠❜✐❡♥t❡ ❛❧❡❛tór✐♦✱ ❞❡✜♥✐❞♦ ❛❝✐♠❛✳

(a) E[logρ0]<0 ⇔ lim

n→∞Sn =∞, P✲q✳❝✳

(b) E[logρ0]>0 ⇔ lim

n→∞Sn =−∞, P✲q✳❝✳

(c) E[logρ0] = 0 ⇔ −∞= lim inf

(15)

✷✳ ❖ ❈❘■❚➱❘■❖ ❉❊ ❘❊❈❖❘❘✃◆❈■❆✴❚❘❆◆❙■✃◆❈■❆ ✼

❉❡♠♦♥str❛çã♦✳ ❋✐①❛❞♦s ω ∈ Ω✱ m−, m+ ∈ N✱ ❞❡♥♦t❛♠♦s ♣♦r Zmm+− ♦ ❝♦♥❥✉♥t♦

[−m−, m+]∩Z✳ ❉❡✜♥✐♠♦s T0 := 0 ❡ ♣❛r❛ z ∈Z\ {0}♦ t❡♠♣♦ ❞❡ ♣❛r❛❞❛ Tz✱ ❞❛❞♦ ♣♦r

Tz := inf{n ≥1 :Sn =z},

❝♦♠ ❛ ❝♦♥✈❡♥çã♦ q✉❡ ♦ í♥✜♠♦ s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ ✈❛③✐♦ é ✐❣✉❛❧ ❛ ∞✳ P❛r❛ t♦❞♦z ∈Zm+

m−✱

νm−,m+,ω(z) := P

z

ω(T−m− < Tm+).

❈♦♠♦ {Sn} é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❤♦♠♦❣ê♥❡❛ ♥♦ t❡♠♣♦✱νm−,m+,ω s❛t✐s❢❛③✿

✭✺✮ νm−,m+,ω(z) =

    

(1−ωz)νm−,m+,ω(z−1) +ωzνm−,m+,ω(z+ 1), z ∈Z

m+−1

m−−1,

1, z =−m−

0, z =m+.

❊♥tã♦νm−,m+,ω é ♦ q✉❡ ❝❤❛♠❛♠♦s ❞❡ ❢✉♥çã♦ ❤❛r♠ô♥✐❝❛✷ ❡ ❡♥❝♦♥tr❛r ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ νm−,m+,ω é ✉♠ ♣r♦❜❧❡♠❛ ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❝♦♥❤❡❝✐❞♦ Pr♦❜❧❡♠❛ ❞❛ ✏❘✉í♥❛ ❞♦ ❆♣♦st❛❞♦r✑✳ ❈♦♥❢♦r♠❡ ♦ ▲❡♠❛ ❇✳✷ ❞♦ ❆♣ê♥❞✐❝❡ ❇✱ ✉♠❛ ❢✉♥çã♦ ❤❛r♠ô♥✐❝❛ ❝♦♠ ❝♦♥❞✐çõ❡s ❞❡ ❝♦♥t♦r♥♦ ❞❛❞❛s é ú♥✐❝❛✳ ❆s ❢✉♥çõ❡s ❞❡✜♥✐❞❛s ♣♦r

Am+,ω(z) :=

    

1 +ρz+1+ρz+1ρz+2+· · ·+ρz+1· · ·ρm+−1, z ∈Z

m+−2

m−

1, z =m+−1,

0, z =m+.

Bm−,ω(z) :=

      

1

ρ−m₋+1···ρz +

1

ρ−m₋+2···ρz +· · ·+

1

ρz, z∈Z m+

m−−2, 1

ρ−m₋+1, z=−m−+ 1,

0, z=−m−.

s❛t✐s❢❛③❡♠

✭✻✮ Am+,ω(z) = 1 +ρz+1Am+,ω(z+ 1) ❡ Bm−,ω(z) = ρz+1Bm−,ω(z+ 1)−1,

✐♠♣❧✐❝❛♥❞♦ q✉❡ ❛s ❝♦♥❞✐çõ❡s ❡♠ ✭✺✮ sã♦ s❛t✐s❢❡✐t❛s ♣❛r❛

νm−,m+,ω(z) =

Am+,ω(z) Am+,ω(z) +Bm−,ω(z)

.

P❡❧❛ ❡①♣r❡ssã♦ ❞❡ νm−,m+,ω✱ ❛ r❡❝♦rrê♥❝✐❛✴tr❛♥s✐ê♥❝✐❛ ❞♦ ♣❛ss❡✐♦ ❡♠ ❛♠❜✐❡♥t❡ ω✱ ♣♦❞❡ s❡r ❧✐❣❛❞❛ à ❝♦♥✈❡r❣ê♥❝✐❛✴❞✐✈❡r❣ê♥❝✐❛ ❞❛s sér✐❡sA∞,ω(0) := lim

m+→∞

Am+,ω(0) ❡ B∞,ω(0) := lim

m−→∞Bm−ω(0)✳ ❉❡✜♥✐♠♦s ❡♥tã♦✱ ♦s ❡✈❡♥t♦s

A₊ :=A∞,ω(0) <∞ ❡ B− :=

B∞,ω(0)<∞ .

❱❡❥❛♠♦s ❛s ❝♦♥❞✐çõ❡s s♦❜r❡ ♦ ❛♠❜✐❡♥t❡ q✉❡ ❞❡t❡r♠✐♥❛rã♦ ❛ ❝♦♥✈❡r❣ê♥❝✐❛✴tr❛♥s✐ê♥❝✐❛ ❞♦ ♣❛ss❡✐♦✿

(16)

✽ ✶✳ ❖ P❆❙❙❊■❖ ❆▲❊❆❚Ó❘■❖ ❊▼ ❆▼❇■❊◆❚❊❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼Z

− ❙❡ω ∈T₊✱ ❡♠ q✉❡T₊:=A₊∩Bc₋✱ ❡♥tã♦ lim

m−→∞mlim+→∞

Pω(T−m− < Tm+) = 0.

❚❡♠♦s q✉❡

{Sn ր ∞} ⊂

[

m−≥1

\

m+≥1

{T−m− ≥Tm+}.

❆❧é♠ ❞✐ss♦✱ ❝♦♠♦(Sn)n≥1 é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ✐rr❡❞✉tí✈❡❧ s♦❜ Pω✱ s❡❣✉❡ q✉❡

♦ ♣❛ss❡✐♦ s❛t✐s❢❛③✿ ♦✉ éPω✲q✳❝✳ r❡❝♦rr❡♥t❡✱ ♦✉ é Pω✲q✳❝✳ tr❛♥s✐❡♥t❡✱ ❥✉st✐✜❝❛♥❞♦ ❛

✐❣✉❛❧❞❛❞❡

{Snր ∞}=

[

m−≥1

\

m+≥1

{T−m− ≥Tm+}, Pω✲q✳❝✳

▲♦❣♦✱

Pω(Snր ∞) = lim

m−→∞mlim+→∞

Pω(T−m− ≥Tm+) = 1.

■ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♦ ❛♠❜✐❡♥t❡ω ♣❡rt❡♥❝❡ àT₊✱ ❡♥tã♦ ♦ ♣❛ss❡✐♦ é tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❞✐r❡✐t❛✳

− ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ω ∈T₋✱ ❡♠ q✉❡T₋ :=Ac

+∩B−✱ s❡❣✉❡ q✉❡

lim

m+→∞ lim

m−→∞Pω(Tm− < Tm+) = 1 ⇒ Pω(Sn ց −∞) = 1.

■ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♦ ❛♠❜✐❡♥t❡ω ♣❡rt❡♥❝❡ àT₋✱ ❡♥tã♦ ♦ ♣❛ss❡✐♦ é tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❡sq✉❡r❞❛✳

− ❋✐①❡ ω∈R✱ ❡♠ q✉❡ R:=Ac

+∩Bc−✳ ❙❡❣✉❡ q✉❡

lim

m−→∞Pω(T−m− ≥Tm+) = 1, ∀ m+ ∈N,

lim

m+→∞

Pω(T−m− < Tm+) = 1, ∀ m− ∈N. ❙❡❣✉❡ q✉❡ ♣❛r❛ t♦❞♦z ∈Z✱ Tz <∞, Pω✲q✳❝✳ ▲♦❣♦✱

Pω(−∞= lim inf

n→∞ Sn≤lim supn→∞

Sn=∞) = 1.

