A Lognormal Central Limit Theorem for Particle Approximations of Normalizing Constants

Introduction

Consider a Markov chain $(X_{n})_{n\geq 0}$ on a measurable state space $(E,\mathcal{E})$, whose transitions are prescribed by a sequence of Markov kernels $\left(M_{n}\right)_{n\geq 1}$, and a collection of positive bounded and measurable functions $(G_{n})_{n\geq 0}$ on $E$. We associate to $\left(M_{n}\right)_{n\geq 1}$ and $(G_{n})_{n\geq 0}$ the sequence of unnormalized Feynman-Kac measures $\left(\gamma_{n}\right)_{n\geq 0}$ on $E$, defined through their action on bounded (real-valued) measurable functions by:

The corresponding sequence of normalized (probability) Feynman-Kac measures $\left(\eta_{n}\right)_{n\geq 0}$ is defined by:

It is easily checked that, for all $n\geq 0$, the normalizing constant $\gamma_{n}(1)$ satisfies

Here and throughout the paper, the notation $\mu(f)$, where $\mu$ is a finite signed measure and $f$ is a bounded function defined on the same space, is used to denote the Lebesgue integral of $f$ with respect to $\mu$, i.e. $\mu(f):=\int f(x)d\mu(x)$. Given a bounded integral operator $K(x,dx^{\prime})$ from $E$ into itself, we denote by $\mu K$ the measure resulting from the action of $K$ on $\mu$, i.e.

For a bounded measurable function $f$ on $E$, we denote by $K(f)$ the (bounded measurable) function resulting from the action of $K$ on $f$, i.e.

Feynman-Kac measures appear in numerous scientific fields including, among others, signal processing, statistics and statistical physics; see , and for many applications. For example, in a non-linear filtering framework, the measure $\eta_{n}$ corresponds to the posterior distribution of the latent state of a dynamic model at time $n$ given the observations collected from time to time $n-1$, and $\gamma_{n}(1)$ corresponds to the likelihood of these very observations. A generic Monte Carlo application has $(\eta_{n})_{n\geq 0}$ corresponding to a sequence of tempered versions of a distribution $\eta$ that we are interested in sampling from using suitable $\eta_{n}$-invariant Markov kernels $M_{n}$, with $(\gamma_{n}(1))_{n\geq 0}$ the resulting sequence of normalizing constants . Two applications are discussed in more details in Sections 1.3.1 and 1.3.2.

A key issue with Feynman-Kac measures is that they are analytically intractable in most situations of interest. Over the past twenty years, particle methods have emerged as the tool of choice to produce numerical approximations of these measures and their associated normalizing constants. We give a brief overview of these methods here, and refer to for a more thorough treatment.

We first observe that the sequence $(\eta_{n})_{n\geq 0}$ admits the following inductive representation: for all $n\geq 1$, one has

Here, $\Phi_{n}$ is the non-linear transformation on probability measures defined by

where, given a bounded positive function $G$ and a probability measure $\mu$ on $E$, $\Psi_{G}$ denotes the Boltzmann-Gibbs transformation:

One then looks for representations of $\Phi_{n}$ of the form:

where $(K_{n,\mu})_{n,\mu}$ is a collection of Markov kernels defined for every time-index $n\geq 1$ and probability measure $\mu$ on $E$. The choice for $K_{n,\mu}$ is far from being unique. One can obviously use $K_{n,\mu}(x,dx^{\prime}):=\Phi_{n}(\mu)(dx^{\prime})$, but there are alternatives. For example, if $G_{n-1}$ takes its values in the interval $]0,1]$, $\Psi_{G_{n-1}}(\mu)$ can be expressed through a non-linear Markov transport equation

with the non-linear Markov transition kernel

Here $\mathcal{F}_{n-1}^{N}$ is the sigma-field generated by the random variables $(\xi_{p}^{(N)})_{0\leq p\leq n-1}$, and $dx:=dx^{1}\times\ldots\times dx^{N}$ stands for an infinitesimal neighborhood of a point $x=(x^{1},\ldots,x^{N})\in E^{N}$.

