On the invariant measure of the random difference equation $X_n=A_n X_n-1+B_n$ in the critical case

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On the invariant measure of the random difference equation X _n = A_nX _n − 1 + B_n in the critical case

Sara Brofferio, Dariusz Buraczewski, Ewa Damek

To cite this version:

Sara Brofferio, Dariusz Buraczewski, Ewa Damek. On the invariant measure of the random difference

equation

_n in the critical case. Annales de l’Institut Henri Poincaré (B)

Probabilités et Statistiques, Institut Henri Poincaré (IHP), 2012, 48 (2), pp.377–395. �10.1214/10-

AIHP406�. �hal-00994864�

❖◆ ❚❍❊ ■◆❱❆❘■❆◆❚ ▼❊❆❙❯❘❊ ❖❋ ❚❍❊ ❘❆◆❉❖▼ ❉■❋❋❊❘❊◆❈❊

❊◗❯❆❚■❖◆ Xn=AnXn−1+Bn ■◆ ❚❍❊ ❈❘■❚■❈❆▲ ❈❆❙❊

❙❆❘❆ ❇❘❖❋❋❊❘■❖✱ ❉❆❘■❯❙❩ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊❲❆ ❉❆▼❊❑

❆❜str❛❝t✳ ❲❡ ❝♦♥s✐❞❡r t❤❡ ❛✉t♦r❡❣r❡ss✐✈❡ ♠♦❞❡❧ ♦♥R^d❞❡✜♥❡❞ ❜② t❤❡ st♦❝❤❛st✐❝ r❡❝✉rs✐♦♥Xn= AnXn−1+Bⁿ✱ ✇❤❡r❡{(Bⁿ, An)}❛r❡ ✐✳✐✳❞✳ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s ✈❛❧✉❡❞ ✐♥R^d×R⁺✳ ❚❤❡ ❝r✐t✐❝❛❧ ❝❛s❡✱

✇❤❡♥E logA1

= 0✱ ✇❛s st✉❞✐❡❞ ❜② ❇❛❜✐❧❧♦t✱ ❇♦✉❣❡r♦❧ ❛♥❞ ❊❧✐❡✱ ✇❤♦ ♣r♦✈❡❞ t❤❛t t❤❡r❡ ❡①✐sts

❛ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ❘❛❞♦♥ ♠❡❛s✉r❡ν ❢♦r t❤❡ ▼❛r❦♦✈ ❝❤❛✐♥{Xn}✳ ■♥ t❤❡ ♣r❡s❡♥t ♣❛♣❡r ✇❡ ♣r♦✈❡

t❤❛t t❤❡ ✇❡❛❦ ❧✐♠✐t ♦❢ ♣r♦♣❡r❧② ❞✐❧❛t❡❞ ♠❡❛s✉r❡ν❡①✐sts ❛♥❞ ❞❡✜♥❡s ❛ ❤♦♠♦❣❡♥❡♦✉s ♠❡❛s✉r❡ ♦♥

R^d\ {0}✳

◆♦✉s ❝♦♥s✐❞ér♦♥s ❧❡ ♠♦❞è❧❡ ❛✉t♦ré❣r❡ss✐❢ s✉rR^d❞é✜♥✐ ♣❛r ré❝✉rr❡♥❝❡ ♣❛r ❧✬éq✉❛t✐♦♥ st♦❝❤❛s✲

t✐q✉❡Xn=AnXn−1+Bn✱ ♦ù{(Bⁿ, An)}s♦♥t ❞❡s ✈❛r✐❛❜❧❡s ❛❧é❛t♦✐r❡s à ✈❛❧❡✉rs ❞❛♥sR^d×R⁺✱

✐♥❞é♣❡♥❞❛♥t❡s ❡t ❞❡ ♠ê♠❡ ❧♦✐✳ ▲❡ ❝❛s ❝r✐t✐q✉❡✱ ❝✬❡st✲à✲❞✐r❡ ❧♦rsq✉❡E logA1

= 0✱ ❛ été ét✉❞✐é

♣❛r ❇❛❜✐❧❧♦t✱ ❇♦✉❣❡r♦❧ ❡t ❊❧✐❡✱ q✉✐ ♦♥t ♠♦♥tré q✉✬✐❧ ❡①✐st❡ ✉♥❡ ❡t ✉♥❡ s❡✉❧❡ ♠❡s✉r❡ ❞❡ ❘❛❞♦♥ν

✐♥✈❛r✐❛♥t❡ ♣♦✉r ❧❛ ❝❤❛î♥❡ ❞❡ ▼❛r❦♦✈{Xn}✳ ❉❛♥s ❝❡ ♣❛♣✐❡r ♥♦✉s ❞é♠♦♥tr♦♥s q✉❡ ❧❛ ♠❡s✉r❡ν✱

❝♦♥✈❡♥❛❜❧❡♠❡♥t ❞✐❧❛té❡✱ ❝♦♥✈❡r❣❡ ❢❛✐❜❧❡♠❡♥t ✈❡rs ✉♥❡ ♠❡s✉r❡ ❤♦♠♦❣è♥❡ s✉rR^d\ {0}✳

✶✳ ■♥tr♦❞✉❝t✐♦♥ ❛♥❞ t❤❡ ♠❛✐♥ r❡s✉❧t

❲❡ ❝♦♥s✐❞❡r t❤❡ ❛✉t♦r❡❣r❡ss✐✈❡ ♣r♦❝❡ss ♦♥R^d✿ X₀^x=x,

X_n^x=AnX_n−1^x +Bn,

✭✶✳✶✮

✇❤❡r❡ t❤❡ r❛♥❞♦♠ ♣❛✐rs{(Bn, An)}^n∈N ✈❛❧✉❡❞ ✐♥R^d×R⁺ ❛r❡ ✐♥❞❡♣❡♥❞❡♥t✱ ✐❞❡♥t✐❝❛❧❧② ❞✐str✐❜✉t❡❞

✭✐✳✐✳❞✳✮ ❛❝❝♦r❞✐♥❣ t♦ ❛ ❣✐✈❡♥ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ µ✳ ❚❤❡ ▼❛r❦♦✈ ❝❤❛✐♥ {X_n^x} ♦❝❝✉rs ✐♥ ✈❛r✐♦✉s

❛♣♣❧✐❝❛t✐♦♥s ❡✳❣✳ ✐♥ ❜✐♦❧♦❣② ❛♥❞ ❡❝♦♥♦♠✐❝s✱ s❡❡ ❬✶❪ ❛♥❞ t❤❡ ❝♦♠♣r❡❤❡♥s✐✈❡ ❜✐❜❧✐♦❣r❛♣❤② t❤❡r❡✳

■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❞❡✜♥❡Xn✐♥ t❤❡ ❣r♦✉♣ ❧❛♥❣✉❛❣❡✳ ▲❡tG❜❡ t❤❡ ✏ax+b✑ ❣r♦✉♣✱ ✐✳❡✳ G=R^d⋊R⁺✱

✇✐t❤ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜②(b, a)·(b^′, a^′) = (b+ab^′, aa^′)✳ ❚❤❡ ❣r♦✉♣G❛❝ts ♦♥ R^d ❜②(b, a)·x=ax+b✱

❢♦r(b, a)∈G❛♥❞x∈R^d✳ ❋♦r ❡❛❝❤n✱ ✇❡ s❛♠♣❧❡ t❤❡ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s(Bn, An)∈G✐♥❞❡♣❡♥❞❡♥t❧②

✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ♠❡❛s✉r❡µ✱ t❤❡♥ X_n^x= (Bn, An)·. . .·(B1, A1)·x✳

❚❤❡ ▼❛r❦♦✈ ❝❤❛✐♥X_n^x✐s ✉s✉❛❧❧② st✉❞✐❡❞ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥E logA1

<0✳ ❚❤❡♥✱ ✐❢ ❛❞❞✐t✐♦♥✲

❛❧❧②E

log⁺|B1|

<∞✱ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡ν ❬✶✻❪✱ ✐✳❡✳ t❤❡ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ν

♦♥R^ds❛t✐s❢②✐♥❣

✭✶✳✷✮ µ∗Gν(f) =ν(f),

❢♦r ❡✈❡r② ♣♦s✐t✐✈❡ ♠❡❛s✉r❛❜❧❡ ❢✉♥❝t✐♦♥f✳ ❍❡r❡

µ∗^Gν(f) = Z

G

Z

Rd

f(ax+b)ν(dx)µ(db da).

❚❤✐s r❡s❡❛r❝❤ ♣r♦❥❡❝t ❤❛s ❜❡❡♥ ♣❛rt✐❛❧❧② s✉♣♣♦rt❡❞ ▼❛r✐❡ ❈✉r✐❡ ❚r❛♥s❢❡r ♦❢ ❑♥♦✇❧❡❞❣❡ ❋❡❧❧♦✇s❤✐♣ ❍❛r♠♦♥✐❝

❆♥❛❧②s✐s✱ ◆♦♥❧✐♥❡❛r ❆♥❛❧②s✐s ❛♥❞ Pr♦❜❛❜✐❧✐t② ✭❝♦♥tr❛❝t ♥✉♠❜❡r ▼❚❑❉✲❈❚✲✷✵✵✹✲✵✶✸✸✽✾✮✳ ❙✳ ❇r♦✛❡r✐♦ ✇❛s ❛❧s♦ s✉♣✲

♣♦rt❡❞ ❜② ❈◆❘❙ ♣r♦❥❡❝t ❆◆❘ ✲ ●●P● ✭❘é❢✳ ❏❈❏❈✵✻ ✶✹✾✵✾✹✮✳ ❉✳ ❇✉r❛❝③❡✇s❦✐ ❛♥❞ ❊✳ ❉❛♠❡❦ ✇❡r❡ ❛❧s♦ s✉♣♣♦rt❡❞

❜② ▼◆✐❙❲ ❣r❛♥ts ◆✷✵✶ ✵✶✷ ✸✶✴✶✵✷✵ ❛♥❞ ◆ ◆✷✵✶ ✸✾✸✾✸✼✳

✶

✷ ❙✳ ❇❘❖❋❋❊❘■❖✱ ❉✳ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊✳ ❉❆▼❊❑

❖♥❡ ♦❢ t❤❡ ♠❛✐♥ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ t❤❡ st❛t✐♦♥❛r② ♠❡❛s✉r❡ ν ✐s ❑❡st❡♥✬s t❤❡♦r❡♠ ❬✶✻❪ ✭s❡❡ ❛❧s♦

