In Section 2.5, we state and prove technical lemmas which are necessary for the main results of the paper

SUPPLEMENTARY MATERIAL OF NONPARAMETRIC BAYESIAN ESTIMATION FOR MULTIVARIATE HAWKES PROCESSES

By Sophie Donnet^∗, Vincent Rivoirard^† and Judith Rousseau^†,‡

INRA, Universit´e Paris Saclay^∗, Universit´e Paris-Dauphine^† and University of Oxford^‡

In this supplement, we provide in Section 1 additional numerical results. Section 2 is devoted to the proofs of Theorems 1 and 2 (Sections2.1 and 2.2). We also prove Corollary 1 in Section 2.3and Lemma1 in Section 2.4. In Section 2.5, we state and prove technical lemmas which are necessary for the main results of the paper. Finally, Section 2.6is devoted to the proofs of results of Section 2.3 of [1].

1. Supplementary material for the simulation Section. In this section, we present two additional elements. First, in Table 1 we present a detailed version of the number of observations resulting from the chosen simulation parameters for the three scenarios.

Neuron k 1 2 3 4 5 6 7 8

K= 2

T = 5 415.96 227.88 T = 10 840.24 457.92 T = 20 1667.16 903.44

K= 8 T = 10 801.72 399.36 799.28 194.16 390.68 397.76 396.24 399.20 T = 20 1601.68 805.68 1599.20 392.36 788.20 796.64 794.40 802.32 K=2 with smoothhk,`

T = 5 428.24 206.60 T = 10 841.64 412.80 T = 20 1731.48 838.40

Table 1

Mean numbers of events on each neuron for the three simulation scenarios (the average is done over all the simulated datasets)

In Figure 1, we then present the posterior density of ν1 and ν2 for one given dataset and the regularized distribution of the posterior mean of ν₁ and ν₂ over the 25 simulated datasets for the regular histogram prior (upper panel) and the random histogram prior (bottom panel). We thus illustrate the fact that both priors provide very similar results.

2. Supplementary material for the proof section.

2.1. Proof of Theorem 1. To prove Theorem 1, we apply the general methodology of Ghosal and van der Vaart [2], with modifications due to the fact that exp(L_T(f)) is the likelihood of the distribution of (N^k)k=1,...,K on [0, T] conditional on G₀ and that the metric d1,T depends on the observations. We set MT =M√

log logT, forM a positive constant. Let

A ={f ∈ F; d_1,T(f₀, f)≤K}

1

(a) Results for the regular histogram prior distribution :On the left, posterior distribution ofν₁ (top) and ν₂ (bottom) withT = 5,T = 10 andT = 20 for one dataset.On the right, regularized distribution of the posterior mean of (ν1, ν2)

Eb

ν`|(N_t^sim)_t∈[0,T]

sim=1...25 over the 25 simulated datasets.

(b)Results for the random histogram prior distribution :On the left, posterior distribution ofν1 (top) andν2

(bottom) withT = 5,T = 10 andT = 20 for one dataset.On the right, regularized distribution of the posterior mean of (ν1, ν2)

Eb

ν`|(N_t^sim)_t∈[0,T]

sim=1...25 over the 25 simulated datasets.

Fig 1: Results ofscenario 1 : influence of the prior distribution (random or regular histogram) on the inference of (ν_k)_k=1,2

and for j≥1, we set

(2.1) S_j ={f ∈ F_T; d_1,T(f, f₀)∈(Kj_T, K(j+ 1)_T]},

whereF_T ={f = ((ν_k)_k,(h_{k,`</sub>)<sub>k,`})∈ F; (h_{k,`</sub>)<sub>k,`}∈ H_T}. So that, for any test functionφ, Π A^c_MTT|N

= R

A^c_{MT T} e^{LT(f)−LT(f0)}dΠ(f) R

Fe^{LT(f)−LT(f0)}dΠ(f) =: N¯T

D_T

≤1Ω^c_T +1

DT< ^{Π(B(T ,T))}

exp(2(κT+1)T 2 T)

+φ1ΩT + e^{2(κT+1)T 2T} Π(B(T, T))

Z

F_T^c

e^{LT(f)−LT(f0)}dΠ(f)

+1ΩT

e^{2(κT+1)T 2T} Π(B(_T, T))