Using the identity (1.3) we can easily obtain a particle approximation $\gamma_{n}^{N}\left(1\right)$ of the normalizing constant $\gamma_{n}\left(1\right)$ by replacing the measures $\left(\eta_{p}\right)_{p=0}^{n-1}$ by their particle approximations $\left(\eta_{p}^{N}\right)_{p=0}^{n-1}$ to get

The main goal of this article is to establish a central limit theorem for $\log$ $\overline{\gamma}_{n}^{N}(1)$ as $n\rightarrow\infty$ when the number of particles $N$ is proportional to $n$. Such a result has been conjectured by Pitt et al. , who provided compelling empirical evidence for it. To our knowledge, the present work gives the first mathematical proof of a result of this type.

2 Statement of the main result

To state our result, we need to introduce additional notations. We start with the convention that $\Phi_{0}(\mu):=\eta_{0}$ for all $\mu$, $K_{0,\mu}(x,\cdot):=\eta_{0}(\cdot)$ for all $x$, and $\mathcal{F}_{-1}^{N}=\{\emptyset,\Omega\}$. For the sake of definiteness, we also let $\eta_{-1}:=\eta_{0}$ and $\eta_{-1}^{N}:=\eta_{0}$. These conventions make (1.4)-(1.6)-(1.9) valid for $n=0$.

Then denote by $V_{n}^{N}$ the centered local error random fields defined, for $n\geq 0$, by

To describe the corresponding covariance structure, let us introduce, for all $n\geq 0$, bounded functions $f_{1},f_{2}$, and probability measure $\mu$, the notation

We then have the following explicit expression for conditional covariances:

It is proved in [3, chapter 9] that, under weak regularity assumptions, $(V_{n}^{N})_{n\geq 0}$ converges in law, as $N$ tends to infinity, to a sequence of $n$ independent, Gaussian and centered random fields $(V_{n})_{n\geq 0}$ with a covariance given by

Note that, with the special choice $K_{n,\mu}(x,\cdot):=\Phi_{n}(\mu)$, (1.14) reduces to

Let us now introduce the family of operators $(Q_{p,n})_{0\leq p\leq n}$ acting on the space of bounded measurable functions, defined by

It is easily checked that $(Q_{p,n})_{0\leq p\leq n}$ forms a semigroup for which $\gamma_{n}=\gamma_{p}Q_{p,n}$.

Finally, we define the Markov kernel $P_{p,n}$ through its action on bounded measurable functions:

It is well-known in the literature that (see for example [3, chapter 9]), for fixed $n$, as $N\to+\infty$, the following convergence in distribution holds under weak regularity assumptions:

Here, we are here interested in the fluctuations of $\overline{\gamma}_{n}^{N}(1)$ as both $n,N\rightarrow\infty$ with $N$ proportional to $n$. It turns out that, in such a regime, the observed behavior is different from that described by (1.19). Indeed, the magnitude of the fluctuations of $\overline{\gamma}_{n}^{N}(1)$ around $1$ does not vanish as $n,N$ go to infinity, and they are described in the limit by a log-normal instead of a normal distribution.

Our result is obtained under specific assumptions that we now list. First, the potential functions are assumed to satisfy

Moreover, we assume that the Dobrushin coefficient of $P_{p,n}$, denoted $\beta(P_{p,n})$, satisfies

for some finite constant $a<+\infty$ and some positive $\lambda>0$. Finally, we assume that the kernels $K_{n,\mu}$ satisfy an inequality of the following form:

for any two probability measures $\mu_{1},\mu_{2}$ on $E$, and any measurable map $f$ with oscillation $\mbox{\rm osc}(f):=\mbox{\rm Sup}_{x,y}|f(x)-f(y)|\leq 1$, where $\kappa$ is a finite constant, and $T_{n}(f,\mu_{2})$ is a measurable map with oscillation $\leq 1$ that may depend on $n,f,\mu_{2}$.

In the rest of the paper, unless otherwise stated, we assume that (1.20)-(1.21)-(1.22) hold.

Several sufficient conditions on the Markov kernels $M_{n}$ under which (1.21) holds are discussed in [3, Section 4.3], as well as in Section 3.4 in . Conditions under which (1.22) is satisfied are given in Section 2.