❬✶✺✱ ✶✸❪✮ s❛②✐♥❣ t❤❛t ✐❢E[A^α₁] = 1✭❛♥❞ s♦♠❡ ♦t❤❡r ❛ss✉♠♣t✐♦♥s ❛r❡ s❛t✐s✜❡❞✮✱ t❤❡♥

ν u:|u|> z

∼Cz^−α ❛s z→+∞,

❢♦r ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tC✳

❍❡r❡ ✇❡ st✉❞② t❤❡ ❝r✐t✐❝❛❧ ❝❛s❡✱ ✇❤❡♥ E

logA1 = 0✳ ❚❤❡♥ Xn ❤❛s ♥♦ ✐♥✈❛r✐❛♥t ♣r♦❜❛❜✐❧✐t②

♠❡❛s✉r❡✳ ❍♦✇❡✈❡r✱ ✐t ✇❛s ♣r♦✈❡❞ ❜② ❇❛❜✐❧❧♦t✱ ❇♦✉❣❡r♦❧ ❛♥❞ ❊❧✐❡ ❬✶❪ t❤❛t ✐❢

• P[A1= 1]<1❛♥❞P[A1x+B1=x]<1❢♦r ❛❧❧x∈R^d✱

• E

(|logA1|+ log⁺|B1|)^2+ε

<∞✱ ❢♦r s♦♠❡ε >0✱

• E[logA1] = 0✳

t❤❡♥ t❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✭✉♣ t♦ ❛ ❝♦♥st❛♥t ❢❛❝t♦r✮ ✐♥✈❛r✐❛♥t ❘❛❞♦♥ ♠❡❛s✉r❡ ν ✱ ✐✳❡✳ ❛ ♠❡❛s✉r❡

s❛t✐s❢②✐♥❣ ✭✶✳✷✮ ✭s❡❡ ❛❧s♦ ❬✷✱ ✸❪✮✳ ❲❡ ✇✐❧❧ s❛② t❤❛t µs❛t✐s✜❡s ❤②♣♦t❤❡s✐s ✭❍✮ ✐❢ ❛❧❧ t❤❡ ❛ss✉♠♣t✐♦♥s

❛❜♦✈❡ ❛r❡ s❛t✐s✜❡❞✳ ❋♦r ♦✉r ♣✉r♣♦s❡ ✇❡ ✇✐❧❧ ♥❡❡❞ ❛♥ ❛❞❞✐t✐♦♥❛❧ ❛ss✉♠♣t✐♦♥✱ ✇❤✐❝❤ ✇✐❧❧ ❜❡ ❝❛❧❧❡❞

❤②♣♦t❤❡s✐s ▼✭δ✮✿

• t❤❡r❡ ❡①✐stsδ >0s✉❝❤ t❤❛tE[A^δ₁+A^−δ₁ +|B1|^δ]<∞✳

❚❤❡ ♠❡❛s✉r❡ν ❛♣♣❡❛rs ✐♥ ❛ ♥❛t✉r❛❧ ✇❛② ✇❤❡♥ ♣r♦❜❧❡♠s r❡❧❛t❡❞ t♦ t❤❡ ♣r♦❝❡ssX_n^x♦r t♦ r❛♥❞♦♠

✇❛❧❦s ♦♥ t❤❡ ❣r♦✉♣G✐♥ t❤❡ ❝r✐t✐❝❛❧ ❝❛s❡ ❛r❡ ✐♥✈❡st✐❣❛t❡❞✳ ▲❡t ✉s ♠❡♥t✐♦♥ t✇♦ ❡①❛♠♣❧❡s✳ ▲❡ P❛❣❡

❛♥❞ P❡✐❣♥é ❬✶✼❪ ♣r♦✈❡❞ t❤❡ ❧♦❝❛❧ ❧✐♠✐t t❤❡♦r❡♠ ❢♦rX_n^x✱ s❛②✐♥❣ t❤❛t ✉♥❞❡r s♦♠❡ ❢✉rt❤❡r ❛ss✉♠♣t✐♦♥s✱

√nE[f(X_n^x)] ❝♦♥✈❡r❣❡s t♦ ν(f) ❢♦r ❛♥② ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❢✉♥❝t✐♦♥ f✳ ❊❧✐❡ ❬✶✵❪ ❞❡s❝r✐❜❡❞ t❤❡

▼❛rt✐♥ ❜♦✉♥❞❛r② ❢♦r t❤❡ ❧❡❢t r❛♥❞♦♠ ✇❛❧❦ ♦♥ t❤❡ ❛✣♥❡ ❣r♦✉♣ ✇✐t❤ t❤❡ ♠❡❛s✉r❡ ν ♣❧❛②✐♥❣ t❤❡

❝❡♥tr❛❧ r♦❧❡✳ ❚❤❡r❡❢♦r❡✱ ✐t ✐s ♥❛t✉r❛❧ t♦ ❛s❦ ❛❜♦✉t ❛ q✉❛♥t✐✜❡❞ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ♠❡❛s✉r❡ν ❛♥❞ t❤❡

❛✐♠ ♦❢ t❤✐s ♣❛♣❡r ✐s t♦ ❛♥s✇❡r t❤✐s q✉❡st✐♦♥✳ ❖✉r ♠❛✐♥ r❡s✉❧t ✐s ❛♥ ❛♥❛❧♦❣✉❡ ♦❢ ❑❡st❡♥✬s t❤❡♦r❡♠ ✐♥

t❤❡ ❝r✐t✐❝❛❧ ❝❛s❡

❚❤❡♦r❡♠ ✶✳✸✳ ❆ss✉♠❡ t❤❛t ❤②♣♦t❤❡s❡s ✭❍✮✱ ▼✭δ✮ ❛r❡ s❛t✐s✜❡❞ ❛♥❞ t❤❡ ❧❛✇ µA♦❢ A1✐s ❛♣❡r✐♦❞✐❝✳

❚❤❡♥ t❤❡r❡ ❡①✐sts ❛ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡Σ♦♥ t❤❡ ✉♥✐t ❜❛❧❧S^d−1⊂R^d ❛♥❞ ❛ str✐❝t❧② ♣♦s✐t✐✈❡ ♥✉♠❜❡r C+ s✉❝❤ t❤❛t t❤❡ ♠❡❛s✉r❡sδ(0,z⁻¹)∗Gν ❝♦♥✈❡r❣❡ ✇❡❛❦❧② ♦♥R^d\ {0}t♦C+Σ⊗^da_a ❛sz→+∞✱ t❤❛t

✐s

z→+∞lim Z

Rd

φ(z⁻¹u)ν(du) =C+

Z

R+

Z

S^d−1

φ(aw)Σ(dw)da a

❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥φ∈Cc(R^d\ {0})✳

■♥ ♣❛rt✐❝✉❧❛r✱ ❢♦r ❡✈❡r② α < β

✭✶✳✹✮ lim

z→∞ν u: αz <|u|< βz

=C+logβ α.

❚❤❡ ✜rst ❡st✐♠❛t❡ ♦❢ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♠❡❛s✉r❡ν ❛t ✐♥✜♥✐t② ✇❛s ❣✐✈❡♥ ❜② ❇❛❜✐❧❧♦t✱ ❇♦✉❣❡r♦❧

❛♥❞ ❊❧✐❡ ❬✶❪✱ ✇❤♦ ♣r♦✈❡❞✱ ❢♦rd= 1❛♥❞ ✉♥❞❡r s♦♠❡ ♥♦♥❞❡❣❡♥❡r❛❝② ❤②♣♦t❤❡s❡s✱ t❤❛t ❢♦r ❡✈❡r②α < β ν (αz, βz]

∼log(β/α)·L(z) ❛sz→+∞,

✇❤❡r❡L✐s ❛ s❧♦✇❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥✳

❚❤❡ s❡❝♦♥❞ ❛✉t❤♦r r❡❝❡♥t❧② ♣r♦✈❡❞ ❬✹❪ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ L(z) ✐s ✐♥ ❢❛❝t ❝♦♥st❛♥t✱ ❜✉t ✐♥ ❛ ♠♦r❡

r❡str✐❝t✐✈❡ s❡tt✐♥❣✿ ❜❡s✐❞❡s t❤❡ ❤②♣♦t❤❡s❡s st❛t❡❞ ❛❜♦✈❡✱ ♦♥❡ ❛ss✉♠❡s ✐♥ ❬✹❪ t❤❛td= 1✱ t❤❡ ❝❧♦s❡❞

s❡♠✐❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✉♣♣♦rt ♦❢ µ✐s t❤❡ ✇❤♦❧❡ ❣r♦✉♣G❛♥❞ t❤❡ ♠❡❛s✉r❡µA ✐s s♣r❡❛❞✲♦✉t✳

▼♦r❡♦✈❡r ♥♦♥❞❡❣❡♥❡r❛❝② ♦❢ t❤❡ ❧✐♠✐t✐♥❣ ❝♦♥st❛♥t C+ ✇❛s ♣r♦✈❡❞ t❤❡r❡ ♦♥❧② ✐♥ t❤❡ ♣❛rt✐❝✉❧❛r ❝❛s❡

✇❤❡♥B1≥ε❛✳s✳

❲❤❡♥ t❤❡ ♠❡❛s✉r❡ µ ✐s r❡❧❛t❡❞ t♦ ❛ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✱ str♦♥❣❡r r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞

r❡❝❡♥t❧② ✐♥ ❬✽✱ ✻❪✳ ◆❛♠❡❧②✱ ❧❡t{µt} ❜❡ t❤❡ ♦♥❡ ♣❛r❛♠❡t❡r s❡♠✐❣r♦✉♣ ♦❢ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s✱ ✇❤♦s❡

✐♥✜♥✐t❡s✐♠❛❧ ❣❡♥❡r❛t♦r ✐s ❛ s❡❝♦♥❞ ♦r❞❡r ❡❧❧✐♣t✐❝ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ♦♥R^d×R⁺✳ ❚❤❡♥ t❤❡r❡ ❡①✐sts