∞

X

j=MT

Z

F_T1f∈Sje^{LT(f)−LT(f0)}(1−φ)dΠ(f) and

E0

Π A^c_MTT|N

≤P0(Ω^c_T) +P0

DT < e^{−2(κT+1)T 2T}Π(B(T, B))

+E0[φ1ΩT] + e^{2(κT+1)T 2T}

Π(B(T, B))



Π(F_T^c) +

∞

X

j=MT

Z

F_T

E0

Ef

1ΩT1f∈Sj(1−φ)|G₀ dΠ(f)



,

since E0

"

Z

F_T^c

e^{LT(f)−LT(f0)}dΠ(f)

#

=E0

"

E0

"

Z

F_T^c

e^{LT(f)−LT(f0)}dΠ(f)|G₀

##

=E0

"

Ef

"

Z

F_T^c

dΠ(f)|G₀

##

= Π(F_T^c), which is controlled by using Assumption (ii). Sincee^{(κT+1)T 2T}e^{LT(f)−LT(f0)}≥1{LT(f)−L_T(f0)≥−(κ_T+1)T ²_T},

P0

DT ≤e^{−2(κT+1)T 2T}Π(B(T, B))

≤P0

Z

B(T,B)

e^{LT(f)−LT(f0)} dΠ(f)

Π(B(_T, B)) ≤e^{−2(κT+1)T 2T}

!

≤P0

Z

B(T,B)

1{^LT(f)−LT(f0)≥−(κT+1)T ²_T}

dΠ(f)

Π(B(_T, B)) ≤e^{−(κT+1)T 2T}

!

≤ E0

hR

B(T,B)1{LT(f)−L_T(f0)<−(κ_T+1)T ²_T}Π(B(^dΠ(fT,B))⁾

i

1−e^{−(κT+1)T 2T}

≤ R

B(T,B)P0 LT(f0)−LT(f)>(κT + 1)T ²_T dΠ(f) Π(B(T, B))

1−e^{−(κT+1)T 2T}

. log log(T) log³(T) T ²_T ,

by using Lemma 2 of Section 4.2 of [1]. Remember we have set ρ⁰_{k,`</sub> := kh<sup>0</sup><sub>k,`}k₁ and ρ_{k,`</sub> := kh<sub>k,`}k₁. Since h_{k,`</sub> and h<sup>0</sup><sub>k,`} are non-negative functions,RA

−sh⁰_{k,`</sub>(u)du≤ρ<sup>0</sup><sub>k,`}, RT−s

0 h⁰_{k,`</sub>(u)du≤ρ<sup>0</sup><sub>k,`},and note that

T d1,T(f, f0) =

K

X

`=1

Z T 0

ν_{`</sub>−ν<sub>`}⁰+

K

X

k=1

Z t⁻ t−A

(h_{k,`</sub>−h<sup>0</sup><sub>k,`})(t−s)dN_s^k

dt

≥

K

X

`=1

Z T 0

ν`−ν<sub>`</sub>⁰+

K

X

k=1

Z t⁻ t−A

(hk,`−h<sup>0</sup><sub>k,`</sub>)(t−s)dN_s^k

! dt

≥

K

X

`=1

T(ν`−ν<sub>`</sub>⁰) + Z T

0 K

X

k=1

Z t⁻ t−A

(hk,`−h<sup>0</sup><sub>k,`</sub>)(t−s)dN_s^k

! dt

,

then for any`= 1, . . . , K, d_1,T(f, f0) ≥

ν_{`</sub>−ν<sub>`}⁰+ 1 T

K

X

k=1

Z T 0

Z t⁻ t−A

(h_{k,`</sub>−h<sup>0</sup><sub>k,`})(t−s)dN_s^kdt

=

ν_{`</sub>−ν<sub>`}⁰+

K

X

k=1

(ρ_{k,`</sub>−ρ<sup>0</sup><sub>k,`})N^k[0, T −A]

T +1

T Z 0

−A

Z A

−s

(h_{k,`</sub>−h<sup>0</sup><sub>k,`})(u)dudN_s^k+ 1 T

Z T⁻ T−A

Z T−s 0

(h_{k,`</sub>−h<sup>0</sup><sub>k,`})(u)dudN_s^k

=

ν_`+

K

X

k=1

ρ_k,`N^k[0, T−A]

T + 1

T Z 0

−A

Z A

−s

h_k,`(u)dudN_s^k+ 1 T

Z T⁻ T−A

Z T−s 0

h_k,`(u)dudN_s^k

− ν_`⁰+

K

X

k=1

ρ⁰_k,`N^k[0, T−A]

T + 1

T Z 0

−A

Z A

−s

h⁰_k,`(u)dudN_s^k+ 1 T

Z T⁻ T−A

Z T−s 0

h⁰_k,`(u)dudN_s^k

! .