■ss♦ s✐❣♥✐✜❝❛ q✉❡ s❡ ♦ ❛♠❜✐❡♥t❡ ω ♣❡rt❡♥❝❡ à R✱ ❡♥tã♦ ♦ ♣❛ss❡✐♦ é r❡❝♦rr❡♥t❡✳

− ❖ ❝❛s♦ω∈A₊∩B₋♥ã♦ s❡rá tr❛t❛❞♦ ❛q✉✐✱ ❥á q✉❡P(A₊∩B₋) = 0✱ ❝♦♠♦ ✈❡r❡♠♦s ❛ s❡❣✉✐r✳

❆ ♣❛rt✐r ❞❡ ❛❣♦r❛✱ ❛ ❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ❝♦♥s✐st❡ ❡♠ ❡st❛❜❡❧❡❝❡r ✉♠❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ♣r♦❜❛❜✐❧✐❞❛❞❡s ❞♦s ❡✈❡♥t♦s T₊✱ T₋ ❡ R ❡ ♦ ✈❛❧♦r E[logρ0]✳ P♦r ✭✻✮ t❡♠♦s q✉❡ A+

❡ B₋ sã♦ ✐♥✈❛r✐❛♥t❡s ❡ ❞❛ ❡r❣♦❞✐❝✐❞❛❞❡ ❞❡ P s❡❣✉❡ q✉❡ P(A₊), P(B₋) ∈ {0,1}✳ ❙❡

P(A₊) = 1✱ ❡♥tã♦ρ1· · ·ρn→0❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡ ❝♦♠♦P é ❡st❛❝✐♦♥ár✐❛✱ρ−1· · ·ρ−n →0

❡♠ ♣r♦❜❛❜✐❧✐❞❛❞❡✳ ▲♦❣♦✱ ❡①✐st❡ ✉♠❛ s✉❜s❡q✉ê♥❝✐❛

nk

Y

i=1

ρ−i

k≥1 t❛❧ q✉❡ ♣❛r❛P q✉❛s❡ t♦❞♦

ω✱ρ−1· · ·ρ−nk →0, q✉❛♥❞♦ k → ∞✳ ❙❡❣✉❡ q✉❡

∞=

∞

X

k=1

nk

Y

i=1

1 ρ−i

≤

∞

X

n=1

n

Y

i=1

1 ρ−i

.

(17)

✸✳ ❆ ▲❊■ ❉❖❙ ●❘❆◆❉❊❙ ◆Ú▼❊❘❖❙ ✾

❖❜s❡r✈❡ q✉❡ P(T₊) =P(A₊∩Bc₋) = P(A₊)−P(A₊∩B₋) ✐♠♣❧✐❝❛ P(A₊)−P(B₋)≤

P(T₊₎_≤_P₍A₊)✳ ❯t✐❧✐③❛♥❞♦ ♦ ❢❛t♦ q✉❡ _P₍A₊_{) = 1}s❡ ❡ s♦♠❡♥t❡ s❡ _P₍B₋) = 0✱ s❡❣✉❡ q✉❡ P(A₊_{) =}_P₍T₊)✳ ▼♦str❛r❡♠♦s q✉❡

✭✼✮ P(A₊) = 1 ⇔ E[logρ0]<0.

❯t✐❧✐③❛r❡♠♦s✱ ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✱ ❞❡✈✐❞♦ ❛ ❑❡st❡♥✱ q✉❡ ♥♦s ❞✐③ q✉❡ s♦♠❛s ❞✐✈❡r❣❡♥t❡s ❞❡ ✈❛r✐á✈❡✐s ❛❧❡❛tór✐❛s ❡st❛❝✐♦♥ár✐❛s ❞✐✈❡r❣❡♠ ♣❡❧♦ ♠❡♥♦s ❧✐♥❡❛r♠❡♥t❡✳ ❯♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦ ❡♥❝♦♥tr❛✲s❡ ♥♦ ❆♣ê♥❞✐❝❡ ❈✳

❚❡♦r❡♠❛ ✶✳✷✳ ❙❡❥❛Y1, Y2,· · · ✉♠❛ s❡q✉ê♥❝✐❛ ❡st❛❝✐♦♥ár✐❛ ❡♠ ✉♠ ❡s♣❛ç♦ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡

(Ω,F_{, P})✳ ❊♥tã♦

n_X∞

i=1

Yi → ∞

o

⇒nlim inf

n→∞

1 n

n

X

i=1

Yi >0

o

, P✲q✳❝✳

❖ ❚❡♦r❡♠❛ ❊r❣ó❞✐❝♦ ❞❡ ❇✐r❦❤♦✛ ✐♠♣❧✐❝❛ q✉❡

✭✽✮ E[logρ0] = lim

n→∞

1 n

n

X

i=1

logρi, P✲q✳❝✳

❙❡ ω ∈ A₊✱ ❡♥tã♦ _lim

n→∞ρ1· · ·ρn = 0 ✐♠♣❧✐❝❛ nlim→∞

n

X

i=1

logρi = −∞✳ ❆ s❡q✉ê♥❝✐❛ {Yi}

❞❡✜♥✐❞❛ ♣♦r Yi := −logρi é ❡st❛❝✐♦♥ár✐❛✳ ▲♦❣♦✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✷ ❡ ♣♦r ✭✽✮✱ s❡❣✉❡ q✉❡

E[logρ0]>0.

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ E[logρ0] < 0 ❡ s❡❥❛ c := −E[logρ0] > 0✳ P♦r ✭✽✮✱

❡①✐st❡ n0(ω) < ∞, P✲q✳❝✳ t❛❧ q✉❡ ♣❛r❛ t♦❞♦ n > n0(ω)✱

1 n

n

X

i=1

logρi ≤ −c

2 ✱ ✐♠♣❧✐❝❛♥❞♦

q✉❡

n

Y

i=1

ρi ≤e

−nc

2 .❙❡❥❛ C₁(ω) :=

n_X0(ω)

n=1

ρ1· · ·ρn✳ ❙❡❣✉❡ q✉❡

∞

X

n=1

ρ1· · ·ρn≤C1(ω) + ∞

X

i=n0(ω)+1

e−2ic <∞, P✲q✳❝✳ ⇒ P(A₊) = 1.

❉❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ♠♦str❛✲s❡ q✉❡P(T₋_{) =}_P₍B₋₎❡_P₍B₋_{) = 1}s❡ ❡ s♦♠❡♥t❡ s❡_E_[log_ρ₀_]_> 0✳ ❈♦♠ ✐ss♦✱ P(R_{) =} _P₍Ac

+∩Bc−) = 1 s❡ ❡ s♦♠❡♥t❡ s❡ E[logρ0] = 0✳

✸✳ ❆ ▲❡✐ ❞♦s ●r❛♥❞❡s ◆ú♠❡r♦s

❙❡❥❛♠ τ0 := 0 ❡ ♣❛r❛ n ∈ N✱ τn := Tn−Tn−1 ❡ τ−n := T−n−T−n+1✳ ❱❡r❡♠♦s ❛ s❡❣✉✐r

q✉❡✱ s♦❜ ❛❧❣✉♠❛s ❤✐♣ót❡s❡s✱ ♣♦❞❡♠♦s ❝♦♥s❡❣✉✐r ✉♠❛ ▲●◆ ♣❛r❛ (τi)i≥1 q✉❡ ♥♦s ❢♦r♥❡❝❡

t❛♠❜é♠ ✉♠❛ ▲●◆ ♣❛r❛ (Si)i≥1✳

▲❡♠❛ ✶✳✸✳ ❙❡ Tn

n →α✱ P✲q✳❝✳✱ ♣❛r❛ ❛❧❣✉♠❛ ❝♦♥st❛♥t❡ α <∞ ❡♥tã♦ Sn

n →

1

(18)

✶✵ ✶✳ ❖ P❆❙❙❊■❖ ❆▲❊❆❚Ó❘■❖ ❊▼ ❆▼❇■❊◆❚❊❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼Z

❉❡♠♦♥str❛çã♦✳ P❛r❛ n∈ N ❞❡✜♥✐♠♦s kn ❝♦♠♦ ♦ ú♥✐❝♦ ✐♥t❡✐r♦ k q✉❡ s❛t✐s❢❛③ Tk ≤

n < Tk+1✳ ◆❡ss❡ ❝❛s♦✱ Sn < k+ 1 ♦✉✱ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡✱

✭✾✮ Sn < kn+ 1.