■◆❱❆❘■❆◆❚ ▼❊❆❙❯❘❊ ■◆ ❚❍❊ ❈❘■❚■❈❆▲ ❈❆❙❊ ✸

❛ ✉♥✐q✉❡ ❘❛❞♦♥ ♠❡❛s✉r❡ν t❤❛t ✐sµt✲✐♥✈❛r✐❛♥t✱ ❢♦r ❛♥②t✳ ▼♦r❡♦✈❡r✱ν ❤❛s ❛ s♠♦♦t❤ ❞❡♥s✐t②ms✉❝❤

t❤❛t

m(zu)∼C(u)z^−d ❛sz→+∞,

❢♦r s♦♠❡ ❝♦♥t✐♥✉♦✉s ♥♦♥③❡r♦ ❢✉♥❝t✐♦♥C ♦♥R^d\ {0}✳

■♥ t❤✐s ♣❛♣❡r ✇❡ ❛❧s♦ ❞❡s❝r✐❜❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♠❡❛s✉r❡ ν ✐♥ t❤❡ ❝❛s❡ ✇❤❡♥ t❤❡ ♠❡❛s✉r❡µA

✐s ♣❡r✐♦❞✐❝✳ ❚❤✐s s✐t✉❛t✐♦♥ ❤❛s ❜❡❡♥ q✉✐t❡ ♥❡❣❧❡❝t❡❞ ✉♣ t♦ ♥♦✇✱ ❛❧s♦ ✐♥ t❤❡ ❝♦♥tr❛❝t✐♥❣ ❝❛s❡✳ ❊✈❡♥

✐❢ ✇❡ ❝❛♥♥♦t ♦❜t❛✐♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ♠❡❛s✉r❡ν ❛t ✐♥✜♥✐t②✱ ✇❡ st✐❧❧ ❤❛✈❡ ❛ ❣♦♦❞ ❡st✐♠❛t✐♦♥ ♦❢

t❤❡ ❛s②♠♣t♦t✐❝ ♦❢ t❤❡ ♠❡❛s✉r❡ ♦❢ t❤❡ ❜❛❧❧ ♦❢ r❛❞✐✉sz

❚❤❡♦r❡♠ ✶✳✺✳ ❙✉♣♣♦s❡ t❤❛t ❤②♣♦t❤❡s❡s ✭❍✮ ❛♥❞ ▼✭δ✮ ❛r❡ s❛t✐s✜❡❞✳ ■❢ t❤❡ ♠❡❛s✉r❡ µA✐s ♣❡r✐♦❞✐❝

♦❢ ♣❡r✐♦❞p✱ ✐✳❡✳ <suppµA>={e^np}n∈Z✱ t❤❡♥ t❤❡ ❢❛♠✐❧② ♦❢ ♠❡❛s✉r❡sδ(0,z⁻¹)∗Gν ✐s ✇❡❛❦❧② ❝♦♠♣❛❝t

❛♥❞ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C+ s✉❝❤ t❤❛t

z→∞lim Z

R^d

φ(z⁻¹u)ν(du) =C+

X

k∈Z

φ(e^pk)

❢♦r ❛♥② ❢✉♥❝t✐♦♥ φ ❜❡❧♦♥❣✐♥❣ t♦ T✱ t❤❡ s✉❜s❡t ♦❢ CC(R^d\ {0}) ❝♦♥s✐st✐♥❣ ♦❢ r❛❞✐❛❧ ❢✉♥❝t✐♦♥s s✉❝❤

t❤❛t P

k∈Zφ(ae^pk) =P

k∈Zφ(e^pk)❢♦r ❛❧❧ a∈R⁺✳ ■♥ ♣❛rt✐❝✉❧❛r ν(u:|u| ≤z)∼ C+

p logz ❛sz→+∞.

❚❤❡ ❝❛s❡ ✇❤❡♥B1 ✐s ♣♦s✐t✐✈❡ ✐s ❛❧s♦ ♦❢ ❛ ♣❛rt✐❝✉❧❛r ✐♥t❡r❡st ✐♥ ❛♣♣❧✐❝❛t✐♦♥s ❛♥❞ ❣❡♥❡r❛❧❧② ❛❧❧♦✇s t♦ ✉s❡ ♠♦r❡ ♣♦✇❡r❢✉❧ t❡❝❤♥✐q✉❡s✳ ■t ✇✐❧❧ ❜❡ t❤❡ s✉❜❥❡❝t ♦❢ ❛ ❢♦rt❤❝♦♠✐♥❣ ♣❛♣❡r✱ ✇❤❡r❡ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ♠♦♠❡♥t ❤②♣♦t❤❡s✐s ♦❢ ❚❤❡♦r❡♠ ✶✳✸ ❝❛♥ ❜❡ ✇❡❛❦❡♥❡❞✳

▲❡t ✉s ♠❡♥t✐♦♥ t❤❛t ❚❤❡♦r❡♠s ✶✳✸ ❛♥❞ ✶✳✺ ❤❛✈❡ ❜❡❡♥ ❛♣♣❧✐❡❞ r❡❝❡♥t❧② t♦ st✉❞② t❛✐❧s ♦❢ ✜①❡❞ ♣♦✐♥ts

♦❢ t❤❡ s♦ ❝❛❧❧❡❞ s♠♦♦t❤✐♥❣ tr❛♥s❢♦r♠ ✐♥ ❛ ❜♦✉♥❞❛r② ❝❛s❡✳ ❍♦✇❡✈❡r ✐♥ t❤✐s ❝♦♥t❡①t ✬❜♦✉♥❞❛r② ❝❛s❡✬

❝♦♥❝❡r♥s ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s ❤❛✈✐♥❣ ✐♥✜♥✐t❡ ♠❡❛♥✳ ❙❡❡ ❬✺❪ ❢♦r ♠♦r❡ ❞❡t❛✐❧s✳

❚❤❡ str✉❝t✉r❡ ♦❢ t❤❡ ♣❛♣❡r ✐s t❤❡ ❢♦❧❧♦✇✐♥❣✳ ❋✐rst ✇❡ ❡st✐♠❛t❡ t❤❡ ❜❡❤❛✈✐♦r ♦❢ ν ❛t ✐♥✜♥✐t② ✐♥

❙❡❝t✐♦♥ ✷ ✉♥❞❡r t❤❡ ✈❡r② ♠✐❧❞ ❤②♣♦t❤❡s✐s ✭❍✮✳ ■♥ ❚❤❡♦r❡♠ ✷✳✶✱ ✇❡ s❤♦✇ t❤❛t δ(0,z⁻¹)∗^Gν(K) ✐s s♠❛❧❧❡r t❤❛♥ CKL(z)✱ ❢♦r ❛❧❧ ❝♦♠♣❛❝t s❡ts K ❛♥❞ ❛ s❧♦✇❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥ L✱ ✐✳❡✳ t❤❡ ❢❛♠✐❧② ♦❢

♠❡❛s✉r❡sδ(0,z⁻¹)∗^Gν/L(z)✐s ✇❡❛❦❧② ❝♦♠♣❛❝t✳ ❲❡ ❛❧s♦ ♣r♦✈❡ t❤❛tR

Rd(1 +|u|)^−γν(du)<∞❢♦r ❛♥② γ >0❛♥❞ ✇❡ ♦❜t❛✐♥ s♦♠❡ ✐♥✈❛r✐❛♥❝❡ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥ts ♦❢δ(0,z⁻¹)∗Gν/L(z)✳

◆❡①t✱ ❛s ✐♥ ❬✹❪✱ ✇❡ r❡❞✉❝❡ t❤❡ ♣r♦❜❧❡♠ t♦ st✉❞② ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡

P♦✐ss♦♥ ❡q✉❛t✐♦♥✳ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ❧❡t µ ❜❡ t❤❡ ❧❛✇ ♦❢−logA1✳ ❚❤❡ ♠❡❛♥ ♦❢ µ ✐s ❡q✉❛❧ t♦ ✵ ❛♥❞

❣✐✈❡♥ ❛ ♣♦s✐t✐✈❡φ∈Cc(R^d\ {0})✇❡ ❞❡✜♥❡ t❤❡ ❢✉♥❝t✐♦♥ ♦♥R✿

✭✶✳✻✮ fφ(x) =δ_(0,e−x)∗Gν(φ) = Z

Rd

φ(e^−xu)ν(du).

❚❤❡♥fφ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥

✭✶✳✼✮ µ∗^Rfφ(x) =fφ(x) +ψφ(x), x∈R,

❢♦r ❛ s♣❡❝✐✜❝ ❢✉♥❝t✐♦♥ψφ✳ ❚❤❡ ❢✉♥❝t✐♦♥ ψφ ♣♦ss❡ss❡s s♦♠❡ r❡❣✉❧❛r✐t② ♣r♦♣❡rt✐❡s ❛♥❞ ✐t ✐s ❡❛s✐❡r t♦

st✉❞② t❤❛♥fφ✳ ❚❤❡ ♠❛✐♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❢♦❧❧♦✇s✿ ❣✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ ψφ ❞❡s❝r✐❜❡ t❤❡

❜❡❤❛✈✐♦r ❛t ✐♥✜♥✐t② ♦❢ ♣♦s✐t✐✈❡ s♦❧✉t✐♦♥s ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✳ ❆♥ ❛♥s✇❡r t♦ t❤✐s r❛t❤❡r ❝❧❛ss✐❝❛❧

q✉❡st✐♦♥ ✇❛s ❣✐✈❡♥ ❜② P♦rt ❛♥❞ ❙t♦♥❡ ❬✶✾❪✱ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛tµ✐s s♣r❡❛❞ ♦✉t✳ ❍♦✇❡✈❡r✱ t❤❡✐r

♠❡t❤♦❞s✱ s❧✐❣❤t❧② ❞❡✈❡❧♦♣❡❞✱ ✇♦r❦ ❢♦r ❣❡♥❡r❛❧ ❝❡♥t❡r❡❞ ♠❡❛s✉r❡µ♦♥R✳ ◆❛♠❡❧② ✇❡ ❝❛♥ ❝♦♥str✉❝t ❛