This implies forf ∈Sj that ν`+

K

X

k=1

ρk,`

N^k[0, T −A]

T ≤ν_`⁰+

K

X

k=1

ρ⁰_k,`N^k[−A, T]

T +K(j+ 1)T

ν`+

K

X

k=1

ρk,`

N^k[−A, T]

T ≥ν_`⁰+

K

X

k=1

ρ⁰_k,`N^k[0, T−A]

T −K(j+ 1)T. (2.2)

On Ω_T,

K

X

k=1

ρ⁰_k,`N^k[−A, T]

T ≤

K

X

k=1

ρ⁰_k,`(µ⁰_k+δ_T),

so that, forT large enough, for all j≥1Sj ⊂ F_j with

F_j :={f ∈ F_T; ν_{`</sub> ≤µ<sup>0</sup><sub>`} + 1 +Kj_T,∀`≤K}, since

(2.3) µ⁰_{`</sub> =ν<sub>`}⁰+

K

X

k=1

ρ⁰_k,`µ⁰_k.

Let (fi)i=1,...,Nj be the centering points of a minimal L1-covering of F_j by balls of radius ζj_T with ζ = 1/(6N₀) (with N₀ defined in Section2of [1]) and defineφ_(j) = maxi=1,...,N_jφ_fi,j whereφ_fi,j is the individual test defined in Lemma 1associated tofi andj (see Section4.1of [1]). Note also that there exists a constant C0 such that

N_j ≤(C₀(1 +j_T)/j_T)^KN(ζj_T/2,H_T,k.k₁)

whereN(ζj_T/2,H_T,k.k₁) is the covering number ofH_T byL1-balls with radiusζj_T/2. There exists C_K such that if j_T ≤ 1 then N_j ≤ C_Ke^−Klog(jT)N(ζj_T/2,H_T,k.k₁) and if j_T > 1 then N_j ≤ CKN(ζjT/2,H_T,k.k₁). Moreover j 7→ N(ζjT/2,H_T,k.k₁) is monotone non-increasing, choosing j≥2ζ₀/ζ, we obtain that

N_j ≤C_K(ζ/ζ₀)^Ke^KlogTe^{x0T 2T},

from hypothesis (iii) in Theorem1. Combining this with Lemma1, we have for all j≥2ζ₀/ζ, E0[1ΩTφ_(j)].N_je^{−T x2(jT∧j22T)}.e^KlogTe^{x0T 2T}e^{−x2T(jT∧j22T)} sup

f∈F_jE0

Ef[1ΩT1f∈S_j(1−φ_(j))|G₀]

.e^{−x2T(jT∧j22T)}, forx₂ a constant. Setφ= maxj≥M_Tφ_(j) withM_T >2ζ₀/ζ, then

E0[1ΩTφ].e^KlogTe^{x0T 2T}







b⁻¹_T c

X

j=M_T

e^{−x2T 2Tj2} + X

j≥⁻¹_T

e^{−T x2Tj}





.e^{−x2T 2TMT2/2}

and ∞

X

j=MT

Z

F_T E0

Ef

1ΩT1f∈S_j(1−φ)|G₀

dΠ(f).e^{−x2T 2TMT2/2}. Therefore, using Assumption (i),

e^{2(κT+1)T 2T} Π(B(_T, B))

∞

X

j=MT

Z

F_TE0

Ef

1ΩT1f∈Sj(1−φ)|G₀

dΠ(f) =o(1) ifM is a constant large enough, which terminates the proof of Theorem 1.

2.2. Proof of Theorem2. The proof of Theorem2 follows the same lines as for Theorem1, except that the decomposition of F_T is based on the sets F_j and H_{T ,i}, i≥1 andj ≥M_T for someM_T >0.

