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Keywords: Markov Decision process, Possibility theory, lexicographic compar- isons, possibilistic qualitative utilities
1 Introduction
The classical paradigm **for** sequential decision making under uncertainty is the one of expected utility-based Markov Decision **Processes** (MDP) [11, 2], which assumes that the uncertain effects of actions can be represented by probability distributions and that utilities are additive. But the EU model is not tailored to problems where uncertainty and preferences are ordinal in essence. Alternatives to the EU-based model have been proposed to handle ordinal preferences/uncertainty. Remaining within the probabilis- tic, quantitative, framework while considering ordinal preferences has lead to quantile- based approaches [17, 15, 8, 18, 9]) Purely ordinal approaches to sequential decision under uncertainty have also been considered. In particular, possibilistic MDPs [13, 12, 4, 1] form a purely qualitative decision model with an ordinal evaluation of plausibility and preference. In this model, uncertainty about the consequences of actions is repre- sented by possibility distributions and utilities are also ordinal. The decision criteria are either the optimistic qualitative utility or its pessimistic counterpart [5]. However, it is now well known that possibilistic decision criteria suffer from the drowning effect [6]. Plausible enough bad or good consequences may completely blur the comparison between policies, that would otherwise be clearly differentiable. [6] have proposed lex- icographic refinements of possibilistic criteria **for** the one-step decision case, in order to remediate the drowning effect. In this paper, we propose an extension of the lexico- graphic preference relations to **stationary** possibilistic MDPs.

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the assumptions of the classical linear model on which standard inference techniques can be ap- plied. Further they provide interesting insights on the structure of Markovian **processes**, and thus have interest by themselves. The intercalary independence property was apparently first given with- out proof by Ogawara (1951) **for** univariate Markov **processes**, while the truncation property was used implicitly by him (again without proof) in the context of univariate autoregressive **stationary** Gaussian **processes**. Ogawara (1951) notes that these results have been stated without proof in Lin- nik (1949). However no proof is given by Ogawara (1951) nor (apparently) by any other author. In this section, we demonstrate and extend these results to multivariate Markov **processes** of order p, allowing **for** non-stationarity and non-normality. In order to keep things as simple as possible, we shall assume that the time index set T contains the positive integers N : T ⊇ N = {1, 2, . . . }.

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Estimating function inference is indispensable **for** many common point process models where the joint intensities are tractable while the likelihood function is not. In this paper we es- tablish asymptotic normality of estimating function estimators in a very general setting of non-**stationary** point **processes**. We then adapt this result to the case of non-**stationary** determi- nantal point **processes** which are an important class of models **for** repulsive point patterns. In practice often first and second order estimating functions are used. **For** the latter it is common practice to omit contributions **for** pairs of points separated by a distance larger than some trun- cation distance which is usually specified in an ad hoc manner. We suggest instead a data-driven approach where the truncation distance is adapted automatically to the point process being fit- ted and where the approach integrates seamlessly with our asymptotic framework. The good performance of the adaptive approach is illustrated via simulation studies **for** non-**stationary** determinantal point **processes**.

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From a theoretical point of view, neither the likelihood method nor the minimum contrast methods **for** DPPs have been studied thoroughly, even in assuming that a spectral method **for** C is known. In this work, we focus on parametric **stationary** DPPs and we prove the strong consistency and the asymptotic normality of the min- imum contrast estimators based on K and g. These questions are in connection with the general investigation of Y. Guan and M. Sherman [10], who study the asymp- totic properties of the latter estimators **for** **stationary** point **processes**. However the setting in [10] has a clear view to Cox **processes** and the assumptions involve both

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bution) parametrically estimated and integrated from 0 to t. The extension in higher dimension requires further developments due to the lack of natural ordering. It may be done **for** point pro- cesses admitting a conditional density with respect to the Poisson process. These point **processes** correspond to the Gibbs measures. The equilibrium in one dimension between the number of **events** and the integrated hazard rate may be replaced in higher dimension by the Campbell equilibrium equation or Georgii-Nguyen-Zessin formula (see Georgii ( 1976 ), Nguyen and Zessin ( 1979a ) and Section 2.3 ), which is the basis **for** defining the class of h −residuals where h represents a test func- tion. In particular, Baddeley et al. Baddeley et al. ( 2005 ) consider different choices of h leading to the so-called raw residuals, inverse residuals and Pearson residuals, and show that they share similarities with the residuals obtained **for** generalized linear models.