❝❧❛ssF(µ)♦❢ ❢✉♥❝t✐♦♥sψ✱ ✇✐t❤ ✇❡❧❧ ❞❡✜♥❡❞ ♣♦t❡♥t✐❛❧ t❤❛t ❝❛♥ ❜❡ ✉s❡❞ t♦ ❞❡s❝r✐❜❡ s♦❧✉t✐♦♥s ♦❢ t❤❡

❝♦rr❡s♣♦♥❞✐♥❣ P♦✐ss♦♥ ❡q✉❛t✐♦♥✳ ❆❧❧ t❤❡ ❞❡t❛✐❧s ✇✐❧❧ ❜❡ ✜❣✉r❡❞ ♦✉t ✐♥ ❙❡❝t✐♦♥ ✸✳

✹ ❙✳ ❇❘❖❋❋❊❘■❖✱ ❉✳ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊✳ ❉❆▼❊❑

❚❤❡ ♥❡①t st❡♣ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ψφ ❜❡❧♦♥❣s t♦F(µ)✳ ❆ ♣r✐♦r✐✱ t❤✐s ✐s ♥♦t tr✉❡ ❢♦r ❛❧❧

❢✉♥❝t✐♦♥φ✳ ❍♦✇❡✈❡r ✇❡ ❛r❡ ❛❜❧❡ t♦ ❝♦♥str✉❝t s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s t❤❛t ❤❛✈❡ t❤❡ ❣♦♦❞ ♣r♦♣❡rt✐❡s ❛♥❞

❛❧❧♦✇ t♦ ❞❡❞✉❝❡ ♦✉r ♠❛✐♥ r❡s✉❧ts ✭❙❡❝t✐♦♥ ✹✮✳

❋✐♥❛❧❧② ✐♥ ❙❡❝t✐♦♥ ✺ ✇❡ ♣r♦✈❡ t❤❛t t❤❡ ❧✐♠✐t ♦❢ν αz <|u|< βz

✐s str✐❝t❧② ♣♦s✐t✐✈❡ ❛♥❞ t❤❡ ♦♥❧②

❤②♣♦t❤❡s✐s ♥❡❡❞❡❞ ❢♦r t❤✐s r❡s✉❧t ✐s ❝♦♥❞✐t✐♦♥ ✭❍✮✳

❚❤❡ ❛✉t❤♦rs ❛r❡ ❣r❛t❡❢✉❧ t♦ ❏❛❝❡❦ ❉③✐✉❜❛➠s❦✐ ❛♥❞ ❨✈❡s ●✉✐✈❛r❝✬❤ ❢♦r st✐♠✉❧❛t✐♥❣ ❞✐s❝✉ss✐♦♥s ♦♥

t❤❡ s✉❜❥❡❝t ♦❢ t❤❡ ♣❛♣❡r✳

✷✳ ❚❤❡ ✉♣♣❡r ❜♦✉♥❞

❚❤❡ ❣♦❛❧ ♦❢ t❤✐s s❡❝t✐♦♥ ✐s t♦ ♣r♦✈❡ ❛ ♣r❡❧✐♠✐♥❛r② ❡st✐♠❛t❡ ♦❢ t❤❡ ♠❡❛s✉r❡ν ❛t ✐♥✜♥✐t②✳ ❲❡ ♣r♦✈❡

t❤❛t✱ ✉♥❞❡r t❤❡ ✈❡r② ♠✐❧❞ ❤②♣♦t❤❡s✐s ✭❍✮ ♦♥ t❤❡ ♠❡❛s✉r❡ µ✱ t❤❡ t❛✐❧ ♠❡❛s✉r❡ ♦❢ ❛ ❝♦♠♣❛❝t s❡t δ(0,z⁻¹)∗ν(K)✐s ❜♦✉♥❞❡❞ ❜② ❛ s❧♦✇❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥ L(z)✱ t❤❛t ✐s ❛ ❢✉♥❝t✐♦♥ ♦♥ R⁺ s✉❝❤ t❤❛t limz→+∞L(az)/L(z) = 1❢♦r ❛❧❧ a >0✳ ❙✉❝❤ ❢✉♥❝t✐♦♥s ❣r♦✇ ✈❡r② s❧♦✇❧②✱ ♥❛♠❡❧② t❤❡② ❛r❡ s♠❛❧❧❡r t❤❛♥z^γ ❢♦r ❛♥②γ >0✱ ✐♥ ❛ ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢+∞✳

❚❤❡♦r❡♠ ✷✳✶✳ ■❢ ❤②♣♦t❤❡s✐s ✭❍✮ ✐s ❢✉❧✜❧❧❡❞✱ t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ s❧♦✇❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥ L ♦♥

R^∗₊ s✉❝❤ t❤❛t t❤❡ ♥♦r♠❛❧✐③❡❞ ❢❛♠✐❧② ♦❢ ♠❡❛s✉r❡s ♦♥ R^d\ {0}

✭✷✳✷✮ δ(0,z⁻¹)∗^Gν

L(z)

✐s ✇❡❛❦❧② ❝♦♠♣❛❝t ❢♦rz≥1✳ ❚❤✉s(1+|x|)^−γ∈L¹(ν)❢♦r ❛♥②γ >0✳ ❋✉rt❤❡r♠♦r❡✱ ❛❧❧ ❧✐♠✐t ♠❡❛s✉r❡s η ❛r❡ ♥♦♥ ♥✉❧❧ ❛♥❞ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ G(µA)✱ t❤❡ ❝❧♦s❡❞ s✉❜✲❣r♦✉♣ ♦❢R^∗

+ ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✉♣♣♦rt ♦❢µA✱ t❤❛t ✐s

δ(0,a)∗Gη=η ∀a∈G(µA).

❚❤✐s t❤❡♦r❡♠ ✐s ❛ ♣❛rt✐❛❧ ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ Pr♦♣♦s✐t✐♦♥ ✺✳✷ ✐♥ ❬✶❪✳

❲❡ ✜rst ♣r♦✈❡ t❤❛t t❤❡ µ✲✐♥✈❛r✐❛♥❝❡ ♦❢ ν ✐♠♣❧✐❡s t❤❛t t❤❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥ts ♦❢ t❤❡ t❛✐❧ ❛r❡

✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢G(µA)✱ ♥❛♠❡❧② ✇❡ ❤❛✈❡

▲❡♠♠❛ ✷✳✸✳ ❙✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❢✉♥❝t✐♦♥L(z)s✉❝❤ t❤❛t t❤❡ ❢❛♠✐❧② ✭✷✳✷✮ ✐s ✇❡❛❦❧② ❝♦♠♣❛❝t

✇❤❡♥z ❣♦❡s t♦+∞✱ t❤❡♥ t❤❡ ❛❝❝✉♠✉❧❛t✐♦♥ ♣♦✐♥tsη ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ G(µA)✳

Pr♦♦❢✳ ▲❡tη ❜❡ ❛ ❧✐♠✐t ♠❡❛s✉r❡ ❛❧♦♥❣ ❛ s❡q✉❡♥❝❡ {zn} ❛♥❞ ✜① ❛ ❢✉♥❝t✐♦♥ φ ∈C_c¹(R^d\ {0})✳ ❲❡

❝❧❛✐♠ t❤❛t t❤❡ ❢✉♥❝t✐♦♥

h(y) =δ(0,y)∗^Gη(φ) = lim

n→∞

δ_(0,zn⁻¹_y)∗^Gν(φ) L(zn)

♦♥R⁺✐sµA✲s✉♣❡r❤❛r♠♦♥✐❝✳ ■♥❞❡❡❞✱ ♦❜s❡r✈❡ t❤❛t ❢♦r ❛❧❧(b, a)∈Gt❤❡r❡ ✐s ❛ ❝♦♠♣❛❝t s❡tK=K(b)

❛♥❞ ❛ ❝♦♥st❛♥tCs✉❝❤ t❤❛t

|φ(z⁻¹(au+b))−φ(z⁻¹(au))|< C|z⁻¹b|1^K(z⁻¹(au))

❢♦r ❛❧❧z >1 ❛♥❞u∈R^d✳ ❚❤❡♥

n→∞lim

|δ_(0,z⁻¹_{n )}∗Gδ(b,a)∗Gν(φ)−δ_(0,z⁻¹_{n )}∗Gδ(0,a)∗Gν(φ)|

L(zn) ≤ lim

n→∞

C|z_n⁻¹b|ν(a⁻¹znK) L(zn)

≤ Cη(a⁻¹K)· lim

n→∞|z_n⁻¹b|= 0,

■◆❱❆❘■❆◆❚ ▼❊❆❙❯❘❊ ■◆ ❚❍❊ ❈❘■❚■❈❆▲ ❈❆❙❊ ✺

❤❡♥❝❡

Z

G

h(ay)µA(da) = Z

G n→∞lim

δ_(0,z−1

n y)∗Gδ(0,a)∗Gν(φ)

L(zn) µ(db da)

= Z

G n→∞lim

δ_(0,zn⁻¹_y)∗^Gδ(b,a)∗^Gν(φ)

L(zn) µ(db da)

≤ lim

n→∞

δ_(0,zn⁻¹_y)∗^Gµ∗^Gν(φ)

L(zn) ❜② ❋❛t♦✉✬s ▲❡♠♠❛

= lim

n→∞

δ_(0,zn⁻¹_y)∗Gν(φ)

L(zn) =h(y).

❙✐♥❝❡ h ✐s ♣♦s✐t✐✈❡ ❛♥❞ ❝♦♥t✐♥✉♦✉s✱ t❤❡♥ ❜② t❤❡ ❈❤♦q✉❡t✲❉❡♥② t❤❡♦r❡♠ h(ay) = h(y) ❢♦r ❡✈❡r②

a∈G(µA)✱ t❤❛t ✐sδ(0,a)∗^Gη(φ) =η(φ)✳

Pr♦♦❢ ♦❢ ❚❤❡♦r❡♠ ✷✳✶✳ ❙t❡♣ ✶✳ ❚❤❡ ✜rst st❡♣ ✐s t♦ ♣r♦✈❡ t❤❛t t❤❡ t❛✐❧ ♦❢ t❤❡ ♠❡❛s✉r❡ν s❛t✐s✜❡s ❛ q✉♦t✐❡♥t t❤❡♦r❡♠✳ ◆❛♠❡❧② t❤❛t t❤❡r❡ ❡①✐sts ❛ ❢❛♠✐❧② ♦❢ ❜♦✉♥❞❡❞ ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❢✉♥❝t✐♦♥s s s✉❝❤ t❤❛tδ(0,z⁻¹)∗Gν(s)✐s str✐❝t❧② ♣♦s✐t✐✈❡ ❢♦r ❛❧❧z≥1 ❛♥❞ t❤❛t ❢♦r ❡✈❡r② ❝♦♠♣❛❝t s❡tK t❤❡r❡ ✐s

❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥tCK s✉❝❤ t❤❛t

✭✷✳✹✮ δ(0,z⁻¹)∗^Gν(K)≤CKδ(0,z⁻¹)∗^Gν(s) ∀z≥1.