We are now in position to state the main result of the paper.

Assume (1.20)-(1.21)-(1.22), and let $v_{n}$ be defined as

Assume that $N$ depends on $n$ in such a way that

One then has the following convergence in distribution:

where $\mathcal{N}\left(u,v\right)$ denotes the normal distribution of mean $u$ and variance $v$.

We believe that Theorem 1.1 may be established under the weaker stability assumptions developed in and , at the price of a significantly increased technical complexity.

Under assumption (1.20), it is easily seen that one always has $\sup_{n}\frac{v_{n}}{n}<+\infty$. If, in addition to (1.20)-(1.21)-(1.22), one assumes that $\liminf_{n\to+\infty}\frac{v_{n}}{n}>0$ instead of the stronger assumption (1.23), the proof of Theorem 1.1 still leads to a lognormal limit theorem of the following form:

This theoretical result was used in to optimize the asymptotic variance of Metropolis-Hastings estimates, for a given computational budget, using proposal distributions based on particle methods. Another straightforward application is to the bias-correction of log-Bayes factors estimates in large datasets. Yet another potential application in the spirit of is that $\sigma^{2}$ provides a criterion which could be used to select between various interacting particle schemes.

3 Some illustrations

Here, we discuss two concrete situations where Theorem 1.1 can be used, and where the variance expression (1.23) can be made more explicit.

In the present context, we have a map $\Phi$ such that $\Phi_{n}=\Phi$ for all $n\geq 1$, and conditions (1.20)-(1.21) ensure that $\Phi$ has a unique fixed point measure $\eta_{\infty}$ such that

Setting $\overline{Q}=Q/\eta_{\infty}Q(1)$, we find that the function $h$ satisfies the spectral equations

The measure $\eta_{\infty}$ is the so-called quasi-invariant or Yaglom measure. Under some additional conditions, the parameter $\lambda$ coincides with the largest eigenvalue of the integral operator $Q$, and $h$ is the corresponding eigenfunction. In statistical physics, $Q$ comes from a discrete-time approximation of a Schrödinger operator, and $h$ is called the ground state function. For a more thorough discussion, we refer the reader to Chapters 2 and 3 in and Chapter 7 in .

In this scenario, the limiting variance $\sigma^{2}$ appearing in (1.24) is given by

In particular, if the Markov kernels used in the particle approximation scheme are given by $K_{\eta}(x,\mbox{\LARGE.})=\Phi(\eta)$, then using (1.15) we find that $\sigma^{2}=\eta_{\infty}\left([h-1]^{2}\right)$. The detailed statement and proof of these results are provided in Section 3.3.

3.2 Non-linear filtering

Let $(X_{n},Y_{n})_{n\geq 0}$ be a Markov chain on some product state space $E_{1}\times E_{2}$ whose transition mechanism takes the form

where $\left(\nu_{n}\right)_{n\geq 0}$ is a sequence of positive measures on $E_{2}$, $\left(M_{n}\right)_{n\geq 0\text{ }}$ is a sequence of Markov kernels from $E_{1}$ into itself, and $\left(g_{n}\right)_{n\geq 0}$ is a sequence of density functions on $E_{2}\times E_{1}$. The aim of non-linear filtering is to infer the unobserved process $\left(X_{n}\right)_{n\geq 0}$ given a realization of the observation sequence $Y=y$. It is easy to check that

using $G_{n}:=g_{n}(y_{n},\mbox{\LARGE.})$ in (1.1). Furthermore, the density denoted $p_{n}(y_{0},\ldots,y_{n})$ of the random sequence of observations $(Y_{0},\ldots,Y_{n})$ w.r.t. to the product measure $\otimes_{0\leq p\leq n}\nu_{p}$ evaluated at the observation sequence, that is the marginal likelihood, is equal to the normalizing constant $\gamma_{n+1}(1)$. In this context, the multiplicative formula (1.3) takes the following form

