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General criteria for the study of quasi-stationarity
Denis VillemonaisCollaborations with Nicolas Champagnat and Cecile Mailler
Universite de Lorraine, Nancy, France
UZH, Stochastische Prozesse, March 27, 2019
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1. Absorbed processes
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A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
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A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
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A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
3 / 21
A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
3 / 21
A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
3 / 21
A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
3 / 21
A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
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A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
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A generic example
Process evolving stochastically in a domain E ⊂R2, absorbed at theboundary
→ ∂ ∉ E unique absorbing point
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Formal defintion
Let (Xt)t∈[0,+∞[ evolving E ∪ {∂}, where ∂ ∉ E is absorbing.
Denoting by τ∂ = inf{t ≥ 0, Xt = ∂} the hitting time of ∂,
Xt = ∂, ∀t ≥ τ∂ almost surely.
Example : Stochastic Lotka-Volterra system in R2
The process (Xt)t∈[0,+∞) describes the population dynamic of twocompeting populations. It evolves in R2+ and satisfies the followingSDE
dX 1t =
√X 1
t dB1t +X 1
t (r− c11X 1t − c12X 2
t )dt,
dX 2t =
√X 2
t dB2t +X 2
t (r− c21X 1t − c22X 2
t )dt.
∂= (0,0) or {0}×R+∪R+× {0} : extinction of the whole populationor extinction of exactly one type.
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Formal defintion
Let (Xt)t∈[0,+∞[ evolving E ∪ {∂}, where ∂ ∉ E is absorbing.
Denoting by τ∂ = inf{t ≥ 0, Xt = ∂} the hitting time of ∂,
Xt = ∂, ∀t ≥ τ∂ almost surely.
Example : Stochastic Lotka-Volterra system in R2
The process (Xt)t∈[0,+∞) describes the population dynamic of twocompeting populations. It evolves in R2+ and satisfies the followingSDE
dX 1t =
√X 1
t dB1t +X 1
t (r− c11X 1t − c12X 2
t )dt,
dX 2t =
√X 2
t dB2t +X 2
t (r− c21X 1t − c22X 2
t )dt.
∂= (0,0) or {0}×R+∪R+× {0} : extinction of the whole populationor extinction of exactly one type.
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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Limiting behaviour
In many interesting cases,
Px0 (Xt ∈ ·) −−−→t→∞ δ∂, ∀x0 ∈ E ∪ {∂}.
Question : What about the distribution of the process conditionednot to be absorbed, namely Px0 (Xt ∈ · | t < τ∂) ?
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2. Quasi-stationary distributions
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Definition
A quasi-stationary distribution (QSD) is a probability measure αon E such that
α= limt→∞Pµ (Xt ∈ ·|t < τ∂)
for some initial probability measure µ on E.
Proposition
A probability measure α is a QSD if and only if, for any t ≥ 0,
α=Pα(Xt ∈ .|t < τ∂).
→ Surveys and book
Meleard, V. 2012, Van Doorn, Pollett 2013Collet, Martınez, San Martın 2013 (prior work)
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Definition
Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that
limt→∞Pµ(Xt ∈ .|t < τ∂) =α.
In the general case :
Existence of a QSD is not true
Existence does not imply uniqueness of a QSD
Uniqueness does not imply attraction of all initialdistributions
Attraction of all initial distributions does not imply uniformconvergence
Question : how to guarantee some or all of the above properties ?
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Definition
Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that
limt→∞Pµ(Xt ∈ .|t < τ∂) =α.
In the general case :
Existence of a QSD is not true
Existence does not imply uniqueness of a QSD
Uniqueness does not imply attraction of all initialdistributions
Attraction of all initial distributions does not imply uniformconvergence
Question : how to guarantee some or all of the above properties ?
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Definition
Let α be a QSD. The domain of attraction of α is the set of initialdistributions µ such that
limt→∞Pµ(Xt ∈ .|t < τ∂) =α.
In the general case :
Existence of a QSD is not true
Existence does not imply uniqueness of a QSD
Uniqueness does not imply attraction of all initialdistributions
Attraction of all initial distributions does not imply uniformconvergence
Question : how to guarantee some or all of the above properties ?