■♥ ♦t❤❡r ✇♦r❞s ✇❡ s❤♦✇ t❤❡ q✉♦t✐❡♥t ❢❛♠✐❧② _δ^{δ(0,z−1 )∗Gν}

(0,z−1 )∗Gν(s) ✐s ✇❡❛❦❧② ❝♦♠♣❛❝t✳

❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s ♣r♦♣❡rt② r❡❧✐❡s ♦♥❧② ♦♥ t❤❡ ❢❛❝t t❤❛t✱ ❜② ❤②♣♦t❤❡s✐s ✭❍✮✱ t❤❡ s✉♣♣♦rt ♦❢µ❝♦♥t❛✐♥s

❛t ❧❡❛st t✇♦ ❡❧❡♠❡♥ts✱ ♦♥❡ ❝♦♥tr❛❝t✐♥❣ ❛♥❞ t❤❡ ♦t❤❡r ❞❡❧❛t✐♥❣R^d✳ ▲❡t ❝❛❧❧ t❤❡♠g+= (b+, a+)❛♥❞

g−= (b−, a−)✇✐t❤a+>1> a−✳

●✐✈❡♥ t✇♦ r❡❛❧ ♥✉♠❜❡rsα❛♥❞β ✇❡ ❝♦♥s✐❞❡r t❤❡ ❛♥♥✉❧✉s C(α, β) ={u∈R^d |α≤ |u| ≤β}.

❖❜s❡r✈❡ t❤❛t ❢♦r ❛❧❧(b, a)∈Gt❤❡ ❢♦❧❧♦✇✐♥❣ ✐♠♣❧✐❝❛t✐♦♥ ❤♦❧❞s u∈C

α+|b|

a ,β− |b| a

⇒ au+b∈C(α, β)

❯s✐♥❣ t❤✐s r❡♠❛r❦ ❛♥❞ t❤❡ ❢❛❝t t❤❛tν ✐s ✐♥✈❛r✐❛♥t ✇✐t❤ r❡s♣❡❝t t♦µ^∗n✱ ♦♥❡ ❝❛♥ ✈❡r✐❢② t❤❛t δ(0,z⁻¹)∗^Gν(C(α, β))≥µ^∗n(U)ν

C

(b,a)∈Umax

αz+|b| a , min

(b,a)∈U

βz− |b| a

✭✷✳✺✮

❢♦r ❛♥②U s✉❜s❡t ♦❢G❛♥❞ n∈N✳

❋✐rst ✇❡ ♣r♦✈❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡ R >0 s✉❝❤ t❤❛tδ_(0,z−1)∗Gν(C(1/R, R))✐s str✐❝t❧② ♣♦s✐t✐✈❡ ❢♦r ❛❧❧z≥1✳

❋✐①z≥1❛♥❞ t❛❦❡n∈Ns✉❝❤ t❤❛taⁿ⁻¹₊ ≤z≤aⁿ₊✳ ❈❧❡❛r❧②✱ ✐❢gⁿ = (b(gⁿ), a(gⁿ))✐s t❤❡n✲t❤ ♣♦✇❡r

♦❢ ❛♥ ❡❧❡♠❡♥tg= (b, a)∈Gt❤❡♥

a(gⁿ) =aⁿ ❛♥❞ b(gⁿ) =

n−1X

i=0

aⁱb= aⁿ−1 a−1 b.

❈♦♥s✐❞❡r t❤❡δ✲♥❡✐❣❤❜♦r❤♦♦❞ ♦❢gⁿ Uδ(gⁿ) =

(b, a)∈G|e^−δ< a⁻¹a(gⁿ)<e^δ ❛♥❞|b−b(gⁿ)|< δ .

✻ ❙✳ ❇❘❖❋❋❊❘■❖✱ ❉✳ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊✳ ❉❆▼❊❑

❖❜s❡r✈❡ t❤❛tµ^∗n(Uδ(gⁿ₊))>0❢♦r ❛❧❧δ >0 ❛♥❞ ❢♦r(b, a)∈Uδ(g₊ⁿ) z/R+|b|

a ≤ aⁿ₊/R+|b(g₊ⁿ)|+δ

e^−δaⁿ₊ ≤e^δ1

R + |b+| a+−1+δ

=:αR

Rz− |b|

a ≥Raⁿ⁻¹₊ − |b(gⁿ₊)| −δ

e^δaⁿ₊ ≥e^−δR

a+ − |b+| a+−1 −δ

=:βR.

❙✐♥❝❡ν ✐s ❛ ❘❛❞♦♥ ♠❡❛s✉r❡ ✇✐t❤ t❤❡ ✐♥✜♥✐t❡ ♠❛ss✱ ✐ts s✉♣♣♦rt ❝❛♥♥♦t ❜❡ ❝♦♠♣❛❝t✳ ❚❤✉s✱ ❢♦r ❛ ✜①❡❞

δ✱ t❤❡r❡ ❡①✐ts ❛ s✉✣❝✐❡♥t❧② ❧❛r❣❡Rs✉❝❤ t❤❛t✿ ν(C(αR, βR))>0.❚❤❡♥ ❜② ✭✷✳✺✮✿

✭✷✳✻✮ δ_(0,z−1)∗Gν(C(1/R, R))≥µ^∗n(Uδ(g₊ⁿ))ν(C(αR, βR))>0

❢♦r ❛❧❧z≥1✳

❋♦rR > 2 ❝♦♥s✐❞❡r t❤❡ ❝♦♠♣❛❝t s❡tsK_±ⁿ =C(2a⁻ⁿ_± /R, a⁻ⁿ_± R/2).◆♦t✐❝❡ t❤❛t ❢♦r δ < log(4/3)✱

(b, a)∈Uδ(gⁿ_±)❛♥❞z > z_±ⁿ := 2R(|b(g_±ⁿ)|+δ)✿ z/R+|b|

a ≤ze^δ1/R+z⁻¹(|b(gⁿ_±)|+δ)

aⁿ_± ≤z2a⁻ⁿ_± R

e^δ1 +z⁻¹R(|b(gⁿ_±)|+δ) 2

≤z2a⁻ⁿ_± R zR− |b|

a ≥ze^−δR−z⁻¹(|b(gⁿ_±)|+δ)

aⁿ_± ≥za⁻ⁿ_± R 2 ·2e^−δ

1−z⁻¹|b(gⁿ_±)|+δ R

≥za⁻ⁿ_± R 2 .

❚❤✉s ❜② ✭✷✳✺✮✿

δ(0,z⁻¹)∗Gν(C(1/R, R))≥µ^∗n(Uδ(g_±ⁿ))ν(C(z 2a⁻ⁿ_± /R, z a⁻ⁿ_± R/2)) =C_K⁻¹n

±δ(0,z⁻¹)∗Gν(K_±ⁿ)

❢♦r ❛❧❧ z > z_±ⁿ✳ ❙✐♥❝❡ δ_(0,z−1) ∗G ν(C(1/R, R)) > 0✱ t❤❡ ❛❜♦✈❡ ✐♥❡q✉❛❧✐t② ❤♦❧❞s ✐♥ ❢❛❝t ❢♦r ❛❧❧

z ≥ 1✱ ♣♦ss✐❜❧② ✇✐t❤ ❛ ❜✐❣❣❡r ❝♦♥st❛♥t CK ❛♥❞ s✉✣❝✐❡♥t❧② ❧❛r❣❡ R✳ ❲❡ ♠❛② ❛ss✉♠❡ t❤❛t R >

2 max{a+,1/a−}✱ t❤❡♥ t❤❡ ❢❛♠✐❧② ♦❢ s❡tsK_±ⁿ ❝♦✈❡rsR^d\ {0}✳

❋✐♥❛❧❧② ♥♦t✐❝❡✱ t❤❛t ❡✈❡r② ❢✉♥❝t✐♦♥s∈Cc(R^d\ {0})❛♥❞ s✉❝❤ t❤❛ts(u)≥1C(1/R,R)(u)s❛t✐s✜❡s

✭✷✳✹✮✳ ■♥❞❡❡❞✱ ❧❡t K ❜❡ ❛ ❣❡♥❡r✐❝ ❝♦♠♣❛❝t s❡t ✐♥R^d\ {0} ❝♦✈❡r❡❞ ❜② ❛ ✜♥✐t❡ ♥✉♠❜❡r ♦❢ ❝♦♠♣❛❝ts {Ki}^i∈I ♦❢ t❤❡ t②♣❡K_±ⁿ✱ t❤❡♥

δ(0,z⁻¹)∗Gν(K)≤X

i∈I

δ(0,z⁻¹)∗ν(Ki)≤

|I|max

i∈I CKi

δ(0,z⁻¹)∗Gν(s)

❢♦r ❛❧❧z≥1✳

❙t❡♣ ✷✳ ▲❡tL(z) =δ_(0,z−1)∗Gν(s)✱ s♦ t❤❛tδ_(0,z−1)∗Gν/L(z)✐s ✇❡❛❦❧② ❝♦♠♣❛❝t ✇❤❡♥z❣♦❡s t♦

+∞✳ ■t r❡♠❛✐♥s t♦ ♣r♦✈❡ t❤❛tL✐s ❛ s❧♦✇❧② ✈❛r②✐♥❣ ❢✉♥❝t✐♦♥✳ ❋✐①a∈G(µA)❛♥❞ ♦❜s❡r✈❡ t❤❛t L(az)

L(z) =δ_(0,a−1)∗Gδ_(0,z−1)∗Gν(s)

L(z) .