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INRA-MIAT, Toulouse, France, email: regis.sabbadin@inra.fr
Abstract. Possibilistic Markov Decision **Processes** offer a compact and tractable way to represent and solve problems of sequential decision under qualitative un- certainty. Even though appealing **for** its ability to handle qualitative problems, this model suffers from the drowning effect that is inherent to possibilistic decision theory. The present paper proposes to escape the drowning effect by extending to **stationary** possibilistic MDPs the lexicographic preference relations defined in [6] **for** non-sequential decision problems and provides a value iteration algorithm to compute policies that are optimal **for** these new criteria.

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distribution of a corresponding Markov chain (we refer the reader to the sur- veys [ 15 , 19 ] and to the book [ 11 ] **for** general references on quasi-**stationary** distributions; basic facts and useful results on quasi-**stationary** distributions are also reminded in Section 3 ). Under a similar setting but using stochastic approx- imation techniques, Benaïm and Cloez [ 3 ] and Blanchet, Glynn and Zheng[ 6 ] independently proved the almost sure convergence of the occupation measure µt toward the quasi-**stationary** distribution of X. These works have since been generalized to the compact state space case by Benaïm, Cloez and Panloup [ 4 ] under general criteria **for** the existence of a quasi-**stationary** distribution **for** X. Continuous time diffusion **processes** with smooth bounded killing rate on com- pact Riemanian manifolds have been recently considered by Wang, Roberts and Steinsaltz [ 21 ], who show that a similar algorithm with weights also converges toward the quasi-**stationary** distribution of the underlying diffusion process. Re- cently, Mailler and Villemonais [ 14 ] have proved such a convergence result **for** **processes** with smooth and bounded killing rate evolving in non-compact (more precisely unbounded) spaces using a measure-valued Pólya process representa- tion of this reinforced algorithm.

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varying sub-linear coeﬃcients, 2) a Lipschitz assumption on the predictors and 3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to diﬀerent moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach **for** deriving an adaptive predictor **for** locally **stationary** time varying autoregres- sive (TVAR) **processes**. It is obtained by aggregating a finite number of well chosen predictors, each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the TVAR coeﬃcients. We show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness. To prove this result, a lower bound is established **for** the minimax rate of the prediction risk **for** the TVAR process. Numerical experiments complete this study. An important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online predic- tion context.

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5 Conclusion
In this paper, we focus on the class of **stationary** determinantal point **processes** and present two estimators of the intensity parameter **for** which we prove asymptotic properties. Among the two estimators, one of them, namely the median-based estimator is tailored to be robust to outliers. The median-based estimator depends on a tuning estimator, the number of blocks into which the original window is divided. The empirical findings show that the results are quite sensitive to this parameter. To correct that sensitivity we propose a combined approach and define the estimator e

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In contrast, our asymptotic results are proved in a very general setting, i.e. **for** a large class of **stationary** marked Gibbs models and test functions. The method employed to prove asymptotic normality is based on a conditional centering assumption, first appeared in [24] **for** the Ising model and generalized to certain spatial point **processes** in [28]. The main restriction that this method induces is only the finite range of the Hamiltonian. There are no limitations on the space of parameters and, in particular, the possible presence of phase transition does not affect the asymptotic behavior of the estimator. Moreover, the test functions may depend on the parameters. This extension seems important to us because, as emphasized in Section 3.2.2, such test functions can lead to quick and/or explicit estimators. All the general hypotheses assumed **for** the asymptotic results are discussed. **For** this, we focus on exponential family models, that is, on models whose interaction function is linear in the parameters. We show that our integrability and regularity assumptions are not restrictive since they are valid **for** a large class of models such as the Multi-Strauss marked point process, the Strauss-disc type point process, the Geyer’s triplet point process, the quermass model and **for** all test functions used as a motivation **for** this work. In the setting of the exponential family models, we also discuss the classical identifiability condition which is required **for** the Takacs-Fiksel procedure. To the best of our knowledge, this is the first attempt to discuss it. We will specially dwell on questions like: what choices of test functions (and how many test functions) lead to a unique minimum of the contrast function? We propose general criteria and provide examples. It seems commonly admitted that to achieve the identification of the Takacs-Fiksel procedure, one should at least choose as many test functions as the number of parameters. As a consequence of our study, it appears that one should generally strictly choose more test functions than the number of parameters to achieve identification.

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[2] R. Boman and J.-P. Ponthot. ALE methods **for** **stationary** solutions of metal forming **processes**. In J.A. Cavas, editor, Second ESAFORM conference on Material Forming, pages 585–589, Guimaraes, Portugal, 1998.
[3] A. N. Brooks and T. J. R. Hughes. Streamline upwind/petrov-galerkin formulations **for** convection dominated flows with particular emphasis on the incompressible navier- stokes equations. Computer Methods in Applied Mechanics and Engineering, 32:199– 259, 1982.