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3. Uniqueness of QSDs and exponential convergence
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Probabilistic approach
Let X evolving in E ∪ {∂} absorbed at ∂.
→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that
Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.
→ Assumption A2 (Harnack inequality)
Pν(t < τ∂)
Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.
Theorem (Champagnat, V. 2016)
A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M 1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥
TV ≤ Ce−γt .
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Probabilistic approach
Let X evolving in E ∪ {∂} absorbed at ∂.
→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that
Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.
→ Assumption A2 (Harnack inequality)
Pν(t < τ∂)
Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.
Theorem (Champagnat, V. 2016)
A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥
TV ≤ Ce−γt .
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Probabilistic approach
Let X evolving in E ∪ {∂} absorbed at ∂.
→ Assumption A1 (Doeblin condition)There exists a probability measure ν and c1 > 0 such that
Px(X1 ∈ ·|1 < τ∂) ≥ c1ν(·), ∀x ∈ E.
→ Assumption A2 (Harnack inequality)
Pν(t < τ∂)
Px(t < τ∂)> c2 > 0, ∀x ∈ E,t ≥ 0.
Theorem (Champagnat, V. 2016)
A1 and A2 ⇔ there exists C > 0, γ> 0 and α ∈M1(E) such that,for all µ ∈M1(E),∥∥Pµ(Xt ∈ ·|t < τ∂)−α∥∥
TV ≤ Ce−γt .
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Comments and examples
A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017
A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process
Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity
or regularity of the boundaries· Not suited for the study of classical models (linear BD,
Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)
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Comments and examples
A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017
A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process
Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity
or regularity of the boundaries· Not suited for the study of classical models (linear BD,
Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)
11 / 21
Comments and examples
A1 and A2 have been used in several situations· general one dimensional diffusion processes· multi-dimensional diffusion processes (with K. Coulibaly-P)· birth and death processes with catastrophes· multi-dimensional birth and death processes· branching/dying Brownian motions· time-inhomogeneous processes· Benaım, Cloez, Panloup 2016, Chazotte, Collet, Meleard 2017
A1 and A2 also imply several interesting properties· Spectral properties of the infinitesimal generator· Uniform convergence of eλ0tPx(t < τ∂) toward an eigenfunction· Existence and exponential ergodicity of the Q-process
Intrinsec limitations· Uniform convergence and uniqueness of QSD=⇒ compact state spaces or entrance boundary at infinity
or regularity of the boundaries· Not suited for the study of classical models (linear BD,
Orstein-Uhlenbeck, AR-1, Galton-Watson, etc...)
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4. A general criterion for the study of quasi-stationarity
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A direct application : perturbed dynamical systems
Let E be a measurable set of Rd with positive Lebesgue measureand let ∂ 6∈ E. Assume that
Xn+1 ={
f (Xn)+ξn if Xn 6= ∂ and f (Xn)+ξn ∈ E,
∂ otherwise,
where f :Rd →Rd is measurable and (ξn)n∈N is an i.i.d.non-degenerate Gaussian sequence in Rd.
Theorem (Champagnat, V. 2017)
If f is locally bounded such that |x|− |f (x)| −−−−−→|x|→+∞
+∞, then there
exists a quasi-stationary distribution νf attracting all initialdistributions on E admitting an exponential moment.
We will denote (later) by θf < 1 the associated eigenvalue :PνQSD (n < τ∂) = θn
f .
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The abstract criterion
Assumption (E). ∃ n1 ∈N, θ1,θ2,c1,c2,c3 > 0, ϕ1,ϕ2 : E →R+ and aprobability measure ν on K ⊂ E such that
(Local A1) ∀x ∈ K , Px(Xn1 ∈ ·) ≥ c1ν(·∩K ).
(Local A2) ∀x ∈ K ,
Pν(n < τ∂)
Px(n < τ∂)≥ c2, ∀n ∈N.
(Aperiodicity) ∀x ∈ K , ∃n4(x) such that
∀n ≥ n4(x), Px(Xn ∈ K ) > 0.