▲❡t{zn}^n∈N❜❡ ❛ s❡q✉❡♥❝❡ s✉❝❤ t❤❛tδ(0, z_n⁻¹)∗^Gν/L(zn)❝♦♥✈❡r❣❡s t♦ s♦♠❡ ❧✐♠✐t ♠❡❛s✉r❡η✳ ❚❤❡♥

❜② ✐♥✈❛r✐❛♥❝❡ ♦❢ t❤❡ ❧✐♠✐t ♠❡❛s✉r❡

n→∞lim L(azn)

L(zn) =δ(0,a⁻¹)∗^Gη(s) =η(s) = 1.

❙✐♥❝❡ ❢♦r ❛♥② s❡q✉❡♥❝❡✱ t❤❡r❡ ❡①✐sts ❛ s✉❜s❡q✉❡♥❝❡ s✉❝❤ t❤❛t t❤❡ ❝♦♥❝❧✉s✐♦♥ ❛❜♦✈❡ ❤♦❧❞s✱ ✐❢ µA

❛♣❡r✐♦❞✐❝✱L✐s s❧♦✇❧② ✈❛r②✐♥❣ ❛♥❞ t❤❡ ♣r♦♦❢ ✐s ❝♦♠♣❧❡t❡❞✳

■❢ µA ✐s ♣❡r✐♦❞✐❝✱ t❤❛t ✐s G(µA) = he^pi✱ t❛❦❡ ❛♥② ❝♦♥t✐♥✉♦✉s ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞ ❢✉♥❝t✐♦♥

s0≥1C(1/R,R)✱ ✐✳❡✳ ❛ ❢✉♥❝t✐♦♥ s❛t✐s❢②✐♥❣ ✭✷✳✹✮✱ ❛♥❞ ❞❡✜♥❡

✭✷✳✼✮ s(u) =

Z

R∗ +

1[e^−p,e^p)(t)s0(u/t)dt t .

■◆❱❆❘■❆◆❚ ▼❊❆❙❯❘❊ ■◆ ❚❍❊ ❈❘■❚■❈❆▲ ❈❆❙❊ ✼

❆♥ ❡❛s② ❛r❣✉♠❡♥t s❤♦✇s t❤❛t ❛❧s♦ s ✐s ✐♥ Cc(R^d\ {0}) ❛♥❞ ✐t ✐s ❜✐❣❣❡r t❤❛♥ s♦♠❡ ♠✉❧t✐♣❧❡ ♦❢

1C(1/R,R)✳ ❲❡ ❝❧❛✐♠ t❤❛tδ(0,a⁻¹)∗^Gη(s) =η(s) ❢♦r ❛❧❧ a∈R^∗

+ ❛♥❞ ♥♦t ♦♥❧② ❢♦r a∈G(µA)✭❛♥❞

t❤✉sL(z)✐s s❧♦✇❧② ✈❛r②✐♥❣✮✳ ■♥ ❢❛❝t✱ ❧❡te^Kp∈G(µA)s✉❝❤ t❤❛te^Kp> ae^p t❤❡♥

δ(0,a⁻¹)∗Gη(s) = Z

Rd

Z

R^∗

+

1[ae^−p,ae^p)(t)s0(u/t)dt t η(du)

= Z

Rd

Z

R∗ +

1[ae^−p,e^Kp)(t)−1[ae^p,e^Kp)(t)

s0(u/t)dt t η(du)

= Z

Rd

Z

R∗ +

1[ae^−Kp,e^p)(t)s0 u/t

−1[ae^−Kp,e^−p)(t)s0 u/t η(du)dt

t

= Z

Rd

Z

R∗ +

1[e^−p,e^p)(t)s0(u/t)dt

t η(du) =η(s),

s✐♥❝❡η ✐sG(µA)✲✐♥✈❛r✐❛♥t✳

✸✳ ❘❡❝✉rr❡♥t ♣♦t❡♥t✐❛❧ ❦❡r♥❡❧ ❛♥❞ s♦❧✉t✐♦♥s ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ❢♦r ❣❡♥❡r❛❧

♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡s

❆s ✐t ❤❛s ❜❡❡♥ ♦❜s❡r✈❡❞ ✐♥ t❤❡ ✐♥tr♦❞✉❝t✐♦♥✱ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ♠❡❛s✉r❡

ν ♦♥❡ ❤❛s t♦ ❝♦♥s✐❞❡r t❤❡ ❢✉♥❝t✐♦♥

fφ(x) = Z

Rd

φ(ue^−x)ν(du) t❤❛t ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥

✭✸✳✶✮ µ∗^Rf =f +ψ

❢♦r ❛ ♣❡❝✉❧✐❛r ❝❤♦✐❝❡ ♦❢ t❤❡ ❢✉♥❝t✐♦♥ψ=ψφ=µ∗^Rfφ−fφ✳

❙t✉❞②✐♥❣ s♦❧✉t✐♦♥s ♦❢ s✉❝❤ ❡q✉❛t✐♦♥ ❢♦r ❛ ❝❡♥t❡r❡❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥R✐s ❛ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠✳

P♦rt ❛♥❞ ❙t♦♥❡ ✐♥ t❤❡✐r ♣❛♣❡rs ❬✶✽✱ ✶✾❪ ❣✐✈❡ ❛♥ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❞❡s❝r✐❜✐♥❣ ❛❧❧ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇

s♦❧✉t✐♦♥s ♦❢ ✭✸✳✶✮ ✐♥ t❡r♠s ♦❢ t❤❡ r❡❝✉rr❡♥t ♣♦t❡♥t✐❛❧ ❦❡r♥❡❧ A ♦❢ t❤❡ ❢✉♥❝t✐♦♥ ψ✳ ❍♦✇❡✈❡r✱ t❤❡② s✉♣♣♦s❡ ❡✐t❤❡r t❤❛t t❤❡ ♠❡❛s✉r❡ ✐s s♣r❡❛❞✲♦✉t ♦r✱ ✐❢ ♥♦t✱ t❤❛t t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ψ✐s ❝♦♠♣❛❝t❧② s✉♣♣♦rt❡❞✳ ❚❤✐s s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s t♦♦ r❡str✐❝t✐✈❡ ✐♥ ♦✉r s❡tt✐♥❣✿ s✉❝❤ ❢✉♥❝t✐♦♥s ❞❡❝❛② t♦♦ s❧♦✇❧②

❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❢✉♥❝t✐♦♥φ ✇♦✉❧❞ ♥♦t ❜❡ ❛ ν✲✐♥t❡❣r❛❜❧❡✳ ❋♦r t❤✐s r❡❛s♦♥✱ t❤❡ r❡s✉❧ts ♦❢ t❤❡

♣r❡✈✐♦✉s ♣❛♣❡r ❬✹❪ ♦♥ t❤❡ ❞❡❝❛② ♦❢ t❤❡ ♠❡❛s✉r❡ ν ✇❡r❡ ♦❜t❛✐♥❡❞ ✉♥❞❡r t❤❡ ❤②♣♦t❤❡s✐s t❤❛t µ ✐s s♣r❡❛❞ ♦✉t✳ ❚♦ ❛✈♦✐❞ t❤✐s r❡str✐❝t✐♦♥ ✇❡ ♥❡❡❞ t♦ ❣❡♥❡r❛❧✐③❡ t❤❡ t❡❝❤♥✐❝s ✉s❡❞ ❜② P♦rt ❛♥❞ ❙t♦♥❡ ❬✶✽❪

t♦ ❛ ❧❛r❣❡r ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥sF(µ)❛ss♦❝✐❛t❡❞ t♦ ❛♥ ❛r❜✐tr❛r② ♠❡❛s✉r❡µ✳

▲❡tµ ❜❡ ❛ ❝❡♥t❡r❡❞ ♣r♦❜❛❜✐❧✐t② ♠❡❛s✉r❡ ♦♥R✇✐t❤ ✜♥✐t❡ s❡❝♦♥❞ ♠♦♠❡♥t σ² =R

Rx²µ(dx)✳ ❲❡

❞❡♥♦t❡ ❜② bµ(θ) = R

Re^ixθµ(dx) ✐ts ❋♦✉r✐❡r tr❛♥s❢♦r♠ ❛♥❞ ❣✐✈❡♥ ❛ ❢✉♥❝t✐♦♥ψ∈L¹(R)✇❡ ❞❡✜♥❡ ✐ts

❋♦✉r✐❡r tr❛♥s❢♦r♠ ❜②ψ(θ) =b R

Re^ixθψ(x)dx✳

▲❡tF(µ)❜❡ t❤❡ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥sψ✱ s✉❝❤ t❤❛t

✭✶✮ ψ✱x²ψ❛♥❞ψb❛r❡ ❡❧❡♠❡♥ts ♦❢L¹(R)✱

✭✷✮ t❤❡ ❢✉♥❝t✐♦♥ _1−b^ψ(−θ)b_µ(θ) ✐sdθ✲✐♥t❡❣r❛❜❧❡ ♦✉ts✐❞❡ ❛♥② ♥❡✐❣❤❜♦r❤♦♦❞ ♦❢ ③❡r♦✳

❚❤❡ s❡❝♦♥❞ ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞ ❡✳❣✳ ✇❤❡♥ t❤❡ ♠❡❛s✉r❡µ✐s ❛♣❡r✐♦❞✐❝ ❛♥❞ψb❤❛s ❛ ❝♦♠♣❛❝t s✉♣♣♦rt

♦r ✇❤❡♥ t❤❡ ♠❡❛s✉r❡ µ ✐s s♣r❡❛❞✲♦✉t ✭s✐♥❝❡ ✐s t❤✐s ❝❛s❡ sup_|θ|>a|µ(θ)b | < 1✮✳ ❚❤✉s✱ t❤❡ s❡t F(µ)

❝♦♥t❛✐♥s t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s ♦♥ ✇❤✐❝❤ P♦rt ❛♥❞ ❙t♦♥❡ ❞❡✜♥❡ t❤❡ r❡❝✉rr❡♥t ♣♦t❡♥t✐❛❧ ❜✉t ✐t ✐s✱ ✐♥

♠❛♥② ❝❛s❡s✱ ❜✐❣❣❡r✳ ■♥ ♣❛rt✐❝✉❧❛r ✇❡ ✇✐❧❧ s❡❡ ✐♥ ▲❡♠♠❛ ✸✳✻✱ t❤❛t ✐❢ t❤❡ ♠❡❛s✉r❡ ❤❛s ❛♥ ❡①♣♦♥❡♥t✐❛❧