For each i ≥ 1, j ≥ M_T, consider S_i,j⁰ a maximal set of ζj_T-separated points in F_j ∩ H_T,i (with a slight abuse of notations) andφi,j = max_f1∈S⁰

i,jφ_f1 with φ_f1 defined in Lemma 1. Then,

|S_i,j⁰ | ≤CK(ζ/ζ0)^Ke^Klog(T)N(ζjT/2,H_{T ,i},k.k₁).

Setting ¯NT ,ij := R

F_T∩H_{T ,i}1f∈Sje^{LT(f)−LT(f0)}dΠ(f), using similar computations as for the proof of Theorem1, we have

E0

Π A^c_M

T_T|N

≤P0(Ω^c_T) +P0

D_T < e^{−2(κT+1)T 2T}Π(B(_T, B))

+ e^{2(κT+1)T 2T}

Π(B(T, B))Π(F_T^c) +E0



1ΩT

+∞

X

i=1 +∞

X

j=MT

φ_ijN¯_T,ij D_T



+ e^{2(κT+1)T 2T} Π(B(_T, B))E0



1ΩT

+∞

X

i=1 +∞

X

j=MT

(1−φ_ij) ¯N_T,ij



.

Assumptions of the theorem allow us to deal with the first three terms. So, we just have to bound the last two ones. Using the same arguments and the same notations as for Theorem 1,

E0



1ΩT

+∞

X

i=1 +∞

X

j=MT

(1−φij) ¯NT ,ij



 =

+∞

X

i=1

Z

F_T∩H_{T ,i} +∞

X

j=MT

E0

h

1ΩT1f∈Sj(1−φij)e^{LT(f)−LT(f0)} i

dΠ(f)

=

+∞

X

i=1

Z

FT∩HT ,i

+∞

X

j=M_T

E0

Ef[1ΩT1f∈Sj(1−φ_ij)|G₀] dΠ(f)

.

+∞

X

i=1

Z

F_T∩H_{T ,i}

dΠ(f)

+∞

X

j=MT

e^{−x2T(jT∧j22T)}.e^{−x2T 2TMT2/2}.

Now, for γ a fixed positive constant smaller thanx2, settingπT ,i= Π(H_{T ,i}), we have E0



1^ΩT

+∞

X

i=1 +∞

X

j=MT

φij

N¯_{T ,ij} DT



≤P0

DT < e^{−2(κT+1)T 2T}Π(B(T, B)) +P0

∃(i, j);√

πT ,iφi,j> e^{−γT(jT∧j22T)}∩ΩT

+

+∞

X

i=1 +∞

X

j=M_T

e^{−γT(jT∧j22T)}√

π_{T ,i}e^{2(κT+1)T 2T} Π(B(T, B))E0

1ΩT

Z

FT

1f∈Sje^{LT(f)−LT(f0)}dΠ(f|HT ,i)

.

Now, P0

∃(i, j);√

π_{T ,i}φ_i,j > e^{−γT(jT∧j22T)}∩Ω_T

≤

+∞

X

i=1

√π_T,i

+∞

X

j=MT

e^{γT(jT∧j22T)}E0[1ΩTφ_i,j]

.

+∞

X

i=1

√π_T,i

+∞

X

j=MT

e^{(γ−x2)T(jT∧j22T)+Klog(T)}N(ζj_T/2,H_T,i,k.k₁)

.e^{(γ−x2)T 2TMT2/2}

+∞

X

i=1

√πT,iN(ζ0T,H_{T ,i},k.k₁) =o(1).

But, we have

E0

1ΩT

Z

F_T 1f∈S_je^{LT(f)−LT(f0)}dΠ(f|H_{T ,i})

≤ 1 and

E0



1ΩT

+∞

X

i=1 +∞

X

j=MT

φij

N¯_T,ij DT



 .

+∞

X

i=1

√πT ,ie^{−γT 2TMT2} e^{2(κT+1)T 2T}

Π(B(T, B))+o(1) =o(1), forM a constant large enough. This terminates the proof of Theorem 2.

2.3. Proof of Corollary1. LetwT →+∞. The proof of Corollary1follows from the usual convexity argument, so that

kfˆ−f₀k₁≤w_Tε_T +E^π

kf−f₀k₁1kf−f0k1>wTεT|N ,

together with a control of the second term of the right hand side similar to the proof of Theorem 3.