❆❧é♠ ❞✐ss♦✱ s❡k ∈Né t❛❧ q✉❡ Tk ≤n < Tk+1✱ ♣♦❞❡♠♦s r❡❡s❝r❡✈❡rn ❝♦♠♦ n=Tk+j ❡♠

q✉❡ 0≤j < Tk+1−Tk✳ ❈♦♠♦ STk =k✱

✭✶✵✮ Sn=ST_kn+j ≥kn−j =kn−(n−Tkn).

▲♦❣♦✱ ♣♦r ✭✾✮ ❡ ✭✶✵✮ s❡❣✉❡ q✉❡ kn

n −

1− Tkn

n

≤ Sn

n ≤ kn

n + 1 n. P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ kn s❡❣✉❡ q✉❡ lim

n→∞

kn

n = lim_n_→∞ n

Tn ❡ ♣♦rt❛♥t♦

1

α ≤lim infn→∞ Sn≤lim sup_n_→∞ Sn≤

1 α.

❱♦❧t❛♥❞♦ ❛♦ ❝❛s♦ ❡♠ q✉❡ P =δ⊗Z

p ✱ ❛ ▲❋●◆ ✐♠♣❧✐❝❛✱P✲q✳❝✳✱

Sn

n →E[S1] = 2p−1 =

    

1−1−_pp 1+1−_pp =

1−E[ρ0]

1+E[ρ0], p6=

1 2

0, p= 1₂.

❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛ é ✉♠❛ ❣❡♥❡r❛❧✐③❛çã♦ ❞❡ss❡ r❡s✉❧t❛❞♦✳

❚❡♦r❡♠❛ ✶✳✹✳ ❙❡❥❛♠ τ1 ❡ τ−1 ♦s t❡♠♣♦s ❞❡ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ❡♠ 1 ❡ −1✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊♥tã♦✸

(a) E[τ1]<∞ ⇒ lim

n→∞

Sn n =

1

E[τ1], P✲q✳❝✳ (b) E[τ−1]<∞ ⇒ lim

n→∞

Sn n =−

1

E[τ−1], P✲q✳❝✳ (c) E[τ1] =∞ ❡ E[τ−1] =∞ ⇒ lim

n→∞

Sn

n = 0, P✲q✳❝✳

❙❡❣✉❡ ❞❡ ✭✹✮✱ ❛ s❡❣✉✐♥t❡ ✈❡rsã♦ q✉❡♥❝❤❡❞ ❞❡ss❡ r❡s✉❧t❛❞♦✿ ❚❡♦r❡♠❛ ✶✳✺✳ P❛r❛ P q✉❛s❡ t♦❞♦ ❛♠❜✐❡♥t❡ω✱

(a) E[τ1]<∞ ⇒ lim

n→∞

Sn n =

1

E[τ1], Pω✲q✳❝✳ (b) E[τ−1]<∞ ⇒ lim

n→∞

Sn n =−

1

E[τ−1], Pω✲q✳❝✳ (c) E[τ1] =∞ ❡ E[τ−1] =∞ ⇒ lim

n→∞

Sn

n = 0, Pω✲q✳❝✳

❉❡✜♥✐♠♦s

Am :=

1 ω0 + m X j=1 1 ω−j

j−1

Y

i=0

ρ−i ❡ Bm :=

1 1−ω0

+

m

X

j=1

1 1−ωj

j−1

Y

i=0

ρ−1i

✸_{❖ ▲❡♠❛ ✶✳✻ ❛ s❡❣✉✐r✱ ✐♠♣❧✐❝❛ q✉❡}_E_[_τ₁_]_❡_E_[_τ

−1]♥ã♦ ♣♦❞❡♠ s❡r ❛♠❜❛s ✜♥✐t❛s✱ ❥á q✉❡P✲q✳❝✳ ❛s sér✐❡s

∞ X

j=1

j−1

Y

i=0

ρ−i ❡ ∞ X

j=1

j−1

Y

i=0

(19)

✸✳ ❆ ▲❊■ ❉❖❙ ●❘❆◆❉❊❙ ◆Ú▼❊❘❖❙ ✶✶

❆ ❞❡♠♦♥str❛çã♦ ❞♦ ❚❡♦r❡♠❛ 1.4 é ❜❛s❡❛❞❛ ♥♦ ❚❡♦r❡♠❛ ✶✳✶✱ ♥♦ ▲❡♠❛ ✶✳✸ ❡ ♥♦s s❡❣✉✐♥t❡s ❧❡♠❛s✿

▲❡♠❛ ✶✳✻✳ ❙❡❥❛♠ A := lim

m→∞Am ❡ B := limm→∞Bm. ❊♥tã♦

(a) E[τ1] =E[A],

(b) E[τ−1] =E[B].

▲❡♠❛ ✶✳✼✳ ❙❡ lim sup

n→∞

Sn =∞, P✲q✳❝✳✱ ❡♥tã♦ (τi)i≥1 é ❡st❛❝✐♦♥ár✐❛ ❡ ❡r❣ó❞✐❝❛ s♦❜ P✳

❞❡♠♦♥str❛çã♦ ❞♦ t❡♦r❡♠❛ ✶✳✹✳ ▼♦str❡♠♦s ♣r✐♠❡✐r♦ q✉❡

✭✶✶✮ E[τ1]<∞=⇒Snր ∞ P✲q✳❝✳

❉❡ ❢❛t♦✱ ♦❜s❡r✈❡ q✉❡ ♣❡❧♦ ▲❡♠❛ ✶✳✻✱ E[τ1] < ∞ ✐♠♣❧✐❝❛ E[A] < ∞✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱

❛ ❝❛✉❞❛ ❞❛ sér✐❡ A t❡♥❞❡ ❛ ③❡r♦✱ ♣♦rt❛♥t♦✱ Qi_j₌₁ρ−j → 0 P✲q✳❝✳ q✉❛♥❞♦ i → ∞✳ ❏á

✈✐♠♦s ✭✈✐❞❡ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ✶✳✷✮ q✉❡ ♣❡❧❛ ❡st❛❝✐♦♥❛r✐❡❞❛❞❡ ❞♦ ❛♠❜✐❡♥t❡✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ E[logρ0] < 0✱ ♦ q✉❡ ❣❛r❛♥t❡ Sn ր ∞ P✲q✳❝✳ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✷✳ ❙❡ E[τ1] < ∞✱

❡♥tã♦ ✭✶✶✮ ❡ ♦ ▲❡♠❛ ✶✳✼ ✐♠♣❧✐❝❛♠ q✉❡ ❛ s❡q✉ê♥❝✐❛(τi)i≥1 é ❡st❛❝✐♦♥ár✐❛ ❡ ❡r❣ó❞✐❝❛✳ ❈♦♠♦

Tn =τ1+· · ·+τn✱ ♦ ❚❡♦r❡♠❛ ❞❡ ❇✐r❦❤♦✛ ♣❛r❛ (τi)i≥1 ✐♠♣❧✐❝❛ q✉❡✿

Tn

n −→E[τ1], P✲q✳❝✳ ▲♦❣♦✱ ♣❡❧♦ ▲❡♠❛ ✶✳✻ s❡❣✉❡ q✉❡

Sn

n −→ 1 E[τ1]

, P✲q✳❝✳,

♦ q✉❡ ♣r♦✈❛ ❛ ♣r✐♠❡✐r❛ ❛✜r♠❛çã♦ ❞♦ t❡♦r❡♠❛✳ ❆ s❡❣✉♥❞❛ é ♣r♦✈❛❞❛ ❞♦ ♠❡s♠♦ ❥❡✐t♦✳ P❛r❛ ❛ t❡r❝❡✐r❛✱ s❡E[τ1] =∞❡E[logρ0]≤0❛ ❞❡♠♦♥str❛çã♦ é ✐❣✉❛❧ ❛♦ ❝❛s♦ ❡♠ q✉❡E[τ1]<∞

❡ s❡ E[τ1] =∞ ❡ E[logρ0]≥0 ❛ ❞❡♠♦♥tr❛çã♦ é ✐❣✉❛❧ ❛♦ ❝❛s♦E[τ−1]<∞.

❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✶✳✻✳ Pr♦✈❛r❡♠♦s ♣r✐♠❡✐r♦ ❛ ❞❡s✐❣✉❛❧❞❛❞❡ E[τ1] ≥ E[A]✳

❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ τ1✿

τ1 =1{S1=+1}+1{S1=−1}(1 +τ

′′

0 +τ1′′),

✭✶✷✮

= 1 +1{S1=−1}(τ

′′

0 +τ1′′).

✭✶✸✮ ❡♠ q✉❡τ′′

0 é ❞❡✜♥✐❞♦ ❞❡ ♠❛♥❡✐r❛ t❛❧ q✉❡ 1 +τ0′′ s❡❥❛ ♦ t❡♠♣♦ ❞❡ ♣r✐♠❡✐r❛ ✈♦❧t❛ ❡♠0✱ ❡ τ1′′

❞❡ ♠❛♥❡✐r❛ t❛❧ q✉❡ 1 +τ′′

0 +τ1′′ s❡❥❛ ♦ t❡♠♣♦ ❞❡ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ❡♠ 1 ✭✈❡❥❛ ❋✐❣✉r❛ ✹✮✳

❋✐①❡ ✉♠ ❛♠❜✐❡♥t❡ ω∈Ω✳ ❈♦♠♦Pω(S1 =−1) = 1−ω0✱ t❡♠♦s ♣♦r ✭✶✸✮✿

Eω[τ1] = 1 + (1−ω0)

Eω[τ0′′|S1 =−1] +Eω[τ1′′|S1 =−1]

.

▼♦str❛r❡♠♦s ❛❣♦r❛ q✉❡ s❡ E[τ1] < ∞✱ ❡♥tã♦ ❡ss❛ ✐❞❡♥t✐❞❛❞❡ ❧❡✈❛ à s❡❣✉✐♥t❡ r❡❧❛çã♦ ❞❡

r❡❝♦rrê♥❝✐❛✿ ♣❛r❛ q✉❛s❡✲t♦❞♦ω✱

✭✶✹✮ Eω[τ1] =

1 ω0

+ρ0Eθ−1_ω[τ₁].

Pr✐♠❡✐r♦✱ ♦❜s❡r✈❡ q✉❡ ❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❛{S1 =−1}✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡τ0′′ s♦❜Pω é ✐❣✉❛❧ à

(20)

✶✷ ✶✳ ❖ P❆❙❙❊■❖ ❆▲❊❆❚Ó❘■❖ ❊▼ ❆▼❇■❊◆❚❊❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼Z

τ′′ 1

1

τ1

−1

τ′′ 0

1 +τ′′ 0

N Z

❋✐❣✉r❛ ✶✳ ❆ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ τ1 ❡♠ ✭✶✷✮✳

t♦❞♦ ❛♠❜✐❡♥t❡ ω✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ Pω(τ0′′ < ∞|S1 = −1) ≥ Pω(τ1 < ∞|S1 = −1) = 1✳

▲♦❣♦✱

Eω[τ₁′′|S1 =−1] = Eω[τ₁′′|S1 =−1, τ0′′<∞] =Eω[τ1],

♣♦✐s✱ ❝♦♥❞✐❝✐♦♥❛❧♠❡♥t❡ ❛ {S1 = −1, τ0′′ < ∞}✱ ❛ ❞✐str✐❜✉✐çã♦ ❞❡ τ1′′ é ✐❣✉❛❧ à ❞✐str✐❜✉✐çã♦

❞❡τ1 s♦❜ Pω✱ ❝♦♠♦ s❡❣✉❡ ❞❛ Pr♦♣r✐❡❞❛❞❡ ❞❡ ▼❛r❦♦✈ ❢♦rt❡✳ ❚❡♠♦s ❡♥tã♦

Eω[τ1] = 1 + (1−ω0)

Eθ−1_ω[τ₁] +E_ω[τ₁]

,

q✉❡ ❞á ✭✶✹✮ ❛♣ós r❡❛rr❛♥❥♦✳

❯s❛♥❞♦ r❡❝✉rs✐✈❛♠❡♥t❡ ✭✶✹✮ ♣❛r❛ ♦s ❛♠❜✐❡♥t❡s θ−1ω✱ θ−2ω✱✳ ✳ ✳ ✱ θ−(m−1)_ω_✱

Eω[τ1] =

1 ω0

+ρ0

₁

ω−1

+ρ−1Eθ−2_ω[τ₁]

=· · ·

= 1 ω0

+

m

X

j=1

1 ω−j

j−1

Y

i=0

ρ−i+Eθ−m−2_ω[τ₁]

m_Y+1

i=0

ρ−i.

❊♠ ♣❛rt✐❝✉❧❛r✱ Eω[τ1] ≥Am(ω)✳ ■♥t❡❣r❛♥❞♦ ❝♦♠ r❡s♣❡✐t♦ ❛ P✱ E[τ1]≥E[Am]✳ ❚♦♠❛♥❞♦

m→ ∞✱ ♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ ✐♠♣❧✐❝❛ E[τ1]≥E[A]✳ ❙❡E[τ1] =∞✱ ❡ss❛

❞❡s✐❣✉❛❧❞❛❞❡ t❛♠❜é♠ é ✈á❧✐❞❛✳

P❛r❛ ♣r♦✈❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥trár✐❛✱ ❝♦♠❡ç❛r❡♠♦s ♠♦str❛♥❞♦ q✉❡ E[τ11{τ1<∞}]≤E[A]✳ ❋✐①❛❞♦ M ∈N✱ ❝♦♠♦

Eω[τ₀′′1{τ1<M}|S1 =−1] =Eθ−1ω[τ11{τ2<M}]≤Eθ−1ω[τ11{τ1<M}] ❡

Eω[τ1′′1{τ1<M}|S1 =−1]≤Eω[τ

′′ 11{τ′′

1<M}|S1 =−1] = Eω[τ11{τ1<M}] s❡❣✉❡ ❞❡ ✭✶✸✮ q✉❡

Eω[τ11{τ1<M}] =P(τ1 < M) + (1−ω0)Eω[(τ

′′

0 +τ1′′)1{τ1<M}|S1 =−1]

≤1 + (1−ω0)

Eθ−1_ω[τ₁1_{_τ₁_<M_}] +E_ω[τ₁1_{_τ₁_<M_}]

.

❊♥tã♦✱

Eω[τ11{τ1<M}]≤ 1 ω0

(21)

✸✳ ❆ ▲❊■ ❉❖❙ ●❘❆◆❉❊❙ ◆Ú▼❊❘❖❙ ✶✸

✐♠♣❧✐❝❛

Eω[τ11{τ1<M}]≤ 1 ω0

+ρ0

₁

ω−1

+ρ−1Eθ−2_ω[τ₁1_{_τ₁_<M_}]

=· · ·

= 1 ω0

+

m

X

j=1

1 ω−j

j−1

Y

i=0

ρ−i+Eθ−m−2_ω[τ₁1_{_τ₁_<M_}]

m_Y+1

i=0

ρ−i.

❙❡ E[A] < ∞✱ ❡♥tã♦ Eh

m

Y

i=0

ρ−i

i

→ 0 q✉❛♥❞♦ m → ∞✱ ♣♦rt❛♥t♦ Eω[τ11{τ1<M}] ≤ E[A].

P❡❧♦ ❚❡♦r❡♠❛ ❞❛ ❈♦♥✈❡r❣ê♥❝✐❛ ▼♦♥ót♦♥❛ s❡❣✉❡ q✉❡Eω[τ11{τ1<∞}]≤E[A].❙❡E[τ1]<∞ ❡♥tã♦ τ1 < ∞, P✲q✳❝✳ ❡ s❡ E[τ1] = ∞ ❡ E[logρo] ≤ 0✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❚❡♦r❡♠❛ ✶✳✶✱

lim sup

n→∞

Sn = ∞, P✲q✳❝✳ ❡ ♣♦rt❛♥t♦✱ τ1 < ∞, P✲q✳❝✳ ❊♠ ❛♠❜♦s✱ E[τ1] = E[τ11{τ1<∞}] ❡ ♣♦rt❛♥t♦✱ E[τ1] = E[A]. P❛r❛ ✜♥❛❧✐③❛r✱ s❡ E[τ1] = ∞ ❡ E[logρo] >0✱ ❡♥tã♦ P(B−) = 1 ❡

∞

X

n=0

1 ρ0· · ·ρ−n

<∞, P✲q✳❝✳ ▲♦❣♦✱ρ0· · ·ρ−n → ∞, P✲q✳❝✳ ✐♠♣❧✐❝❛♥❞♦ q✉❡ E[A] =∞.