(Lyapunov) θ1 < θ2, ϕ1 ≥ 1, supK ϕ1 <∞, infK ϕ2 > 0, ϕ2 ≤ 1,
E(ϕ1(X1)) ≤ θ1ϕ1(x)+ c21K (x), ∀x ∈ E
E(ϕ2(X1)) ≥ θ2ϕ2(x), ∀x ∈ E.
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The abstract criterion
Assumption (E). ∃ n1 ∈N, θ1,θ2,c1,c2,c3 > 0, ϕ1,ϕ2 : E →R+ and aprobability measure ν on K ⊂ E such that
(Local A1) ∀x ∈ K , Px(Xn1 ∈ ·) ≥ c1ν(·∩K ).
(Local A2) ∀x ∈ K ,
Pν(n < τ∂)
Px(n < τ∂)≥ c2, ∀n ∈N.
(Aperiodicity) ∀x ∈ K , ∃n4(x) such that
∀n ≥ n4(x), Px(Xn ∈ K ) > 0.
(Lyapunov) θ1 < θ2, ϕ1 ≥ 1, supK ϕ1 <∞, infK ϕ2 > 0, ϕ2 ≤ 1,
E(ϕ1(X1)) ≤ θ1ϕ1(x)+ c21K (x), ∀x ∈ E
E(ϕ2(X1)) ≥ θ2ϕ2(x), ∀x ∈ E.
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Main result
Theorem (Champagnat, V. 2017)
If Assumption E is satisfied, then ∃α ∈ (0,1), C > 0 such that
∣∣Eµ [f (Xn) | n < τ∂
]−νQSD(h)∣∣≤ Cαnµ(ϕ1)
µ(ϕ2)
for all µ ∈M1(E) such that µ(ϕ1)/µ(ϕ2) <+∞ and all measurablefunction f : E →R such that |f | ≤ϕ1.
Moreover, ∃η : E →R+ and θ0 ≥ θ2 such that, for all |f | ≤ϕ1,∣∣θ−n0 Ex[f (Xn)]−η(x)α(f )
∣∣≤ Cαnϕ1(x), ∀n ≥ 0,
Note : the constants C and α only depend on quantities appearingin Assumption (E).
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Main result
Theorem (Champagnat, V. 2017)
If Assumption E is satisfied, then ∃α ∈ (0,1), C > 0 such that
∣∣Eµ [f (Xn) | n < τ∂
]−νQSD(h)∣∣≤ Cαnµ(ϕ1)
µ(ϕ2)
for all µ ∈M1(E) such that µ(ϕ1)/µ(ϕ2) <+∞ and all measurablefunction f : E →R such that |f | ≤ϕ1.
Moreover, ∃η : E →R+ and θ0 ≥ θ2 such that, for all |f | ≤ϕ1,∣∣θ−n0 Ex[f (Xn)]−η(x)α(f )
∣∣≤ Cαnϕ1(x), ∀n ≥ 0,
Note : the constants C and α only depend on quantities appearingin Assumption (E).
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Main ingredients of the proof
1 The main point of the proof is to use Hairer and Mattingly[2011] to prove that∥∥∥δxSn0n
0,n0n −δySn0n0,n0n
∥∥∥TV
≤ Cαn(2+ψn0n(x)+ψn0n(y))
for some time dependent Lyapunov function ψn and thesemi-group
Sn+mn,n+1f (x) = EXn=x
(f (Xn+1) | n+m < τ∂
)
2 In order to do so, we prove various estimates onEx(ϕ1(Xn))/Ex(ϕ2(Xn)), on Px(Xn ∈ K | n+m < τ∂) and so on.For instance, one central estimate is, ∀θ ∈ (θ1/θ2,1),
Ex(ϕ1(Xn) | n < τ∂)) ≤ Ex(ϕ1(Xn)1n<τ∂)
Ex(ϕ2(Xn)1n<τ∂)≤
(θnϕ1(x)
ϕ2(x)
)∨Cθ.