For time-homogeneous models $(g_{m},M_{m})=(g,M)$ associated to an ergodic process $Y$ satisfying a random environment version of Assumption (1.21), the ergodic theorem implies that the normalized log-likelihood function converges to the entropy of the observation sequence

where $q(Y_{0}~{}|~{}Y_{m},~{}m<0)$ is the conditional density of the random variable $Y_{0}$ w.r.t. the infinite past. In Section 3.4, we shall prove the existence of a limiting measure $\eta_{\infty}^{Y}$, and function $h^{Y}$ such that

where $q_{0,n}((Y_{0},\ldots,Y_{n})|x)$ stands for the conditional density of $(Y_{0},\ldots,Y_{n})$ given $X_{0}=x$. Similar type results have been recently established in using slightly more restrictive assumptions. In this situation, the limiting variance $\sigma^{2}$ appearing in (1.24) satisfies

where $\theta$ denotes the shift operator, and, if the Markov kernels used by the particle approximation scheme are given by $K_{n,\eta}(x,\mbox{\LARGE.})=\Phi_{n}(\eta)$ associated to the potential $G_{n}:=g_{n}(Y_{n},\mbox{\LARGE.})$, then using (1.15) we obtain

The detailed statement and proof of these results are provided in Section 3.4.

4 Notations and conventions

5 Organization of the paper

The key result, Theorem (1.1), is established in Section 4. The key idea is to expand $\log{\overline{\gamma}_{n}^{N}(1)}$ in terms of local fluctuation terms of the form $V_{k}^{N}$. Broadly speaking, the contribution of quadratic terms in the expansion amounts to an asymptotically deterministic bias term whose fluctuations are controlled with variance bounds, while the contribution of linear terms is treated by invoking the martingale central limit theorem.

Regularity of the covariance function

We first note that, in the special case where $K_{n,\eta}(x,\mbox{\LARGE.})=\Phi_{n}(\eta)$ for all $x$, Property (1.22) is in fact a consequence of (1.20) and (1.21). Indeed, we can then write

Observe that (1.22) immediately implies the following Lipschitz-type property:

Note that there is no loss of generality in assuming that $\mu_{2}(f_{1})=\mu_{2}(f_{2})=0$, so that $\|f_{i}\|\leq\mbox{\rm osc}(f_{i})\leq 1$. Thus, using

the desired conclusion follows from (2.1).

We also state the easily checked Lipschitz type bound, valid for all $f_{1},f_{2},\phi_{1},\phi_{2}\in\mathcal{B}_{b}(E)$

Feynman-Kac semigroups

We denote by $(\Phi_{p,n})_{0\leq p\leq n}$ the semigroup of nonlinear operators acting on probability measures defined by

see for example [3, chapter 4]. We also set

Note that $\overline{Q}_{n,n+1}(1)=G_{n}/\eta_{n}(G_{n})=\overline{G}_{n}$, and that

We will use the fact that the semigroup $Q_{p,n}$ satisfies a decomposition similar to (1.3): for any probability measure $\mu$ on $E$, one has that

Also, combining (1.17) and (3.4), we can write

For any $0\leq p\leq n$ and any $f\in\mbox{\rm Osc}(E)$, we have

In addition, for any $\mu,\nu\in\mathcal{P}(E)$ we have

Proof: Using the decomposition (3.4), we have

From the identity $\log{u}-\log{v}=\int_{0}^{1}\frac{(u-v)}{u+t(v-u)}~{}dt$, valid for any $u,v>0$, we deduce the inequality

with $\widetilde{G}_{q}:=G_{q}/\mbox{\rm osc}(G_{q})$ (and the convention that $\widetilde{G}_{q}:=1$ if $G_{q}$ is constant), and $\widetilde{g}_{q}:={\mbox{\rm osc}(G_{q})}/{\inf G_{q}}\leq g_{q}-1$.

This ends the proof of the l.h.s. of (3.6). The proof of the r.h.s. of (3.6) comes from the following expression for $d_{p,n}(f)$:

which implies, using the fact that $\|\overline{Q}_{p,n}(1)\|\leq g_{p,n}$, that

From [3, Section 4.3], see also Proposition 3.1 in , we have

2 Limiting semigroup

We now state a general theorem on the convergence of $\overline{Q}_{p,n}(1)$ when $n\to+\infty$.