❞❡♠♦♥str❛çã♦ ❞♦ ▲❡♠❛ ✶✳✼✳ ❆ ❤✐♣ót❡s❡ lim sup

n→∞ Sn =

∞✱ ✐♠♣❧✐❝❛ q✉❡ ❛ s❡q✉ê♥❝✐❛

(τi)i≥1 ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ P✲q✳❝✳ ❙❡❥❛♠ U ❛ ♠❡❞✐❞❛ ✉♥✐❢♦r♠❡ ❡♠ [0,1]✱ U := U⊗N✱ eG ❛

σ✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣❡❧♦s ❝✐❧✐♥❞r♦s ❡♠ [0,1]N _❡ _Pe _{❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ❞❛❞❛ ♣♦r} e_P _:= _P _⊗_U

❡♠ (Ω×[0,1]N_,_F_×_Ge_)✳

❈♦♥str✉✐r❡♠♦s ❡♠ZN ✉♠❛ ❈❛❞❡✐❛ ❞❡ ▼❛r❦♦✈ ❤♦♠♦❣ê♥❡❛ ♥♦ t❡♠♣♦ (Sen)n≥1 ∈ZN s♦❜ Pe✱

❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ❋✐①❛❞♦s (ω, ξ)∈ Ω×[0,1]N_{✱ s❡❥❛♠} _Se

0 := 0 ❡ ♣❛r❛ n ≥ 0 ❞❡✜♥✐♠♦s

e

Sn+1 := Sen +Yn✱ ❡♠ q✉❡ Yn := 1{ξn+1 ≤ ωSne } −1{ξn+1 > ωSne }, ♣❛r❛ t♦❞♦ n ∈ N. ❙❡

e

Pω(·) é ❛ ❞✐str✐❜✉✐çã♦ ❞❡(Sen)n≥1 ❝♦♥❞✐❝✐♦♥❛❞❛ ❛♦ ❛♠❜✐❡♥t❡ ω✱ ❡♥tã♦

e

Pω(Sen+1 =x+ 1|Sen=x) =Peω(Yn+1 = +1|Sen =x) = U(ξn+1 ≤ωx) =ωx.

■ss♦ ✐♠♣❧✐❝❛ q✉❡(Sen)n≥1t❡♠ ❛ ♠❡s♠❛ ❞✐str✐❜✉✐çã♦ q✉❡(Sn)n≥1✳ ❙❡❥❛A∈σ(τ1, τ2,· · ·)✱ ✐✳❡✱

A ={(ω, ξ) : (τ1, τ2,· · ·)∈ B}✱ ❡♠ q✉❡ B ⊂ RN✳ ❉❡✜♥✐♠♦s θAˆ :={(ω, ξ) : (τ2, τ3,· · ·)∈

B}. ▼♦str❛r❡♠♦s q✉❡ s❡ A é ✐♥✈❛r✐❛♥t❡✱ ♦✉ s❡❥❛✱ s❡ θAˆ = A ❡♥tã♦ eP(A) ∈ {0,1}✳ ❙❡❥❛

f(ω, ξ) := 1A(ω, ξ)∈ {0,1}. P❡❧♦ ❚❡♦r❡♠❛ ❞❡ ❋✉❜✐♥✐✱

e

P(A) =

Z

f(ω, ξ)dPe=

Z Z

f(ω, ξ)U(dξ)P(dω). ❋✐①❛❞♦ ω ∈ Ω, ❝♦♠♦ A = ˆθk_A_{✱ ♣❛r❛ t♦❞♦} _k _≥ _1✱ _A _{♥ã♦ ❞❡♣❡♥❞❡ ❞❡} _ξ

1, ξ2,· · · , ξk, ♣❛r❛

t♦❞♦ k ≥ 1✳ ■ss♦ s✐❣♥✐✜❝❛ q✉❡ f(ω,·) é σ✲(ξk+1, ξk+2,· · ·) ♠❡♥s✉rá✈❡❧✱ ♣❛r❛ t♦❞♦ k ≥1 ❡

♣♦rt❛♥t♦✱ f(ω,·) éT∞✱ ❡♠ q✉❡ T∞ _{é ❛} _σ_{✲á❧❣❡❜r❛ ❝❛✉❞❛❧✱ ❞❡✜♥✐❞❛ ♣♦r}

T∞_:= \

k≥1

σ✲(ξk+1, ξk+2,· · ·).

(22)

✶✹ ✶✳ ❖ P❆❙❙❊■❖ ❆▲❊❆❚Ó❘■❖ ❊▼ ❆▼❇■❊◆❚❊❙ ❆▲❊❆❚Ó❘■❖❙ ❊▼Z

Z

a(ω)P(dω)✱ ❝♦♠ a(ω)∈ {0,1}. ❚❡♠♦s q✉❡

a(θω) = f(θω,·) = 1A(θω,·) = 1_θAˆ (ω,·) = 1A(ω,·) = a(ω).

▲♦❣♦✱ a(·) é ✐♥✈❛r✐❛♥t❡ s♦❜ θ ❡ ❝♦♠♦ P é ❡r❣ó❞✐❝❛✱ a(ω) é ❝♦♥st❛♥t❡ P✲q✳❝✳ ❊♥tã♦ a(ω) =

0 (♦✉ 1)P✲q✳❝✳✱♦ q✉❡ ✐♠♣❧✐❝❛ P(e A)∈ {0,1}.

❙❡P é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✱ ❡♥tã♦✿

E[A] =Eh 1 ω0

i

+Eh ρ0 ω−1

i

+· · ·+Ehρ0· · ·ρ−m ω−m−1

i

+· · ·

=Eh 1 ω0

i

+Eh 1 ω−1

i

E[ρ0] +· · ·+E

h ₁

ω−m−1

i

E[ρ0]· · ·E[ρ−m] +· · ·

=Eh 1 ω0

i

(1 +E[ρ0] +E[ρ0]2+· · ·+E[ρ0]m+· · ·)

▲♦❣♦✱ s❡ E[ρ0]<1✱ ❡♥tã♦✱ ♦ ▲❡♠❛ ✶✳✻ ✐♠♣❧✐❝❛

E[τ1] =

1 +E[ρ0]

1−E[ρ0]

.

❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ E[ρ−10 ]<1✱ ❡♥tã♦

E[τ−1] =

1 +E[ρ−1₀ ] 1−E[ρ−1₀ ].

❈♦♠ ✐ss♦✱ ♦ ❚❡♦r❡♠❛ ✶✳✹ ♣❛r❛ P ♣r♦❞✉t♦ t❡♠ ❛ s❡❣✉✐♥t❡ ✈❡rsã♦✿ ❚❡♦r❡♠❛ ✶✳✽✳ ❙❡❥❛ P ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ❡♠ (Ω,F)✳ ❊♥tã♦✿

(a) E[ρ0]<1⇒ lim

n→∞

Sn n =

1−E[ρ0]

1+E[ρ0], P✲q✳❝✳ (b) E[ρ−1₀ ]<1 ⇒ lim

n→∞

Sn n =−

1−E[ρ−₀1]

1+E[ρ−₀1], P✲q✳❝✳

(c) E[ρ0]−1 ≤1≤E[ρ−10 ] ⇒ _nlim_→∞ Snn = 0, P✲q✳❝✳

❋✐①❛♠♦s ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ P ❡ ❞❡♥♦t❛r❡♠♦s ♣♦r vP ♦ ✈❛❧♦r ❞♦ ❧✐♠✐t❡ lim n→∞