✽ ❙✳ ❇❘❖❋❋❊❘■❖✱ ❉✳ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊✳ ❉❆▼❊❑

♠♦♠❡♥t✱ t❤❡♥F(µ)❛❧✇❛②s ❝♦♥t❛✐♥s s♦♠❡ ❢✉♥❝t✐♦♥s t❤❛t ❞❡❝❛② ❡①♣♦♥❡♥t✐❛❧❧②✳ ❚❤❛t ✇✐❧❧ ❜❡ s✉✣❝✐❡♥t t♦ ♣r♦✈❡ ♦✉r ♠❛✐♥ t❤❡♦r❡♠ ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥✳

▲❡tJ(ψ) =R

Rψ(x)dx❛♥❞K(ψ) =R

Rxψ(x)dx✱ t❤❡♥ ✇❡ ❤❛✈❡✿

❚❤❡♦r❡♠ ✸✳✷✳ ❆ss✉♠❡ t❤❛tψ, g∈ F(µ)✱g ✐s ♣♦s✐t✐✈❡ ❛♥❞ s✉❝❤ t❤❛t J(g) = 1✳ ❚❤❡♥ ✿

• ❚❤❡ r❡❝✉rr❡♥t ♣♦t❡♥t✐❛❧

Aψ(x) := lim

λր1

J(ψ)

X∞

n=0

λⁿµ^∗n∗g(0)− X∞

n=0

λⁿµ^∗n∗ψ(x)

,

✐s ❛ ✇❡❧❧ ❞❡✜♥❡❞ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥✳

• Aψ ✐s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ✭✸✳✶✮✳

• ■❢ J(ψ)≥0✱ t❤❡♥Aψ ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❛♥❞

✭✸✳✸✮ lim

x→±∞

Aψ(x)

x =±σ⁻²J(ψ).

• ■❢ J(ψ) = 0✱ t❤❡♥Aψ ✐s ❜♦✉♥❞❡❞ ❛♥❞ ❤❛s ❛ ❧✐♠✐t ❛t ✐♥✜♥✐t②

✭✸✳✹✮ lim

x→±∞Aψ(x) =∓σ⁻²K(ψ).

❚❤❡ ♣r♦♦❢ ♦❢ t❤✐s r❡s✉❧t ✐s r❛t❤❡r t❡❝❤♥✐❝❛❧ ❛♥❞ ❢♦❧❧♦✇s t❤❡ ✐❞❡❛s ♦❢ ❬✶✽❪ ❛♥❞ ❬✶✾❪✳ ❆ s❦❡t❝❤ ♦❢ t❤❡

♣r♦♦❢ ✐s ♣r♦♣♦s❡❞ ✐♥ t❤❡ ❛♣♣❡♥❞✐① ❢♦r r❡❛❞❡r ❝♦♥✈❡♥✐❡♥❝❡✳

❆ ❞✐r❡❝t ❝♦♥s❡q✉❡♥❝❡ ♦❢ t❤❡ ♣r❡✈✐♦✉s t❤❡♦r❡♠✱ ✐s t❤❡ ❢♦❧❧♦✇✐♥❣ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ t❤❡ ❜♦✉♥❞❡❞

s♦❧✉t✐♦♥s ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ✿

❈♦r♦❧❧❛r② ✸✳✺✳ ■❢ J(ψ) = 0✱ t❤❡♥ ❡✈❡r② ❝♦♥t✐♥✉♦✉s s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ❜♦✉♥❞❡❞ ❢r♦♠

❜❡❧♦✇ ✐s ♦❢ t❤❡ ❢♦r♠

f =Aψ+h

✇❤❡r❡h✐s ❝♦♥st❛♥t ✐❢µ✐s ❛♣❡r✐♦❞✐❝✱ ❛♥❞ ✐t ✐s ♣❡r✐♦❞✐❝ ♦❢ ♣❡r✐♦❞p✐❢ t❤❡ s✉♣♣♦rt ♦❢µ✐s ❝♦♥t❛✐♥❡❞ ✐♥

pZ✳ ❚❤✉s ❡✈❡r② ❝♦♥t✐♥✉♦✉s s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥ ✐s ❜♦✉♥❞❡❞ ❛♥❞ t❤❡ ❧✐♠✐t ♦❢ f(x)❡①✐sts

✇❤❡♥x❣♦❡s t♦+∞❛♥❞x∈G(µ)✳

❈♦♥✈❡rs❡❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ ❜♦✉♥❞❡❞ s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✱ t❤❡♥ Aψ ✐s ❜♦✉♥❞❡❞ ❛♥❞

J(ψ) = 0✳ ■♥ ♣❛rt✐❝✉❧❛r t❤❡ ✜rst ♣❛rt ♦❢ ❝♦r♦❧❧❛r② ✐s ✈❛❧✐❞✳

Pr♦♦❢✳ ▲❡tJ(ψ) = 0 ❛♥❞ ❛ss✉♠❡ t❤❛tf ✐s ❛ ❝♦♥t✐♥✉♦✉s s♦❧✉t✐♦♥ ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✳ ❙✐♥❝❡

µ∗f =f+ψ ❛♥❞ µ∗Aψ=Aψ+ψ,

t❤❡ ❢✉♥❝t✐♦♥ h=f −Aψ ✐s µ✲❤❛r♠♦♥✐❝✳ ■t ✐s ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇ ❜❡❝❛✉s❡ ❜♦t❤ −Aψ ❛♥❞ f ❛r❡

❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✳ ❚❤❡r❡❢♦r❡ ❜② t❤❡ ❈❤♦q✉❡t✲❉❡♥② t❤❡♦r❡♠ ❬✾❪✱h(x+y) =h(x)❢♦r ❛❧❧y ✐♥ t❤❡

❝❧♦s❡❞ s✉❜❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ s✉♣♣♦rt ♦❢µ✳

❈♦♥✈❡rs❡❧②✱ s✉♣♣♦s❡ t❤❛t t❤❡r❡ ❡①✐sts ❛ ❜♦✉♥❞❡❞ s♦❧✉t✐♦♥ f0 ♦❢ t❤❡ P♦✐ss♦♥ ❡q✉❛t✐♦♥✳ ❚❤❡♥

Aψ−f0 ✐s µ✲❤❛r♠♦♥✐❝ ❛♥❞ ❜♦✉♥❞❡❞ ❢r♦♠ ❜❡❧♦✇✱ ❛♥❞ s♦ t❤❡ ❈❤♦q✉❡t✲❉❡♥② t❤❡♦r❡♠ ✐♠♣❧✐❡s t❤❛t Aψ✐s ❜♦✉♥❞❡❞✳ ❚❤✉s

x→∞lim Aψ(x)

x = 0

❛♥❞ ❜② ✭✸✳✸✮✱ ✇❡ ❞❡❞✉❝❡J(ψ) = 0✳

❆s ❛♥♥♦✉♥❝❡❞ ✇❡ ♥❡❡❞ t♦ ❝♦♥str✉❝t ❛ ❝❧❛ss ♦❢ ❢✉♥❝t✐♦♥s ✐♥ F(µ)t❤❛t ✇✐❧❧ ❜❡ ✉s❡❞ ❧❛t❡r ♦♥ ❛♥❞

t❤❛t ❤❛✈❡ t❤❡ s❛♠❡ t②♣❡ ♦❢ ❞❡❝❛② ❛t ✐♥✜♥✐t② ❛sµ✿

■◆❱❆❘■❆◆❚ ▼❊❆❙❯❘❊ ■◆ ❚❍❊ ❈❘■❚■❈❆▲ ❈❆❙❊ ✾

▲❡♠♠❛ ✸✳✻✳ ▲❡tY ❜❡ ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ✇✐t❤ t❤❡ ❧❛✇ µ✱ t❤❡♥ t❤❡ ❢✉♥❝t✐♦♥

r(x) =E[|Y +x| − |x|]

✐s ♥♦♥♥❡❣❛t✐✈❡ ❛♥❞

b

r(θ) =C· bµ(−θ)−1 θ²

❢♦rθ6= 0✳ ▼♦r❡♦✈❡r ✐❢E[e^δY + e^−δY]<∞✱ t❤❡♥r(x)≤Ce^−δ1|x| ❢♦rδ1< δ✳

❍❡♥❝❡ r❜❡❧♦♥❣s t♦F(µ)❛♥❞ ❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥ζ∈L¹(R)s✉❝❤ t❤❛tx²ζ ✐s ✐♥t❡❣r❛❜❧❡ t❤❡ ❝♦♥✈♦✲

❧✉t✐♦♥ r∗^Rζ ❜❡❧♦♥❣s t♦F(µ)✳

Pr♦♦❢✳ ❖❜s❡r✈❡ t❤❛t✱ s✐♥❝❡EY = 0✱ ❢♦rx≥0 ✇❡ ❝❛♥ ✇r✐t❡

r(x) =E[(Y +x)−2(Y +x)1Y+x≤0−x] =−2E[(Y +x)1Y+x≤0].