We write E^π

kf −f0k₁1kf−f₀k₁>wTεT|N

≤E^π h

kf−f0k₁1AL1(wTεT)^c1A_εT|Ni +E^π

h

kf −f0k₁1A^c_εT|Ni

and since R

kf−f₀k₁dΠ(f)≤ kf₀k₁+R

kfk₁dΠ(f)<∞, P0

E^π

hkf −f₀k₁1A_L1(w_Tε_T)^c1A_εT|Ni

> w_Tε_T

≤P0 Ω^c_1,T +P0

D_T < e^{−c1T 2T}

+e^{c1T 2T} wTεT

Z

AL1(wTεT)^c

kf −f0k₁E0[Pf(Ω1,T ∩ {d_1,T(f0, f)≤εT})|G₀]dΠ(f)

≤o(1) +o(1) Z

kf −f₀k₁dΠ(f) =o(1),

where the last inequality comes from the proof of Theorem3. Similarly, using the proof of Theorem1, P0

E^π

hkf−f₀k₁1A^c_εT|Ni

> w_Tε_T

≤P0(Ω^c_T) +P0

D_T < e^{−c1T 2T}

+E0[1ΩTφ]

+ e^{c1T 2T} w_Tε_T

Z

AL1(wTεT)^c

kf −f0k₁E0

h Ef

h

(1−φ)1ΩT1{d_1,T(f0,f)>εT}

i

|G₀i dΠ(f)

≤o(1) +o(1) Z

kf−f0k₁dΠ(f),

and P0(kfˆ−f0k₁>3wTεT) =o(1). Since this is true for anywT →+∞, this terminates the proof.

2.4. Proof of Lemma 1. For the sake of completeness, we recall the statement of Lemma 1.

Lemma 1. Let j≥1, f₁∈ F_j and define the test φ_f1,j = max

`=1,...,K

1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f0)≥jT T/8}∨1{N<sup>`</sup>(A<sup>c</sup><sub>1,`</sub>)−Λ<sup>`</sup>(A<sup>c</sup><sub>1,`</sub>;f0)≥jT T/8}

, with for all ` ≤ K, A<sub>1,`</sub> = {t ∈ [0, T]; λ^{`</sup><sub>t</sub>(f<sub>1</sub>) ≥ λ<sup>`}_t(f₀)}, Λ<sup>`</sup>(A<sub>1,`</sub>;f₀) = RT

0 1A1,`(t)λ<sup>`</sup>_t(f₀)dt and Λ<sup>`</sup>(A<sup>c</sup><sub>1,`</sub>;f0) =RT

0 1A^c<sub>1,`</sub>(t)λ<sup>`</sup>_t(f0)dt. Then E0[1ΩTφ_f1,j] + sup

kf−f1k1≤jT/(6N0)

E0

Ef

1ΩT1f∈Sj(1−φ_f1,j)|G₀

≤(2K+ 1) max

` e<sup>−x1,`T jT(</sup>

√

µ⁰_`∧j_T),

with N0 is defined in Section 2 of [1] and x1,`= min

36,1/(4096µ⁰_`),1/

1024K

q µ⁰_`

.

Proof of Lemma 1. Let j ≥1 and f1 = ((ν_k¹)_k=1,...,K,(h¹<sub>`,k</sub>)k,`=1,...,K) ∈ F_j. Let ` ∈ {1, . . . , K} and let

φj,A1,` =1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f0)≥jT _T/8}. By using (2.3), observe that on the event Ω_T,

Z _T

0

λ<sup>`</sup><sub>s</sub>(f<sub>0</sub>)ds=ν<sub>`</sub>⁰T+

K

X

k=1

Z _T

0

Z _s⁻

s−A

h⁰_k,`(s−u)dN_u^kds

≤ν_`⁰T+

K

X

k=1

Z T⁻

−A

Z T 0

1u<s≤A+uh⁰_k,`(s−u)dsdN_u^k and for T large enough,

(2.4)

Z T 0

λ<sup>`</sup><sub>s</sub>(f0)ds≤ν<sub>`</sub>⁰T +

K

X

k=1

ρ⁰_{k,`</sub>N<sup>k</sup>[−A, T]≤2T µ<sup>0</sup><sub>`}.