Sn

n ✳ ❙❡❣✉❡

❞❛ ❉❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❏❡♥s❡♥ q✉❡

− ❙❡ E[logρ0] < 0✱ ❡♥tã♦ E[logρ0−1] > 0 ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱ E[ρ−10 ] > 1✳ ▲♦❣♦✱

vP ≥0✳

− ❙❡E[logρ0]>0✱ ❡♥tã♦ E[ρ−10 ]<1✳ ▲♦❣♦✱ vP ≤0✳

− ❙❡E[logρ0] = 0✱ ❡♥tã♦ E[logρ−10 ] = 0✳ ▲♦❣♦✱ E[ρ−10 ]> 1 ❡ E[ρ0] >1✱ ♣♦rt❛♥t♦✱

vP = 0✳

❈♦♥s✐❞❡r❡ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s ❝♦rr❡s♣♦♥❞❡♥t❡ à ♠❡❞✐❞❛ δ_E⊗_[Z_ω₀_]✱ ❝♦♠ ✈❡❧♦❝✐❞❛❞❡ vP := 2E[ω0]−1✳ P♦r ✭✸✮ t❡♠♦s q✉❡ s❡ E[ρ0] < 1✱ ♦✉ s❡❥❛✱ vP > 0✱ ❡♥tã♦ vP > 0✳ ❉❡

❢♦r♠❛ ❛♥á❧♦❣❛✱vP <0✱ ✐♠♣❧✐❝❛vP <0✳ ❊♥tr❡t❛♥❞♦✱ é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ❡①❡♠♣❧♦s ❡♠ q✉❡

vP = 0 ❡vP 6= 0✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✱ q✉❛♥❞♦ P é ❞❛❞❛ ♣♦rP =α⊗Z, ❡♠ q✉❡α❛ss✉♠❡ ♦s

(23)

✸✳ ❆ ▲❊■ ❉❖❙ ●❘❆◆❉❊❙ ◆Ú▼❊❘❖❙ ✶✺

◗✉❛♥❞♦ P =α⊗Z_, ❡♠ q✉❡ _α ❛ss✉♠❡ ♦s ✈❛❧♦r❡s ₀_,₆❡ ₀_,₀₀₁ ❝♦♠ ♣r♦❜❛❜✐❧✐❞❛❞❡ _p ❡ ₁₋_p✱

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r ✈ár✐♦s ❝❛s♦s ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✈❛❧♦r ❞❡ p✿

− ❖ ♣❛ss❡✐♦ é tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❡sq✉❡r❞❛ s❡ p < 0,944✱ r❡❝♦rr❡♥t❡ s❡ p = 0,944 ❡ tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❞✐r❡✐t❛ s❡p >0,944✳

− ❆ ✈❡❧♦❝✐❞❛❞❡ vP é ♥❡❣❛t✐✈❛ s❡ p < 0,667✱ ✐❣✉❛❧ ❛ ③❡r♦ s❡ p ∈ [0,667,0,998] ❡

♣♦s✐t✐✈❛ s❡ p >0,998✳

− ❆ ✈❡❧♦❝✐❞❛❞❡vP é ♥❡❣❛t✐✈❛ s❡ p <0,833✱ ✐❣✉❛❧ ❛ ③❡r♦ s❡ p= 0,833 ❡ ♣♦s✐t✐✈❛ s❡

(24)

(25)

❈❆Pí❚❯▲❖ ✷

❖ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡s ❉❡s✈✐♦s ◗✉❡♥❝❤❡❞

❱✐♠♦s ♥♦ ❚❡♦r❡♠❛ ✶✳✺ q✉❡ s❡ E[logρ0] ≤ 0✱ ❡♥tã♦ ♣❛r❛ P q✉❛s❡ t♦❞♦ ❛♠❜✐❡♥t❡ ω✱ ♦

♣❛ss❡✐♦ ❛❧❡❛tór✐♦ Sn✱ s♦❜ ❛ ♠❡❞✐❞❛ q✉❡♥❝❤❡❞ Pω✱ ♣♦ss✉✐ ✉♠❛ ✈❡❧♦❝✐❞❛❞❡ ❛ss✐♥tót✐❝❛

Sn

n → 1 E[τ1]

, Pω✲q✳❝✳

◆♦ss♦ ✐♥t❡r❡ss❡ ♥❡st❡ ❝❛♣ít✉❧♦ é ❡st✉❞❛r ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ Sn

n ❡♠ t♦r♥♦ ❞❡st❡ ✈❛❧♦r ❛ss✐♥✲

tót✐❝♦✳ ■st♦ é✱ q✉❡r❡♠♦s ❡st✉❞❛r ♣❛r❛ C ⊂ [−1,1] ❡ t♦❞❛ r❡❛❧✐③❛çã♦ tí♣✐❝❛ ❞♦ ❛♠❜✐❡♥t❡ ω✱ ♦ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ Pω Sn_n ∈ C

✱ ❝♦♠ ♦ ♦❜❥❡t✐✈♦ ❞❡ ♦❜t❡r ✉♠ P●❉ ♣❛r❛

Sn

n✱ q✉❛♥❞♦ P é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✳ ❊♥✉♥❝✐❛r❡♠♦s ❛ s❡❣✉✐r ♦ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ ❞❡st❡

❝❛♣ít✉❧♦✳

❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛ P ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✳ ❊♥tã♦✱ ♣❛r❛ P✲q✉❛s❡ t♦❞♦ ❛♠❜✐❡♥t❡ ω✱

Sn n

n∈N s❛t✐s❢❛③ ✉♠ Pr✐♥❝í♣✐♦ ❞❡ ●r❛♥❞❡ ❉❡s✈✐♦s ✭P●❉✮ s♦❜ Pω✳ ■st♦ é✱ ❡①✐st❡ ✉♠❛

❢✉♥çã♦ I : [−1,1] → [0,∞)✱ s❡♠✐✲❝♦♥tí♥✉❛ ✐♥❢❡r✐♦r♠❡♥t❡✱ I 6= ∞✱ ❡ Ω∗ _⊂ _Ω _{t❛❧ q✉❡}

P(Ω∗_{) = 1} _{❡ t❛❧ q✉❡ ♣❛r❛ t♦❞♦} _ω _∈_Ω∗_{✿ P❛r❛ t♦❞♦ ❢❡❝❤❛❞♦} _F _⊂_[₋₁_,_1]_,

lim sup

n→∞

1

n logPω

Sn n ∈F

≤ −inf

v∈FI(v),

❡ ♣❛r❛ t♦❞♦ ❛❜❡rt♦ A⊂[−1,1]✱

lim inf

n→∞

1

n logPω

Sn n ∈A

≥ −inf

v∈AI(v).

❯♠ ❡s❜♦ç♦ ❞❛ ❢✉♥çã♦ t❛①❛I ❡♥❝♦♥tr❛✲s❡ ♥❛ ❋✐❣✉r❛ ✸✱ ♥♦s ❝❛s♦s ❡♠ q✉❡ ♦ ♣❛ss❡✐♦ é r❡❝♦r✲ r❡♥t❡ ♦✉ tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❞✐r❡✐t❛✳ ◆♦ ❞❡❝♦rr❡r ❞♦ ❝❛♣ít✉❧♦✱ ♦❜t❡r❡♠♦s r❡s✉❧t❛❞♦s q✉❡ ♥♦s ♣❡r♠✐t✐rã♦ ♦❜t❡r ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢✉♥çã♦ I✱ ✈❡r✐✜❝❛❞❛s ♥♦ ▲❡♠❛ ✷✳✶✹✱ t❛✐s ❝♦♠♦ ❞❡r✐✈❛✲ ❜✐❧✐❞❛❞❡ ❡ ❝♦♥✈❡①✐❞❛❞❡✳ ◆❛ s❡çã♦ ✷✳✸✳✶✱ ✐♥t❡r♣r❡t❛r❡♠♦s ❡ ❝♦♠♣❛r❛r❡♠♦s ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛ ❢✉♥çã♦ t❛①❛ ❝♦♠ ♦ ❝❛s♦ ❞♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s✳

❙❡Tné ♦ t❡♠♣♦ ❞❡ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ♥♦ ♣♦♥t♦n∈N✱ ♦❜t❡r❡♠♦s ♣r✐♠❡✐r♦ ✉♠ P●❉ q✉❡♥❝❤❡❞

♣❛r❛ ❛ s❡q✉ê♥❝✐❛ Tn

n✳ ❈♦♠♦ Tn

n ∈ [1,∞)✱ ❡st❡ P●❉ s❡rá ❢r❛❝♦✱ ♥♦ s❡♥t✐❞♦ q✉❡ ❛ ❝♦t❛

s✉♣❡r✐♦r s♦❜r❡ ♦ ❞❡❝❛✐♠❡♥t♦ ❡①♣♦♥❡♥❝✐❛❧ ❞❡ Pω Sn_n ∈ F

✈❛❧❡ s♦♠❡♥t❡ ♣❛r❛ ❝♦♥❥✉t♦s F ⊂[1,∞) ❝♦♠♣❛❝t♦s✳

❖❜s❡✈❡ q✉❡ Tn =τ1+· · ·+τn ❡♠ q✉❡ τi é ♦ t❡♠♣♦ ❣❛st♦ ❡♥tr❡ ❛ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ❡♠i−1