The following bound holds for all $0\leq p\leq n$:

where the limiting function $\overline{Q}_{p,\infty}(1)$ is defined through the following series:

Proof of Theorem 3.2: We first check that the function $\overline{Q}_{p,\infty}(1)$ is well defined, using the fact that, as in the proof of Lemma 3.1,

Using the identity $e^{u}-e^{v}=(x-y)~{}\int_{0}^{1}e^{tu+(1-t)v}dt$, we finally check that

thanks to the fact that $\|\overline{Q}_{p,n}(1)\|\leq g_{p,n}\leq g$. This ends the proof of (3.10).

3 The time-homogeneous case

Here we consider the special case of time-homogeneous models, where there exist $G,M,K$ such that $G_{n}=G$ for all $n\geq 0$, and $M_{n}=M$ and $K_{n}=K$ for all $n\geq 1$.

Our assumptions imply the existence of a unique fixed point $\eta_{\infty}=\Phi(\eta_{\infty})$ towards which $\eta_{n}$ converges exponentially fast: for all $n\geq 0$,

In this situation, Theorem 3.2 leads to a precise description of the asymptotic behavior of the variance term $v_{n}$ appearing in Theorem 1.1. To state it, consider the fixed point measure $\eta_{\infty}$ introduced in (3.12), and define the function $h$ by

In the stationary version of the model where $\eta_{0}:=\eta_{\infty}$, $h$ corresponds to the limiting function $\overline{Q}_{0,\infty}(1)$ whose existence is asserted by Theorem 3.2. In this situation, it turns out that, by stationarity, $\overline{Q}_{n,\infty}(1)=h$ for all $n\geq 1$.

One has the following bound for all $p\geq 0$:

where we use the notation $\mbox{\rm Cov}_{\eta}$ to denote the common value of $\mbox{\rm Cov}_{p,\eta}$ for $p\geq 1$.

An alternative spectral characterization of the map $h$ is given in the following corollary. In the homogeneous case, $Q_{p,p+1}$ does not depend on $p$, so we use the simpler notation $Q$.

In the homogeneous case, the $(\eta_{\infty}Q(1),h)$ is characterized as the unique pair $(\zeta,f)$ such that $Q(f)=\zeta f$ and $\eta_{\infty}(f)=1$.

Proof of Proposition 3.3: Using the exponential convergence to $\eta_{\infty}$ stated in (3.12), and the Lipschitz property (3.7), we have that

We conclude as in the proof of Theorem 3.2.

Using the Lipschitz property (2.2), and the fact that, for all $p,n$, $\|\overline{Q}_{p,n}(1)\|\leq g$, we see that replacing each $\eta_{p-1}$ in the l.h.s. of (3.14) by $\eta_{\infty}$ leads to a $O(1/n)$ error term. Then, using Theorem 3.2 and (2.3), we see that we can replace each $\overline{Q}_{p,n}(1)$ term by $\overline{Q}_{p,\infty}(1)$ in the l.h.s. of (3.14), and commit no more than a $O(1/n)$ overall error. Finally, (3.13), allows us to replace each $\overline{Q}_{p,\infty}(1)$ by $h$, again with an overall $O(1/n)$ error term.

We consider the stationary version of the model where we start with $\eta_{0}:=\eta_{\infty}$.

Let us first check that one indeed has $\eta_{\infty}(h)=1$ and $Q(h)=\eta_{\infty}(Q(1))h$. By Theorem 3.2, we have that

Since by construction, $\eta_{\infty}\overline{Q}_{0,n}(1)=1$, (3.15) yields that $\eta_{\infty}(h)=1$. Then, due to stationarity, one has $\overline{Q}_{p,n}=\overline{Q}^{n-p}$, with $\overline{Q}(f):=Q(f)/\eta_{\infty}Q(1)$, so that one can also deduce from (3.15) that $\overline{Q}(h)=h$, which yields that $Q(h)=\eta_{\infty}(Q(1))h$.

Now consider a pair $(\zeta,f)$ such that $Q(f)=\zeta f$ and $\eta_{\infty}(f)=1$, and let us show that $\zeta=\eta_{\infty}Q(1)$ and $f=h$.

and we deduce from (3.4) and the stationarity of $\eta_{\infty}$ that

Using the fact that $\Phi(\eta_{\infty})=\eta_{\infty}$, we have the identity

Since $Q(f)=\zeta f$ and $\eta_{\infty}(f)=1$, we immediately deduce that $\zeta=\eta_{\infty}(Q(1))$.