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Main ingredients of the proof
1 The main point of the proof is to use Hairer and Mattingly[2011] to prove that∥∥∥δxSn0n
0,n0n −δySn0n0,n0n
∥∥∥TV
≤ Cαn(2+ψn0n(x)+ψn0n(y))
for some time dependent Lyapunov function ψn and thesemi-group
Sn+mn,n+1f (x) = EXn=x
(f (Xn+1) | n+m < τ∂
)
2 In order to do so, we prove various estimates onEx(ϕ1(Xn))/Ex(ϕ2(Xn)), on Px(Xn ∈ K | n+m < τ∂) and so on.For instance, one central estimate is, ∀θ ∈ (θ1/θ2,1),
Ex(ϕ1(Xn) | n < τ∂)) ≤ Ex(ϕ1(Xn)1n<τ∂)
Ex(ϕ2(Xn)1n<τ∂)≤
(θnϕ1(x)
ϕ2(x)
)∨Cθ.
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Application 1. Convergence of a reinforced algorithm
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Consider the process (Yt)t≥0 in Rd evolving as follows :
→ Y evolves following the SDE
dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)
→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1
t
∫ t0 δYs ds.
Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)
Assume that limsupx→+∞⟨b(x),x⟩
|x| <−32‖κ‖
1/2∞. Then
the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞almost surely,
1
t
∫ t
0δYs ds
weak−−−−→t→+∞ νQSD.
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Consider the process (Yt)t≥0 in Rd evolving as follows :
→ Y evolves following the SDE
dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)
→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1
t
∫ t0 δYs ds.
Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)
Assume that limsupx→+∞⟨b(x),x⟩
|x| <−32‖κ‖
1/2∞. Then
the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞
almost surely,1
t
∫ t
0δYs ds
weak−−−−→t→+∞ νQSD.
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Consider the process (Yt)t≥0 in Rd evolving as follows :
→ Y evolves following the SDE
dYt = dBt +b(Yt)dt, Y0 ∈Rd (1)
→ and, with rate κ(Yt) ≥ 1, the process jumps with respect to itsempirical occupation measure 1
t
∫ t0 δYs ds.
Theorem (Champagnat, V. 2017 ; Mailler, V. 2018)
Assume that limsupx→+∞⟨b(x),x⟩
|x| <−32‖κ‖
1/2∞. Then
the solution to (1) admits a unique QSD νQSD with anexponential moment of order ‖κ‖1/2∞almost surely,
1
t
∫ t
0δYs ds
weak−−−−→t→+∞ νQSD.
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Application 2. Law of large numbers for branching processes
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Consider a branching version of the perturbed dynamical systemXn+1 = f (Xn)+ξn with the following branching mecanism :
→ starting from any particle at position x0 ∈ E, N0 ∼ Geom(p) isdrawn and the particle gives birth to N0 iid particles with thesame law as f (x0)+ξ0.
→ each new particle evolve and give birth as above,independently from the others. Denote by Popn the set ofparticles in the population at time n.
Theorem (V. 2019*)
If f is locally bounded such that
|x|− |f (x)| −−−−−→|x|→+∞
+∞,
and if θf /(1−p) > 1, then there exists a random variable W∞ ∈ L2
and three constants θ0 > 0,C > 0,α ∈ (0,1) such that
E
(∣∣∣∣∣θ−n0
∑xk∈Popn
g(xk)−W∞νQSD(g)
∣∣∣∣∣2)
≤ Cαn exp(x0)
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Consider a branching version of the perturbed dynamical systemXn+1 = f (Xn)+ξn with the following branching mecanism :
→ starting from any particle at position x0 ∈ E, N0 ∼ Geom(p) isdrawn and the particle gives birth to N0 iid particles with thesame law as f (x0)+ξ0.
→ each new particle evolve and give birth as above,independently from the others. Denote by Popn the set ofparticles in the population at time n.
Theorem (V. 2019*)
If f is locally bounded such that
|x|− |f (x)| −−−−−→|x|→+∞
+∞,
and if θf /(1−p) > 1, then there exists a random variable W∞ ∈ L2
and three constants θ0 > 0,C > 0,α ∈ (0,1) such that
E
(∣∣∣∣∣θ−n0
∑xk∈Popn
g(xk)−W∞νQSD(g)
∣∣∣∣∣2)
≤ Cαn exp(x0)
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Thank you !
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