❡ ❛ ♣r✐♠❡✐r❛ ✈✐s✐t❛ ❡♠ i✳ ❆♣❡s❛r ❞❡ ♥ã♦ s❡r❡♠ ✐❞❡♥t✐❝❛♠❡♥t❡ ❞✐st✐❜✉í❞❛s✱ ❛s ✈❛r✐á✈❡✐s τi

sã♦ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ❯t✐❧✐③❛r❡♠♦s ❡♥tã♦ ✉♠ ♠ét♦❞♦ ♣❛r❡❝✐❞♦ ❝♦♠ ❛ ♣r♦✈❛ ❞♦ ❚❡♦r❡♠❛ ❞❡ ❈r❛♠ér✱ ❜❛s❡❛❞♦ ♥❛ ❞❡✜♥✐çã♦ ❞❛ s❡❣✉✐♥t❡ ❢✉♥çã♦ ❣❡r❛❞♦r❛ ❞❡ ♠♦♠❡♥t♦s✱

✭✶✺✮ ϕ(r, ω) :=Eω[erτ11{τ1<∞}], ∀ r∈R ❡ω ∈Ω, ♥❛ s✉❛ ❡s♣❡r❛♥ç❛ ❡♠ r❡❧❛çã♦ ❛♦ ❛♠❜✐❡♥t❡✱

✭✶✻✮ Λ(r) :=E[logϕ(r,·)], ∀ r∈R,

(26)

✶✽ ✷✳ ❖ P❘■◆❈❮P■❖ ❉❊ ●❘❆◆❉❊❙ ❉❊❙❱■❖❙ ◗❯❊◆❈❍❊❉

❡ ♥❛ tr❛♥s❢♦r♠❛❞❛ ❞❡ ▲❡❣❡♥❞r❡ ❞❡ Λ✱ ❞❛❞❛ ♣♦r

✭✶✼✮ Λ∗(u) := sup

r∈R

{ru−Λ(r)}, ∀ u∈R.

❈♦♠ ✐ss♦✱ ♦ P●❉ ♣❛r❛ Sn

n s❡rá ♦❜t✐❞♦ ✈✐❛ ❛s s❡❣✉✐♥t❡s ✐♥❝❧✉sõ❡s ♣❛r❛ ❝❡rt♦s ✈❛❧♦r❡s ❞❡ n✱

δ ❡ ǫ✱ ❡♠ q✉❡ Bδ(v) :={z ∈R;|z−v|< δ}✿

n

Sn n ≥v

o

⊂nT⌊vn⌋ ⌊vn⌋ ≤

1

v +ǫ

o

❡ n1−ǫ v <

T⌊vn⌋ ⌊vn⌋ <

1−ǫ₂

v

o

⊂nSn

n ∈Bδ(v)

o

.

❈♦♠♦ Sn

n ∈ [−1,1]✱ s❡rá ♥❡❝❡ssár✐♦ ♦❜t❡r ✉♠ P●❉ ❢r❛❝♦ t❛♠❜é♠ ♣❛r❛ T−n

n ✳ ❆s ♣r♦♣r✐❡✲

❞❛❞❡s ❞❛s ❢✉♥çõ❡sϕ,Λ❡ Λ∗ _{s❡rã♦ ✐♠♣♦rt❛♥t❡s ♥❛ ♣r♦✈❛ ❞♦ P●❉ ❢r❛❝♦ ♣❛r❛} Tn

n✱ ♣♦rt❛♥t♦✱

✐♥✐❝✐❛r❡♠♦s ♦ ❝❛♣ít✉❧♦ ❝♦♠ ♦ ❡st✉❞♦ ❞❡st❛s ❢✉♥çõ❡s ♥❛ ❙❡çã♦ ✷✳✶✳ ◆♦ ♣ró①✐♠♦ ❝❛♣ít✉❧♦✱ ♣r❡❝✐s❛r❡♠♦s ❞❡ ❛❧❣✉♠❛s ❞❡st❛s ♣r♦♣r✐❡❞❛❞❡s t❛♠❜é♠ ♣❛r❛P ❡st❛❝✐♦♥ár✐❛ ❡✴♦✉ ❡r❣ó❞✐❝❛✱ ♣♦r ✐ss♦✱ ❛❧❣✉♥s ❧❡♠❛s sã♦ ♠❛✐s ❣❡r❛✐s ❞♦ q✉❡ ♦ ♥❡❝❡ssár✐♦ ♥❡st❡ ❝❛♣ít✉❧♦✳ ◆❛ ❙❡çã♦ ✷✳✷ ♦❜t❡r❡♠♦s ♦ P●❉ ❢r❛❝♦ ♣❛r❛ Tn

n ❡ ♣❛r❛ T−n

n ✳ ❊st❡s P●❉s s❡rã♦ ❡♥tã♦ ❥✉♥t❛❞♦s ♣❛r❛ ♣r♦✈❛r

♦ ❚❡♦r❡♠❛ ✷✳✶ ♥❛ ❙❡çã♦ ✷✳✸✳

❙✉♣♦r❡♠♦s ❛té ♦ ✜♠ ❞♦ ❝❛♣ít✉❧♦ q✉❡ E[log(ρ0)]≤0✳ ■ss♦ s✐❣♥✐✜❝❛✱ ♣❡❧♦ ❚❡♦r❡♠❛ ✶✳✶ q✉❡

♦ ❛♠❜✐❡♥t❡ é t❛❧ q✉❡ ♦ ♣❛ss❡✐♦ s❡❥❛ r❡❝♦rr❡♥t❡ ♦✉ tr❛♥s✐❡♥t❡ ♣❛r❛ ❛ ❞✐r❡✐t❛✳

❊♥tã♦✱τ1 <∞✱P✲q✳❝✳ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ▲❡♠❛ ✶✳✻ ❡ϕ(r, ω) =Eω[erτ1]✳ ❖ ❝❛s♦E[log(ρ0)]≥

0, tr❛t❛✲s❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

❆♣r❡s❡♥t❛r❡♠♦s ❛ ♣r♦✈❛ ♦❜t✐❞❛ ♣♦r ❈♦♠❡ts✱ ●❛♥t❡rt ❡ ❩❡✐t♦✉♥✐ ❬❈●❩✵✵❪✳ ●r❡✈❡♥ ❡ ❞❡♥ ❍♦❧❧❛♥❞❡r ❬●❞❍✾✹❪ t❛♠❜é♠ ♦❜t✐✈❡r❛♠ ✉♠❛ ♣r♦✈❛ ♣❛r❛ ♦ P●❉ q✉❡♥❝❤❡❞ ❡ ❱❛r❛❞❤❛♥ ❬❱❛r✵✸❪ ♦❜t❡✈❡ ✉♠❛ ♣r♦✈❛ t❛♥t♦ ♣❛r❛ ♦ P●❉ q✉❡♥❝❤❡❞ q✉❛♥t♦ ♣❛r❛ ♦ ❛♥♥❡❛❧❡❞✳

✶✳ Pr♦♣r✐❡❞❛❞❡s ❞❡ ϕ✱ Λ ❡ Λ∗

P❛r❛ P ✉♠❛ ♠❡❞✐❞❛ ❞❡ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❡♠ (Ω,F)✱ ❞❡♥♦t❛r❡♠♦s ♣♦r _P_x ❛ ♠❛r❣✐♥❛❧ ❞❡ _P ❡♠ x∈Z✳ ❉❡✜♥✐♠♦s ♦ s✉♣♦rt❡ ❞❡ P0 ❝♦♠♦

s✉♣♣(P0) :={ω0 ∈[−1,1] :P0(Bǫ(ω0))>0, ∀ ǫ >0}.

❙❡❣✉❡ q✉❡ s✉♣♣(P0) é ❝♦♠♣❛❝t♦✳ ❉❡♥♦t❛♠♦s ♣♦r ωmin ♦ í♥✜♠♦ ❞❡ s✉♣♣(P0) ❡ ωmax ♦

s✉♣r❡♠♦ ❞❡ s✉♣♣(P0)✳ ❈♦♠♦P é ❡❧í♣t✐❝❛ s❡❣✉❡ q✉❡

ωmin >0 ❡ ωmax<1.

