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Moderate deviation principles for weakly interacting particle systems


Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of \(l_2\) (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.

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Author information

Correspondence to Amarjit Budhiraja.

Additional information

The research is supported in part by the National Science Foundation (DMS-1305120), the Army Research Office (W911NF-14-1-0331) and DARPA (W911NF-15-2-0122).

Appendix: Proof of Remark 3.2(iii)

Appendix: Proof of Remark 3.2(iii)

Suppose that for some \(n \in {\mathbb {N}}\), \(\sum _{m=1}^\infty [a(m)]^{2n} < \infty \). We need to show that \(a(m) \sqrt{m} \Vert \mu ^m(0)-p(0) \Vert \rightarrow 0\) almost surely. To simplify the notation, we will abbreviate \(\mu ^m(0), p(0), \mu ^m_i(0), p_i(0)\) as \(\mu ^m, p, \mu ^m_i, p_i\). It follows from Markov’s inequality that for \({\varepsilon }> 0\),

$$\begin{aligned} {\mathbb {P}}(a(m) \sqrt{m} \Vert \mu ^m-p \Vert > {\varepsilon })\le & {} \left( \frac{a(m) \sqrt{m}}{{\varepsilon }}\right) ^{2n} {\mathbb {E}}\Vert \mu ^m-p \Vert ^{2n} \\= & {} \frac{[a(m)]^{2n}}{{\varepsilon }^{2n}} m^n {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^n. \end{aligned}$$

Since \(\sum _{m=1}^\infty [a(m)]^{2n} < \infty \), by Borel-Cantelli lemma, it suffices to show that for every \(n \in {\mathbb {N}}\) there exists some \(\hat{\gamma }_n \in (0,\infty )\) such that

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^n \le \frac{\hat{\gamma }_n}{m^n}. \end{aligned}$$

We will prove (6.1) when \(n=2\) in detail and then sketch the argument for \(n > 2\). First write

$$\begin{aligned} \mu _i^m-p_i = \frac{1}{m} \sum _{j=1}^m {\varvec{1}}_{\{\xi _j=i\}} - p_i \equiv \frac{1}{m} \sum _{j=1}^m Y_{ij}, \end{aligned}$$

where \(Y_{ij} \doteq {\varvec{1}}_{\{\xi _j=i\}} - p_i\). Note that for all \(\alpha , \beta \in {\mathbb {N}}\),

$$\begin{aligned} |Y_{ij}|\le & {} 1, \quad {\mathbb {E}}Y_{ij} = 0, \quad {\mathbb {E}}Y_{ij}^{2\alpha } \le {\mathbb {E}}Y_{ij}^2 \le p_i, \\ {\mathbb {E}}Y_{ij}^{2\alpha } Y_{kl}^{2\beta }\le & {} {\mathbb {E}}Y_{ij}^2 Y_{kl}^2 \le p_i p_k \quad \text { for } i \ne k. \end{aligned}$$

So we have

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^2= & {} \frac{1}{m^4} {\mathbb {E}}\left[ \sum _{i=1}^\infty \sum _{j,j'=1}^m Y_{ij} Y_{ij'} \right] ^2\nonumber \\= & {} \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{j,j',l,l'=1}^m Y_{ij} Y_{ij'} Y_{kl} Y_{kl'}. \end{aligned}$$

From independence of \(\{\xi _j\}\) it follows that \(Y_{ij}\) and \(Y_{kl}\) are independent for \(j \ne l\). Hence

$$\begin{aligned} {\mathbb {E}}Y_{ij} Y_{ij'} Y_{kl} Y_{kl'} \ne 0 \text{ only } \text{ if } j,j',l,l' \text{ are } \text{ matched } \text{ in } \text{ pairs } \text{(e.g. } j=j' \text{ and } l=l'). \end{aligned}$$