Pr♦❝❡❡❞✐♥❣ ❛♥❛❧♦❣♦✉s❧② ❢♦rx <0✱ ✇❡ ♦❜t❛✐♥

✭✸✳✼✮ r(x) =

−2E[(Y +x)1Y+x≤0] ❢♦r x≥0 2E[(Y +x)1Y+x>0] ❢♦r x <0

❚❤✉s t❤❡ ❢✉♥❝t✐♦♥r✐s ♥♦♥♥❡❣❛t✐✈❡✳

❚❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢x ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ✐♥ t❤❡ s❡♥s❡ ♦❢ ❞✐str✐❜✉t✐♦♥s✳ ▲❡ta(x) =|x| ❛♥❞

♦❜s❡r✈❡ t❤❛tr= (µ−δ0)∗a✳ ❚❤❡♥ba(θ) =_θ^C2✱ ❤❡♥❝❡br(θ) =C·^{bµ(−θ)−1}θ² ✳

❚♦ ❡st✐♠❛t❡ t❤❡ ❞❡❝❛② ♦❢r✇❡ ✉s❡ ✭✸✳✼✮✳ ❋♦rx≥0✱ ✇❡ ✇r✐t❡

|r(x)|= 2E[|Y +x|1Y+x≤0] = 2 Z

x+y≤0|x+y|µ(dy)≤2 Z

R

|x+y|e^−δ0(x+y)µ(dy)≤Ce^−δ1x,

❢♦r s♦♠❡ ❝♦♥st❛♥tsδ1< δ0< δ✳

❋✐♥❛❧❧②✱ ✐❢ψ=r∗ζ ✇✐t❤ζ ❛♥❞x²ζ✐♥ L¹(R)✱ t❤❡♥ ✐t ✐s ❡❛s✐❧② ❝❤❡❝❦❡❞ t❤❛t ❜♦t❤ ψ❛♥❞x²ψ❛r❡

✐♥t❡❣r❛❜❧❡✳ ❙✐♥❝❡ψ(θ) =b r(θ)bb ζ(θ) =C^{bµ(−θ)−1}_θ2 ζ(θ)b ❛♥❞bζ✈❛♥✐s❤❡s ❛t ✐♥✜♥✐t②✱ ψ∈ F(µ)

✹✳ Pr♦♦❢s ♦❢ ❚❤❡♦r❡♠s ✶✳✸ ❛♥❞ ✶✳✺ ✲ ❡①✐st❡♥❝❡ ♦❢ t❤❡ ❧✐♠✐t

❖✉r ❛✐♠ ✐s t♦ ❛♣♣❧② t❤❡ r❡s✉❧ts ♦❢ ❙❡❝t✐♦♥ ✸ ❛♥❞ ❢♦r t❤✐s ♣✉r♣♦s❡ ✇❡ ♥❡❡❞ t♦ s❤♦✇ t❤❛t ψφ ✐s s✉✣❝✐❡♥t❧② ✐♥t❡❣r❛❜❧❡✳ ❚❤❡ ✉♣♣❡r ❜♦✉♥❞ ♦❢ t❤❡ t❛✐❧ ♦❢ν❣✐✈❡♥ ✐♥ ❙❡❝t✐♦♥ ✷ ✇✐❧❧ ❣✉❛r❛♥t❡❡ ✐♥t❡❣r❛❜✐❧✐t②

❢♦r ♣♦s✐t✐✈❡ x✳ ❚♦ ❝♦♥tr♦❧ t❤❡ ❢✉♥❝t✐♦♥ ❢♦r x ♥❡❣❛t✐✈❡ ✇❡ ♥❡❡❞ t♦ ♣❡rt✉r❜ s❧✐❣❤t❧② t❤❡ ♠❡❛s✉r❡sµ

❛♥❞ν ✐♥ ♦r❞❡r t♦ ❤❛✈❡ ♠♦r❡ ✐♥t❡❣r❛❜✐❧✐t② ♥❡❛r ✵✳ ❚❤✐s ✐s ✐♥❝❧✉❞❡❞ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛ ♣r♦✈❡❞ ✐♥

❬✹❪ ✭▲❡♠♠❛ ✹✳✶✮✳

▲❡♠♠❛ ✹✳✶✳ ❋♦r ❛❧❧x0∈R^d t❤❡ tr❛♥s❧❛t❡❞ ♠❡❛s✉r❡ν0=δx0∗^Rdν ✐s t❤❡ ✉♥✐q✉❡ ✐♥✈❛r✐❛♥t ♠❡❛s✉r❡

♦❢ µ0=δ(x0,1)∗^Gµ∗^Gδ(−x0,1) ❛♥❞ ✐t ❤❛s t❤❡ s❛♠❡ ❜❡❤❛✈✐♦r ❛s ν ❛t ✐♥✜♥✐t②✱ t❤❛t ✐s✿

x→+∞lim Z

Rd

φ(ue^−x)ν(du)− Z

Rd

φ(ue^−x)ν0(du)

= 0

❢♦r ❡✈❡r② ❢✉♥❝t✐♦♥φ∈C_c¹(R^d\{0})✳ ❋✉rt❤❡r♠♦r❡ t❤❡r❡ ✐sx0∈R^ds✉❝❤ t❤❛t t❤❡ ♠❡❛s✉r❡ν0s❛t✐s✜❡s

✭✹✳✷✮ Z

Rd

1

|u|^γν0(du)<∞❢♦r ❛❧❧ γ∈(0,1)✳

❯s✐♥❣ ✭✹✳✷✮ ✇❡ ❝❛♥ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ψφ ❞❡❝❛②s q✉✐❝❦❧② ❛t ✐♥✜♥✐t②✱ ❛s ✐t ✐s ♣r♦✈❡❞ ✐♥

t❤❡ ❢♦❧❧♦✇✐♥❣ ❧❡♠♠❛✳

✶✵ ❙✳ ❇❘❖❋❋❊❘■❖✱ ❉✳ ❇❯❘❆❈❩❊❲❙❑■ ❆◆❉ ❊✳ ❉❆▼❊❑

▲❡♠♠❛ ✹✳✸✳ ❆ss✉♠❡ t❤❛t ❤②♣♦t❤❡s❡s ✭❍✮ ❛♥❞ ▼✭δ✮ ❛r❡ s❛t✐s✜❡❞✳ ❋✉rt❤❡r♠♦r❡ ❛ss✉♠❡ t❤❛t t❤❡

❢✉♥❝t✐♦♥ |u|^−γ ✐sν(du)✲✐♥t❡❣r❛❜❧❡ ❢♦r ❛❧❧γ∈(0,1)✳ ▲❡tφ❜❡ ❛ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥ ♦♥R^d s✉❝❤ t❤❛t

|φ(u)| ≤ C(1 +|u|)^−β ❢♦r s♦♠❡ β, C > 0✳ ❚❤❡♥ fφ ❛♥❞ µ∗fφ ❛r❡ ✇❡❧❧ ❞❡✜♥❡❞ ❛♥❞ ❝♦♥t✐♥✉♦✉s✳

❋✉rt❤❡r♠♦r❡ ✐❢ φ✐s ▲✐♣s❝❤✐t③✱ t❤❡♥

✭✹✳✹✮ Z

R

Z

G

Z

Rd

φ(e^−x(au+b))−φ(e^−xau)ν(du)µ(db da)dx <∞

❛♥❞

|ψφ(x)| ≤Ce^−ζ|x|,

❢♦rζ < min{δ/4, β,1}✳

Pr♦♦❢✳ ■❢ζ <min{β,1}✱ t❤❡♥

|fφ(x)| ≤ Z

Rd

φ(e^−xu)ν(du)≤ Z

Rd

C

e^−ζx|u|^ζν(du)≤Ce^ζx.

■❢ ✇❡ s✉♣♣♦s❡ ❛❧s♦ζ≤δ✱ ✇❡ ❤❛✈❡ t❤❛t

|µ∗fφ(x)| ≤ Z

R

f_φ(x+y)µ(dy)≤Ce^ζx Z

R+

a^−ζµA(da)≤Ce^ζx.

❚❤✉s ψφ = µ∗^Rfφ−fφ ✐s ✇❡❧❧ ❞❡✜♥❡❞✱ ❝♦♥t✐♥✉♦✉s ❛♥❞ |ψφ(x)| ≤ Ce^ζx✱ t❤❛t ❣✐✈❡s t❤❡ r❡q✉✐r❡❞

❡st✐♠❛t❡s ❢♦r ♥❡❣❛t✐✈❡x✳ ■♥ ♦r❞❡r t♦ ♣r♦✈❡ ✭✹✳✹✮ ✇❡ ❞✐✈✐❞❡ t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦ ♣❛rts✳ ❋♦r ♥❡❣❛t✐✈❡

x✇❡ ✉s❡ t❤❡ ❡st✐♠❛t❡s ❣✐✈❡♥ ❛❜♦✈❡✿

Z 0

−∞

Z

G

Z

Rd

φ(e^−x(au+b))−φ(e^−xau)ν(du)µ(db da)dx

≤ Z 0

−∞

Z

R^d

φ(e^−xu)ν(du)dx+ Z 0

−∞

Z

G

Z

R^d

φ(e^−xau)ν(du)µ(db da)dx

≤ Z 0

−∞

f_|φ|(x)dx+ Z 0

−∞

µ∗f|φ|(x)dx <∞.

❚♦ ❡st✐♠❛t❡ t❤❡ ✐♥t❡❣r❛❧ ♦❢|φ(e^−xau)−φ(e^−x(au+b))|❢♦rx♣♦s✐t✐✈❡✱ ✇❡ ✉s❡ t❤❡ ▲✐♣s❝❤✐t③ ♣r♦♣❡rt②

♦❢φt♦ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐♥❡q✉❛❧✐t② ❢♦r0≤θ≤1

|φ(s)−φ(r)| ≤C|s−r|^θ max

ξ∈{|s|,|r|}

1 (1 +ξ)^β(1−θ).

❆❣❛✐♥ ✇❡ ❞✐✈✐❞❡ t❤❡ ✐♥t❡❣r❛❧ ✐♥t♦ t✇♦ ♣❛rts✳ ❋✐rst ✇❡ ❝♦♥s✐❞❡r t❤❡ ✐♥t❡❣r❛❧ ♦✈❡r t❤❡ s❡t ✇❤❡r❡

|au+b| ≥ ¹2a|u|✳ ❲❡ ❝❤♦♦s❡ θ <min{δ/2,1}✱ γ <min{θ/2, β(1−θ)}✳ ❚❤❡♥✱ ✐♥ ✈✐❡✇ ♦❢ ▼✭δ✮✱ ✇❡

❤❛✈❡

Z Z

|au+b|≥¹₂|au|

φ(e^−xau)−φ(e^−x(au+b))µ(db da)ν(du)

≤ Z

G

Z

Rd

C|e^−xb|^θ

(1 +|e^−xau|)^β(1−θ)ν(du)µ(db da)≤ Z

G

Z

Rd

C|e^−xb|^θ

|e^−xau|^γν(du)µ(db da)

≤Ce^{−(θ−γ)x} Z

G|b|^θ|a|^−γµ(db da) Z

Rd|u|^−γν(du)

≤Ce^{−(θ−γ)x} Z

G |b|^2θ+|a|^−2γ

µ(db da)≤Ce^−γx.