Let j ≤ q

µ⁰_{`</sub><sup>−1</sup><sub>T</sub> and x = x1j<sup>2</sup>T <sup>2</sup><sub>T</sub>, for x1 a constant. We use inequality (7.7) of [4], with τ = T, H<sub>t</sub>= 1<sub>A1,`}(t),v= 2T µ⁰<sub>`</sub> and M<sub>T</sub> =N<sup>`</sup>(A<sub>1,`</sub>)−Λ<sup>`</sup>(A_1,`;f₀).So,

P0

n

N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f0)≥√

2vx+x 3

o

∩ΩT

≤e^{−x1j2T 2T}.

Ifx1≤1/(1024µ⁰_`) and x1 ≤36, we have that

(2.5) √

2vx+x 3 = 2

q

µ⁰_`x1jT T +x1j²T ²_T

3 ≤2

q µ⁰_`x1

1 +

√x₁ 6

jT T ≤ jT T

8 . Then

P0

N<sup>`</sup>(A<sub>1,`</sub>)−Λ<sup>`</sup>(A<sub>1,`</sub>;f₀)≥ jT _T 8

∩Ω_T

≤e^{−x1j2T 2T}. Ifj ≥q

µ⁰_`⁻¹_T , we apply the same inequality but withx=x0jT T with x0 = q

µ⁰_` ×x1. Then,

√

2vx+x 3 = 2

r µ⁰_`x₁

q

µ⁰_`j_TT + x₁

q

µ⁰_`jT _T

3 ≤2

q

µ⁰_`x₁jT _T + x₁

q

µ⁰_`jT _T

3 ≤ jT _T

8 , where we have used (2.5). It implies

P0

N<sup>`</sup>(A<sub>1,`</sub>)−Λ<sup>`</sup>(A<sub>1,`</sub>;f₀)≥ jT _T 8

∩Ω_T

≤e^{−x0jT T}.

Finally E0

1ΩTφj,A1,`

≤e^{−x1T jT(}

√

µ⁰_`∧jT).Now, assume that Z

A_1,`

(λ^{`</sup><sub>t</sub>(f1)−λ<sup>`}_t(f0))dt≥ Z

A^c_1,`

(λ^{`</sup><sub>t</sub>(f0)−λ<sup>`}_t(f1))dt.

Then

(2.6) kλ^{`</sup>(f1)−λ<sup>`}(f0)k₁

2 :=

RT

0 |λ^{`</sup><sub>t</sub>(f1)−λ<sup>`}_t(f0)|dt

2 ≤

Z

A1,`

(λ^{`</sup><sub>t</sub>(f1)−λ<sup>`}_t(f0))dt.

Letf = ((ν_k)_k=1,...,K,(h<sub>`,k</sub>)k,`=1,...,K)∈S_j satisfying kf−f₁k₁ ≤ζj_T for someζ >0. Then, kλ^{`</sup>(f)−λ<sup>`}(f1)k₁ ≤T|ν_{`</sub>−ν<sub>`}¹|+

Z T 0

Z t⁻ t−A

X

k

(h_{k,`</sub>−h<sup>1</sup><sub>k,`})(t−u)dN_u^k

dt

≤T|ν_{`</sub>−ν<sub>`}¹|+X

k

Z _T

0

Z _t⁻

t−A

|(h_{k,`</sub>−h<sup>1</sup><sub>k,`})(t−u)|dN_u^kdt

≤T|ν_{`</sub>−ν<sub>`}¹|+ max

k N^k[−A, T]X

k

kh_{k,`</sub>−h<sup>1</sup><sub>k,`}k₁ ≤T N0kf−f1k₁ (2.7)

and kλ^{`</sup>(f)−λ<sup>`}(f1)k₁ ≤T N0ζjT. Since f ∈Sj, there exists`(depending on f) such that kλ<sup>`</sup>(f)−λ^`(f0)k₁ ≥jT T.

This implies in particular that ifN₀ζ <1,

kλ^{`</sup>(f1)−λ<sup>`}(f0)k₁≥ kλ^{`</sup>(f)−λ<sup>`}(f0)k₁−T N0ζjT ≥(1−N0ζ)T jT. We then have

Λ<sup>`</sup>(A<sub>1,`</sub>;f)−Λ<sup>`</sup>(A<sub>1,`</sub>;f₀) = Λ<sup>`</sup>(A<sub>1,`</sub>;f)−Λ<sup>`</sup>(A<sub>1,`</sub>;f₁) + Λ<sup>`</sup>(A<sub>1,`</sub>;f₁)−Λ<sup>`</sup>(A<sub>1,`</sub>;f₀)

