Journal of Statistical Physics

, Volume 144, Issue 1, pp 150–170 | Cite as

Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability



For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows.

We consider an unobservable centered stationary Gaussian process X t with covariance function K(u,θ)=E[X t X t+u ], parametrized by an unknown vector θ which lies in a compact subset Θ of ℝ p . We assume that the only observable data are generated by centered stationary processes \(Y_{t}^{\varepsilon }\), indexed by a scale separation parameter ε>0. These approximating processes have arbitrary probability distributions, exponentially decaying covariances, and are assumed to converge to X t in L 4 as ε→0. We show how to construct estimators of the underlying parameter vector θ which depend only on the observable data \(Y_{t}^{\varepsilon }\), and converge to the true parameter values as ε→0.

We study adaptive subsampling schemes involving [N(ε)+k(ε)]→∞ observations \(V_{n} = Y^{\varepsilon }_{n \Delta(\varepsilon )}\) extracted from the approximating process \(Y^{\varepsilon }_{t}\) by subsampling at time intervals Δ(ε)→0. We focus on parameter estimators which are smooth functions of subsampled empirical covariance estimators \(\hat{r}_{k}(N,\Delta)\) associated to non vanishing time lags k(ε)Δ(ε) tending to fixed positive limits as ε→0.

We show that provided lim  ε→0 N(ε)Δ(ε)=+∞, these subsampled approximate covariance estimators converge in L 2 to the true covariance function K(u,θ) of X t for all u,θ. Applying a generic version of the method of moments suitably boosted up by adequately adjusted multiple subsampling schemes, we show that this implies, in a very wide range of situations, the existence of consistent estimators \(\hat{\theta}(\varepsilon )\) of the unknown parameter vector θ, based only on adequately subsampled approximate data \(Y^{\varepsilon }_{t}\).


Gaussian processes Empirical covariance estimators Adaptive sub-sampling Indirect observability Non vanishing lags 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Ecole Normale SuperieureParisFrance

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