As a consequence, the fact that $Q(f)=\eta_{\infty}(Q(1))f$ implies that, for all $n\geq 1$, one has

On the other hand, given two bounded functions $f_{1},f_{2}$, we have that

Letting $n\rightarrow\infty$, (3.12) and Theorem 3.2 yield

Using $f_{1}:=f$ and $f_{2}:=h$, we deduce that $f=h$.

4 The random environment case

Specifically, we consider a family $(M_{s})_{s\in S}$ of Markov kernels on $E$, a family $(G_{s})_{s\in S}$ of positive bounded functions on $E$.

We then use $K^{y}_{n,\mu}:=K_{(y_{n-1},y_{n}),\mu}$ for all $n\geq 1$.

4.2 Contraction properties

Rewriting (3.2) and (3.7) in the present context, we have that, for all $y$,

with the constant $b$ defined in (3.6). Using (3.16), we have

Arguing as in , we conclude that for any $f\in\mathcal{B}_{b}(E)$, and any $\mu\in\mathcal{P}(E)$, $\Phi_{0,n}^{\theta^{-n}(y)}(\mu)(f)$ is a Cauchy sequence, so that $\boldsymbol{\Phi}_{0,n}^{\theta^{-n}(y)}(\mu)$ weakly converges to a measure $\eta_{\infty}^{y}$, as $n\to\infty$. In addition, for any $n\geq 0$, we have

and exponential convergence to equilibrium

We now restate the conclusion of Theorem 3.2 in the present context : for all $0\leq p\leq n$, one has that

where the limiting function $\overline{Q}^{y}_{p,\infty}(1)$ is defined through the series:

Setting $q:=q-p$ in the definition, we rewrite

Combining this bound with (3.18), we deduce that

We conclude as in the proof of Theorem 3.2.

Arguing as in the proof of Corollary 3.14, then applying the ergodic theorem, we deduce the following asymptotic behavior for the variance $v_{n}$.

Fluctuation analysis

In addition to the local error fields $V^{N}_{n}$ defined in (1.12), we consider the global error fields $W^{N}_{n}$ defined by

We now quote key moment estimates on $V^{N}_{n}$ and $W^{N}_{n}$, see [3, chapter 4] or [5, chapter 9]. Under our assumptions, one has that, for all $n\geq 0$, $N\geq 1$, all $f\in\mbox{\rm Osc}(E)$ and $m\geq 1$,

2 Expansion of the particle estimate of log-normalizing constants

Starting from the product-form expression (1.11), we apply a second-order expansion for the logarithm of each factor. Using (4.3), we have that, for all $n\geq 0$ and $N\geq 1$,

where, for all $m\geq 1$, the remainder term satisfies the moment $||C(n,N)||_{m}\leq c(m)$.

3 Second order perturbation formulae

We derive an expansion of $W_{n}^{N}(f)$ in terms of local error terms $V_{p}^{N}$ introduced in (1.12), up to an error term of order $\frac{1}{N}$. The key result we prove is the following.

For all $n\geq 0$, $N\geq 1$ and any function $f\in\mbox{\rm Osc}(E)$,

and where the remainder measure $R_{n}^{N}$ is such that, for all $m\geq 1$,

To prove Theorem 4.1, we start with the following exact decomposition of $W_{n}^{N}(f)$ into a first term of order $1$ involving the $V_{p}^{N}$ for $p=0,...,n$ plus a remainder term of order $1/\sqrt{N}$.

For all $n\geq 0$, $N\geq 1$ and any function $f\in\mbox{\rm Osc}(E)$, we have the decomposition

Note that, under our assumptions, the remainder term satisfies for all $m\geq 1$

Decomposing $1/\eta_{p}^{N}(\overline{G}_{p})$ into a term of order $1$ plus a term of order $1/\sqrt{N}$ as follows

we refine Theorem 4.2 into the following decomposition, which now has an error term of order $1/N$.