≥ −kλ^{`</sup>(f)−λ<sup>`}(f1)k₁+ Z

A1,`

(λ^{`</sup><sub>t</sub>(f1)−λ<sup>`}_t(f0))dt

≥ −kλ^{`</sup>(f)−λ<sup>`}(f1)k₁+kλ^{`</sup>(f1)−λ<sup>`}(f0)k₁ 2

≥ −T N₀ζj_T + (1−N₀ζ)T j_T

2 = (1/2−3N₀ζ/2)T j_T. Taking ζ= 1/(6N0) leads to

Ef

1f∈Sj(1−φj,A1,`)1ΩT|G₀

= Ef

h

1f∈Sj1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f0)<jT T/8}1ΩT|G₀i

≤ Ef

h

1f∈S_j1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f)≤−jT _T/8}1ΩT|G₀i

≤ Ef

h

1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f)≤−jT _T/8}1ΩT|G₀i .

Note that we can adapt inequality (7.7) of [4], withH_t=1A1,`(t) to the case of conditional probability givenG₀ since the processEtdefined in the proof of Theorem 3 of [4], being a supermartingale, satisfies Ef[E_t|G₀]≤E₀ = 1 and, given that from (2.2) and (2.4),

Z T 0

λ<sup>`</sup><sub>s</sub>(f)ds≤ν<sub>`</sub>T+

K

X

k=1

ρ_{k,`</sub>N<sup>k</sup>[−A, T]≤2T µ<sup>0</sup><sub>`}+K(j+ 1)T _T =: ˜v forT large enough, we obtain

Ef

h

1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f)≤−√ 2˜vx−^x

3}1ΩT|G₀i

≤e^−x. We use the same computations as before, observing that ˜v=v+K(j+ 1)T _T.

Ifj ≤ q

µ⁰_`⁻¹_T we set x=x1j²T ²_T, forx1 a constant. Then,

√

2˜vx+x

3 ≤ √

2vx+x 3 +p

2K(j+ 1)T Tx

≤ 2 q

µ⁰_`x1jT T + x₁j²T ²_T

3 +p

2K(j+ 1)Tx1jT T

≤ 2 q

µ⁰_`x1

1 +

√x₁ 6

jT T + 2p

KjTx1jT T

≤ 2 q

µ⁰_`x1

1 +

√x₁ 6

+ 2

r K

q µ⁰_`x1

! jT T.

Therefore, if x₁ ≤min

36,1/(4096µ⁰_`),1/

1024K q

µ⁰_` , then

√

2˜vx+ x

3 ≤ jT _T 8 . Ifj ≥q

µ⁰_`⁻¹_T , we setx=x₀jT _T withx₀ = q

µ⁰_` ×x₁. Then,

√

2˜vx+x

3 ≤ √

2vx+x 3 +p

2K(j+ 1)T _Tx

≤ 2 r

µ⁰_`x₁ q

µ⁰_`j_TT+ x1

q

µ⁰_`jT T

3 +

r

2K(j+ 1)T _T q

µ⁰_`x₁jT _T

≤ 2 q

µ⁰_`x1jT T + x₁

q µ⁰_`jT _T

3 + 2

r K

q

µ⁰_`x1jT T ≤ jT T

8 . Therefore,

Ef

h

1{N<sup>`</sup>(A1,`)−Λ<sup>`</sup>(A1,`;f)≤−jT _T/8}1ΩT|G₀i

≤e^{−x1T jT(}

√

µ⁰_`∧jT). Now, if

Z

A1,`

(λ^{`</sup><sub>t</sub>(f<sub>1</sub>)−λ<sup>`}_t(f₀))dt <

Z

A^c_1,`

(λ^{`</sup><sub>t</sub>(f<sub>0</sub>)−λ<sup>`}_t(f₁))dt, then

Z

A^c_1,`

(λ^{`</sup><sub>t</sub>(f<sub>1</sub>)−λ<sup>`}_t(f₀))dt≥ kλ^{`</sup>(f<sub>1</sub>)−λ<sup>`}(f₀)k₁ 2

and the same computations are run withA_{1,`</sub>playing the role ofA<sup>c</sup><sub>1,`}. This ends the proof of Lemma 1.