❙✉♣♦r❡♠♦s s❡♠♣r❡ q✉❡ s✉♣♣(P0)∩ 0,1₂

6

=∅❡ q✉❡ s✉♣♣(P0)∩

₁

2,1

6

=∅✱ ♦✉ s❡❥❛✱

✭✶✽✮ ωmin ≤

1

2 ❡ ωmax≥ 1 2.

▲❡♠❛ ✷✳✷✳ ❙❡❥❛♠ P ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ❡ ϕ ❞❡✜♥✐❞❛ ♣♦r ✭✶✺✮✳ ❊♥tã♦✱ ♣❛r❛ t♦❞♦ r >0✱ ϕ(r, ω) = ∞✱ ♣❛r❛ P✲q✳t✳ω✳

❉❡♠♦♥str❛çã♦✳ ❯s❛♥❞♦ ❛ ❞❡❝♦♠♣♦s✐çã♦ ✭✶✷✮✱ t❡♠♦s q✉❡ ϕ(r, ω) =er_ω

0+ (1−ω0)Eω[er(1+τ

′′

0+τ1′′)|S

1 =−1]

=er_ω

0+ (1−ω0)erϕ(r, θ−1ω)ϕ(r, ω).

(27)

✶✳ P❘❖P❘■❊❉❆❉❊❙ ❉❊ ϕ✱ Λ❊ Λ∗ _✶✾

P❛r❛ r > 0 ✜①♦✱ ❞❡✜♥✐♠♦s Ar := {ω : ϕ(r, ω) = ∞}✳ ❙❡ θ−1ω ∈ Ar✱ ♦ ❧❛❞♦ ❞✐r❡✐t♦

❞❡ ✭✶✾✮ é ✐♥✜♥✐t♦✱ ❧♦❣♦✱ ϕ(r, ω) t❛♠❜é♠ é ❡ ♣♦rt❛♥t♦✱ ω ∈ Ar✳ ▲♦❣♦✱ Ar é ✉♠ ❡✈❡♥t♦

✐♥✈❛r✐❛♥t❡✳ ❈♦♠♦ P é ❡r❣ó❞✐❝❛✱ P(Ar) ∈ {0,1} ❡ ♣❛r❛ ♠♦str❛r♠♦s q✉❡ P(Ar) = 1✱ é

s✉✜❝✐❡♥t❡ ✈❡r✐✜❝❛r q✉❡ P(Ar)>0✳ P❛r❛ N ∈N✱ ❞❡✜♥✐♠♦s

BN :={ω:ωx ≤ 1₂,∀x= 0,−1,· · · ,−N}.

❉❛❞♦ ω∈Ω✱ s❡❥❛ωN _{♦ ❛♠❜✐❡♥t❡ ❡♠ q✉❡}

ωN x =

(

1

2, s❡ x∈ {0,−1,· · · ,−N},

ωx, ❝❛s♦ ❝♦♥trár✐♦.

❯♠ ❛❝♦♣❧❛♠❡♥t♦ ♣❡r♠✐t❡ ❝♦♠♣❛r❛r ❛ ♣r♦❜❛❜✐❧✐❞❛❞❡ ❞♦ ❡✈❡♥t♦ ❝r❡s❝❡♥t❡ 1{τ1<n} ♣❛r❛ ♦ ♣❛ss❡✐♦ ♥♦ ❛♠❜✐❡♥t❡ω∈BN ❡ ♣❛r❛ ♦ ♣❛ss❡✐♦ ♥♦ ❛♠❜✐❡♥t❡ωN✿ ♣❛r❛ t♦❞♦n∈N❡ω∈BN✱

Eω[1{τ1<n}]≤EωN[1{τ1<n}]✱ ❧♦❣♦✱

Eω[τ1] = ∞

X

n=1

Pω(τ1 ≥n) = ∞

X

n=1

(1−Eω[1{τ1<n}])

≥

∞

X

n=1

(1−EωN[1_{_τ₁_<n_}]) =

∞

X

n=1

PωN(τ₁ ≥n) =E_ωN[τ₁].

❈♦♠♦ ex _≥_x_❡ _{r >}_{0✱ s❡❣✉❡ q✉❡}

ϕ(r, ω) = Eω[erτ1]≥rEω[τ1]≥rEωN[τ₁] ≥rEωN[τ₁1_{_T₁_<T

−N}]≡rEωSRW[τ11{T1<T−N}],

❡♠ q✉❡ ωSRW _{é ♦ ❛♠❜✐❡♥t❡ q✉❡ ❞❡s❝r❡✈❡ ♦ ♣❛ss❡✐♦ ❛❧❡❛tór✐♦ s✐♠♣❧❡s s✐♠étr✐❝♦✱ q✉❡ s❛✲}

t✐s❢❛③ EωSRW[τ₁] = ∞✱ ❝♦♠♦ ❢♦✐ ✈✐st♦ ♥❛ ■♥tr♦❞✉çã♦✳ ❈♦♠♦ τ₁ < ∞, P✲q✳❝✳✱ t❡♠♦s q✉❡

lim

N→∞EωSRW[τ11{T1<T−N}] = EωSRW[τ1]✳ ▲♦❣♦✱ s❡ K = 1

⌊erωmin⌋ + 1✱ ❡①✐st❡ N0 = N0(r)

t❛❧ q✉❡ s❡ ω ∈ BN0✱ ❡♥tã♦ ϕ(r, ω) ≥ K. P♦rt❛♥t♦✱ s❡ ω ∈ BN0+1✱ ❡♥tã♦ θ

−1_ω _∈ _B

N0 ❡ ϕ(r, θ−1_ω₎_≥_K_✳

❋✐①❡ ω ∈BN0+1✳ ❙❡❣✉❡ ❞❡ ✭✶✾✮ q✉❡

ϕ(r, ω)≥(1−ω0)erϕ(r, θ−1ω)ϕ(r, ω)

≥(1−ω0)erKϕ(r, ω)≥ ⌊erωmin⌋Kϕ(r, ω).

❙❡ϕ(r, ω)<∞,❡♥tã♦K ≤ 1

⌊erωmin⌋✱ ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ▲♦❣♦✱ϕ(r, ω) =∞✱ ♦ q✉❡ ✐♠♣❧✐❝❛

BN0+1 ⊂ Ar ❡ ♣♦rt❛♥t♦✱ P(Ar) ≥ P(BN0+1). ❆❧é♠ ❞✐ss♦✱ P(BN0+1) > 0✱ ❥á q✉❡ P é

♣r♦❞✉t♦ ❡ ❡st❛♠♦s s✉♣♦♥❞♦ ✭✶✽✮✳ ▲♦❣♦✱ P(Ar)>0.

❈♦♠ ✐ss♦✱ t❡♠♦s q✉❡ Λ ≡ ∞ ❡♠ (0,∞)✱ q✉❛♥❞♦ P é ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦✳ ❖ ♣ró①✐♠♦ ❧❡♠❛ ❢♦r♥❡❝❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ Λ ♣❛r❛ ✈❛❧♦r❡s ♥ã♦ ♣♦s✐t✐✈♦s ❞❡r✳

▲❡♠❛ ✷✳✸✳ ❙❡❥❛♠ P ✉♠❛ ♠❡❞✐❞❛ ♣r♦❞✉t♦ ❡ Λ ❞❡✜♥✐❞❛ ♣♦r ✭✶✻✮✳ ❊♥tã♦✱ ✭❛✮ r7→Λ(r) é ❝♦♥✈❡①❛✱ Λ(0) = 0 ❡ lim

r→−∞Λ(r) =−∞.

✭❜✮ r 7→ Λ(r) é ❞✉❛s ✈❡③❡s ❞❡r✐✈á✈❡❧ ❡♠ (−∞,0). ◆❡st❡ ✐♥t❡r✈❛❧♦✱ Λ′ _{❝r❡s❝❡ ❡} _Λ′ _∈

(1,E[τ1])✳ P❛r❛ u∈(1,E[τ1]) ❡①✐st❡ ✉♠ ú♥✐❝♦ r(u)∈(−∞,0)✱ t❛❧ q✉❡