For all $n\geq 0$, $N\geq 1$ and any function $f\in\mbox{\rm Osc}(E)$, we have the decomposition

where the remainder term is such that, for all $m\geq 1$, $||\mathcal{R}_{n}^{N}(f)||_{m}\leq c(m)$.

Proof: Using (4.8), we obtain (4.9) with the remainder term

Note that, for any $m\geq 1$, we have that

We are now ready to derive Theorem 4.1, by replacing the $W^{N}_{p}$ terms appearing in the previous corollary by their expansions in terms of the $V^{N}_{p}$ provided by Theorem 4.2. Here is the proof of Theorem 4.1.

4 Fluctuations of local random fields

As mentioned in Section 1.2, when $N$ goes to infinity, the fields $(V^{N}_{n})_{n\geq 0}$ converge in distribution to a sequence of independent centered Gaussian random fields $(V_{n})_{n\geq 0}$ whose covariances are characterized by

We recall that for any $n\geq 1$, $q\geq 1$, and any $q-$tensor product function

the $q$-moments of a centered Gaussian random field $V$ are given by the Wick formula

where $\pi(q)$ denotes the set of pairings of $\{1,\ldots,q\}$, i.e. the set of partitions $\boldsymbol{i}$ of $\{1,\ldots,q\}$ into pairs $\boldsymbol{i}_{1}=\{i_{1},i_{2}\},\ldots,\boldsymbol{i}_{q/2}=\{i_{q-1},i_{q}\}$. Notice that when $q$ is odd, both sides of the above formula are equal to zero.

In the following, we give quantitative bounds on the convergence speed for product-form functionals of the fields $V_{n}^{N}$.

One has the following bound, valid for any $f=(f_{i})_{1\leq i\leq p}\in\mbox{\rm Osc}(E)^{p}$, integers $a=(a_{i})_{1\leq i\leq p}$, $n\geq 0$ and $N\geq 1$:

To prove the proposition, we use the following lemma.

Consider a sequence of $N$ independent random variables $(Z_{i})_{1\leq i\leq N}$ with distributions $\left(\mu_{i}\right)_{1\leq i\leq N}$ on $E$, and define the empirical random fields $V^{N}$ for $f\in\mbox{\rm Osc}(E)$ by

Finally, let $\overline{V}^{N}$ denote a centered Gaussian random field with covariance function defined for any $f,\phi\in\mbox{\rm Osc}(E)$ by

For any $1\leq q\leq N$, and any $q-$tensor product function

where $\rho(q):=1$ for even $q$, and $\rho(q):=1/2$ for odd $q$.

Each term in the above r.h.s. such that an index $j_{i}$ appears exactly once in the list $(j_{1},\ldots,j_{q})$ must be zero, so the only terms that may contribute to the sum are those for which every index appears at least twice. In the case where $q$ is odd, the number of such combinations of indices is bounded above by $c(q)N^{(q-1)/2}$, for some finite constant $c(q)<\infty$ depending only on $q$. Since each expectation is bounded in absolute value by 1, we are done.

Now assume that $q$ is even. Consider a pairing $\boldsymbol{i}$ of $\{1,\ldots,q\}$ given by $\boldsymbol{i}_{1}=\{i_{1},i_{2}\},\ldots,\boldsymbol{i}_{q/2}=\{i_{q-1},i_{q}\}$, and a combination of indices $j_{1},\ldots,j_{q}$ such that $j_{a}=j_{b}$ whenever $a,b$ belong to the same pair, while $j_{a}\neq j_{b}$ otherwise. Denoting by $k_{r}$ the value of $j_{a}$ when $a\in\boldsymbol{i}_{r}$, and using independence, we see that the contribution of this combination to the sum is

Every combination of indices in which every index appears exactly twice is of the form we have just described. Then, the number of combinations in which every index appears at least twice, but that are not of the previous form, is $O(N^{q/2-1})$. As a consequence

(a detailed proof of this formula is provided in Proposition 8.6.1 in ). Now note that

where the last identity uses the Wick formula (4.10).

We end the proof of (4.11) using the fact that $0\leq\left(1-(N)_{p}/N^{p}\right)\leq(p-1)^{2}/N$, for any $p\leq N$. This ends the proof of the lemma.