2.5. Technical lemmas.

2.5.1. Control of the number of occurrences of the process on a fixed interval.

Lemma 2. For any M ≥1, for any α > 0, there exists a constant C_α only depending on f₀ such that for any T >0, the set

Ω˜T = (

`∈{1,...,K}max sup

t∈[0,T]

N^`([t−A, t))≤CαlogT )

satisfies

P0( ˜Ω^c_T)≤T^−α and for any 1≤m≤M

E0

"

`∈{1,...,K}max sup

t∈[0,T]

N^`([t−A, t))m

×1_Ω˜c T

#

≤2T^−α/2, for T large enough.

Proof. For the first part, we split the interval [−A;T] into disjoint intervals of length A and we use Proposition 2 of [4]. For the second part, we set

X := max

`∈{1,...,K} sup

t∈[0,T]

N^`([t−A, t))

×1Ω˜^c_T ≥0 and the equality

E0[X^m] =

Z +∞

0

mx^m−1P0(X > x)dx

=

Z CαlogT 0

mx^m−1P0(X > x)dx+ Z +∞

CαlogT

mx^m−1P0(X > x)dx

≤ m(CαlogT)^m−1

Z CαlogT 0

P0( ˜Ω^c_T)dx+ Z +∞

CαlogT

mx^m−1P0(X > x)dx

≤ m(CαlogT)^mT^−α+ Z +∞

CαlogT

mx^m−1P0(X > x)dx.

Furthermore, for T large enough, Z +∞

CαlogT

mx^m−1P0(X > x)dx ≤

Z +∞

CαlogT

mx^m−1P0 max

`∈{1,...,K} sup

t∈[0,T]

N^`([t−A, t))

> x

! dx

≤

Z +∞

CαlogT

mx^m−1P0 max

`∈{1,...,K} sup

t∈[0,e^x/Cα]

N^`([t−A, t))

> x

! dx

≤

Z +∞

CαlogT

mx^m−1exp(−αx/C_α)dx≤T^−α/2.

2.5.2. Control of N[0, T]. Letk∈ {1, . . . , K}. We have the following result.

Lemma 3. For any k∈ {1, . . . , K}, for allα >0 there existsδ0>0 such that P0

N^k[0, T] T −µ⁰_k

≥δ₀

r(logT)³ T

!

=O(T^−α).

Proof of Lemma 3. We use Proposition 3 of [4] and notations introduced for this result. We de- noteN[−A,0) the total number of points ofN in [−A,0), all marks included. Letδ_T :=δ₀p

(logT)³/T, withδ₀ a constant. We have

(2.8)

P0

N^k[0, T] T −µ⁰_k

> δ_T

≤P0

N^k[0, T]− Z T

0

λ^k_t(f₀)dt

>T δ_T 2

! +P0

Z T 0

[λ^k_t(f₀)−µ⁰_k]dt

> T δ_T 2

!

and we observe that

λ^k_t(f0) =ν_k⁰+ Z t⁻

t−A K

X

`=1

h⁰<sub>`,k</sub>(t−s)dN<sub>s</sub><sup>`</sup>=Z◦S_t(N),

withZ(N) =λ^k₀(f₀), whereSis the shift operator introduced in Proposition 3 of [4]. We then have Z(N)≤b(1 +N[−A,0))

with

b= max

k max{ν_k⁰,max

` kh<sup>0</sup><sub>`,k</sub>k_∞}.

So, for anyα >0, the second term of (2.8) isO(T^−α) forδ₀ large enough depending onαandf₀. The first term is controlled by using Inequality (7.7) of [4] withτ =T,x=x₀T δ²_T,H_t= 1,v=µ⁰_kT+T δ_T/2 and

M_T =N^k[0, T]− Z T

0

λ^k_t(f0)dt.

We take x₀ a positive constant such that q

8µ⁰_kx₀<1,so that, forT large enough T δ_T

2 ≥√

2vx+x/3.

Therefore, we have P0

|M_T|> T δT

2

≤ P0

|M_T| ≥√

2vx+x/3 and Z T

0

λ^k_t(f0)dt≤v

+P0

Z T 0

λ^k_t(f0)dt > v

≤ 2 exp(−x) +P0

Z T 0

[λ^k_t(f0)−µ⁰_k]dt

> T δT

2

≤ 2 exp(−x₀δ₀²(logT)³) +O(T^−α) =O(T^−α), which terminates the proof.