ψ(t) dt = 0 and ψ(0) = ψ(1) = 0.
Remark 3 The choice of a wavelet satisfying Assumption W ( ∞) is quite restricted because of the required smoothness of ψ. **For** instance, the function ψ(t) = (t 2
− t + a) exp(−1/t(1 − t)) and a ≃ 0.23087577 satisfies Assumption W ( ∞). The class of ”wavelet” checking Assumption W (5/2) is larger. **For** instance, ψ can be a dilated Daubechies ”mother” wavelet of order d with d ≥ 6 to ensure the smoothness of the function ψ.It is also possible to apply the following theory to ”essentially” compactly supported ”mother” wavelet like the Lemari´e-Meyer wavelet. Note that it is not necessary to choose ψ being a ”mother” wavelet associated to a multi-resolution analysis of L 2 (R) as in the recent paper of Moulines et al. (2007). The whole theory can be developed without this assumption, in which case the choice of ψ is larger.

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particular, we recover the results of Knobloch and Partzsch [14], who proved that (1.1) holds **for** a class of diffusion **processes** evolving in R d (d ≥ 3), us- ing two sided estimates combined with non-trivial spectral properties of the infinitesimal generator of X. We actually prove that the two sided esti- mates are sufficient **for** diffusion **processes** in R d but also **for** general Markov **processes**, while some spectral properties can be recovered from our results. Our second result (Theorem 3.1) is based on gradient estimates of the Dirichlet semi-group obtained by Wang [22] and Priola and Wang [21]. The gradient estimates of [22] hold **for** Brownian motions with C 1 drift evolving in bounded manifolds with C 2 boundary ∂M and killed when they hit ∂M .

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The intercalary independence property was apparently first given without proof by Ogawara 1951 for univariate Markov processes, while the truncation property was used implicitly by him a[r]

that tends to 1−γ γ ǫ when k tends to ∞. In other words, we can see here that the problem of “computing a (non **stationary**) approximately-optimal policy” is not harder than that of “computing approximately the value of some fixed policy”. Since the respective asymptotic errors are 1−γ γ ǫ and 1−γ 1 ǫ, it seems even simpler ! Proof of Theorem 2. The value of π k,m satisfies:

3.3.2 Daubechies Wavelets
The Daubechies wavelet filters represent a collection of wavelets that improve the frequency- domain characteristics of the Haar wavelet and may be still interpreted as generalized differences of adjacent averages (Daubechies, 1992). Daubechies derived these wavelets from the criteria of a compactly supported function with the maximum number of van- ishing moments. In general, there are no explicit time-domain formulae **for** this class **for** wavelet filters (when possible, filter coefficients will be provided). Daubechies first chose an extremal phase factorization, whose resulting wavelet we denote by D(L) where L is the length of the filter. An alternative factorization leads to the least asymmetric class of wavelets, which we denote by LA(L). Shann and Yen (1999) provided exact values **for** both the extremal phase and least asymmetric wavelets of length L ∈ {8, 10}. Longer extremal phase and least asymmetric wavelet filters do not have a closed form and have been tabulated by, **for** example, Daubechies (1992) and Percival and Walden (2000). **For** Daubechies wavelets, the number of vanishing moments is half the filter length, thus the Haar wavelet has a single vanishing moment, the D(4) wavelet has two vanishing moments, and the D(8) and LA(8) wavelets both have four vanishing moments. One implication of this property is that longer wavelet filters may produce **stationary** wavelet coefficients vectors from "higher degree" non-**stationary** stochastic **processes**.

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et al. ( 2015 ) . More precisely, using aggregation techniques introduced in the context of
individual sequences prediction (see e.g. Cesa-Bianchi and Lugosi ( 2006 )) and statistical learning (see e.g. Barron ( 1987 )), one can aggregate su fficiently many predictors in order to build a minimax predictor which adapts to the unknown smoothness β of the time vary- ing parameter of a TVAR process. However, a crucial requirement in Giraud et al. ( 2015 ) is to rely on β-minimax-rate sequences of predictors **for** any β > 0. Our main contribution here is to fill this gap, hence achieving to solve the problem of the adaptive minimax-rate linear forecasting of locally **stationary** TVAR **processes** with coe fficients of any (unknown, arbitrarily large) H¨older smoothness index.

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