Given an even number $q$ and a collection of functions $(f_{i})_{1\leq i\leq q}\in\mbox{\rm Osc}(E)^{q}$, for any $n\geq 0$ and $N\geq 1$, we have

We end the proof of (4.12) using the bound

We now come to the proof of Proposition 4.4.

where $a:=a_{p}$. Given $\mathcal{F}_{a-1}^{N}$, we let $\overline{V}_{a}^{N}$ be a sequence of Gaussian random fields with covariance function defined for any $f,\phi\in\mbox{\rm Osc}(E)$ by

On the other hand, combining (4.12) with Wick’s formula (4.10)

One then concludes by iterating the argument.

5 Expansion of the particle estimates continued

We now plug the expansions obtained in Section 4.3 into the development obtained in (4.4), which leads, after some rearrangement, to the following.

For any $n\geq 0$, $N\geq 1$, we have the second order decomposition

and some remainder term such that $||C_{2}(n,N)||_{m}\leq c(m)$, for all $m\geq 1$.

By Theorem 4.1, we may replace $W_{q}^{N}$ by $W_{q}^{N}$ in the linear terms of the expression we want to expand, i.e. the l.h.s. of (4.14), while committing at most an error of the form

On the other hand, using the cruder expansion provided by Theorem 4.2, we may replace $W_{q}^{N}$ by just $\sum_{p=0}^{q}V_{p}^{N}\left[d_{p,q}(\overline{G}_{q})\right]$ in the quadratic terms appearing in the l.h.s. of (4.14), and commit an overall error of the form

By the definition of $W_{q}^{N}$ given in (4.5), we have

It remains to analyze the quadratic part, which we write as

Recalling that $\overline{Q}_{q,q}(1)=1$, we conclude that

The next step is to show that both centered terms $U_{n}^{N}$ and $Y_{n}^{N}$ yield negligible contributions in (4.14).

For any $n\geq 0$, and any $N\geq 1$, we have that

First consider replacing each $V_{k}^{N}$ by the corresponding $V_{k}$ in the above expectations. By Proposition 4.4 together with (3.6), the overall error is bounded by

The only possibility to have a non-zero term is when either $k=k^{\prime}$ and $l=l^{\prime}$ or $k=l^{\prime}$ and $k^{\prime}=l$. Restricting summation to this subset of indices, we obtain that

With a similar argument, we also obtain the following result.

Now, we consider the remaining term in (4.14), i.e.

and show that it can be replaced by its expectation up to a negligible random term.

For any $n\geq 0$, $N\geq 1$, we have the following bound :

Observe that, whenever $k\neq k^{\prime}$, the terms in the above two sums coincide. Therefore, it remains to bound the contribution in both sums of the terms that have $k=k^{\prime}$. In both expressions, the corresponding sum is bounded above in absolute value by

Proof: Recalling that $\overline{Q}_{p,n}(1)-1=\sum_{p\leq k<n}\left(\overline{Q}_{p,k+1}-\overline{Q}_{p,k}\right)$, we prove that

Replacing each $V_{k}^{N}$ by $V_{k}$ in the expectation of $H_{n}^{N}$, we obtain

To control the error introduced by the replacement, we use Proposition 4.4, (3.3) and (3.6), so that the overall error can be bounded above by

6 Central limit theorem

This section established the proof of theorem 1.1. Using Proposition (4.4), the decomposition (4.14), and Propositions (4.8), (4.9), (4.10) and (4.11), we obtain

with $\varepsilon_{n}^{N}$ going to zero in probability as $n$ goes to infinity. Thus, to prove the theorem, it remains to show that

converges in distribution to a standard normal. We do so using the central limit theorem for martingale difference arrays (see e.g. ). The martingale property just comes from the fact that, for any $q\geq 0$ and any bounded function $f_{q}$, one has

converges to $1$ in probability. One easily checks from the definition that

The last point to be checked is the asymptotic negligibility condition, that is, for all $\epsilon>0$, we have to prove that

goes to zero in probability. By Schwarz’s inequality and (4.2), the expectation of this expression is bounded above by

References