Using this observation, (6.2) can be written as

$$\begin{aligned} \frac{1}{m^4} {\mathbb {E}}\sum _{r=1}^5 \sum _{(i,k,j,j',l,l') \in {\mathcal {H}}_r} Y_{ij} Y_{ij'} Y_{kl} Y_{kl'} \equiv \sum _{r=1}^5 {\mathcal {T}}_r^m, \end{aligned}$$

where \({\mathcal {H}}_1\), \({\mathcal {H}}_2\), \({\mathcal {H}}_3\), \({\mathcal {H}}_4\) and \({\mathcal {H}}_5\) are collections of \((i,k) \in {\mathbb {N}}^2\) and \((j,j',l,l') \in \{1,\cdots ,m\}^4\) such that \(\{ j = j' \ne l = l' \}\), \(\{ j = l \ne j' = l' \}\), \(\{ j = l' \ne j' = l \}\), \(\{ j = j' = l = l', i = k \}\) and \(\{ j = j' = l = l', i \ne k \}\), respectively. For \({\mathcal {T}}_1^m\), it follows from independence of \(\{\xi _j\}\) that

$$\begin{aligned} {\mathcal {T}}_1^m = \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l=1 \\ j \ne l \end{array}}^m Y_{ij}^2 Y_{kl}^2 = \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l=1 \\ j \ne l \end{array}}^m {\mathbb {E}}Y_{ij}^2 {\mathbb {E}}Y_{kl}^2 \le \frac{1}{m^2} \sum _{i,k=1}^\infty p_i p_k = \frac{1}{m^2}. \end{aligned}$$

For \({\mathcal {T}}_2^m\), using independence of \(\{\xi _j\}\) and Cauchy-Schwarz inequality we have

$$\begin{aligned} {\mathcal {T}}_2^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m Y_{ij} Y_{il'} Y_{kj} Y_{kl'} = \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m {\mathbb {E}}\left( Y_{ij} Y_{kj} \right) {\mathbb {E}}\left( Y_{il'} Y_{kl'} \right) \\&\le \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m \sqrt{{\mathbb {E}}Y_{ij}^2 {\mathbb {E}}Y_{kj}^2 {\mathbb {E}}Y_{il'}^2 {\mathbb {E}}Y_{kl'}^2} \le \frac{1}{m^2} \sum _{i,k=1}^\infty p_i p_k = \frac{1}{m^2}. \end{aligned}$$

Similarly, \({\mathcal {T}}_3^m \le \frac{1}{m^2}\). For \({\mathcal {T}}_4^m\) and \({\mathcal {T}}_5^m\), we have

$$\begin{aligned} {\mathcal {T}}_4^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{i=1}^\infty \sum _{j=1}^m Y_{ij}^4 \le \frac{1}{m^3} \sum _{i=1}^\infty p_i = \frac{1}{m^3}, \\ {\mathcal {T}}_5^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{\begin{array}{c} i,k=1 \\ i \ne k \end{array}}^\infty \sum _{j=1}^m Y_{ij}^2 Y_{kj}^2 \le \frac{1}{m^3} \sum _{\begin{array}{c} i,k=1 \\ i \ne k \end{array}}^\infty p_i p_k \le \frac{1}{m^3}. \end{aligned}$$

Combining above estimates we can bound (6.2) by \(\frac{3}{m^2}+\frac{2}{m^3}\). This proves (6.1) when \(n=2\).

Note that the above argument relies crucially on (6.3), which helps reduce the 4-fold summation over \(j,j',l,l'\) in (6.2) to one that is at most 2-fold so that we have (6.1) when \(n=2\). When \(n > 2\), the right hand side of (6.2) will consist of a 2n-fold summation over the second subscript of Y’s. It is easy to check that an observation analogous to (6.3) holds and reduces the 2n-fold summation to one that is at most n-fold. From this we can deduce the desired bound (6.1) and complete the proof.

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Budhiraja, A., Wu, R. Moderate deviation principles for weakly interacting particle systems. Probab. Theory Relat. Fields 168, 721–771 (2017). https://doi.org/10.1007/s00440-016-0723-3

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  • Moderate deviations
  • Large deviations
  • Laplace principle
  • Variational representations
  • Weakly interacting jump-diffusions
  • Nonlinear Markov processes
  • Mean field asymptotics
  • Schwartz distributions
  • Poisson random measures

Mathematics Subject Classification

  • 60F10
  • 60K35
  • 60J75
  • 60J60