June 4-6, 2018

## Organised by:

• Paul Doukhan, AGM Cergy-Pontoise
• Jennifer Denis, AGM Cergy-Pontoise
• Flora Koukiou, LPTM, Cergy-Pontoise
• Eva Loecherbach, AGM Cergy-Pontoise
• Nathalie Picard, THEMA, Cergy-Pontoise
• Jean-Luc Prigent, THEMA, Cergy-Pontoise
• Joseph Rynkiewicz SAMM, Paris 1, La Sorbonne

## Aims of the Conference

The techniques of time series are widely used for the prediction of random phenomena. Other related questions such as for example calibration, resampling or risk management are other important issues.

Stationarity is usually assumed because of a fundamental technical feature: it makes possible to use the ergodic theorem in order to derive consistent estimations. This idea of ergodicity is thus beyond any asymptotical theory for large samples.

The aim of the conference is to consider models which dont meet the stationarity condition. Indeed, the models should fi t the considered data sets, and beyond the local stationarity (see e.g. Dahlhaus), possible models invoking periodicities, exogenous data or shape constraints as monotonicity need further considerations. Applications to real problems and data sets should fit the considered mathematical models.

The emblematic global warming problem is of a vital importance; checking whether it really exists should rely on a test of hypothesis, and relevant models are needed; one may think of models whose dynamics is driven by a sequence of parameters t)1 ≤ t ≤ n, for θt = (θt(1), … ,θt(d)) over a period of observation {1, … ,n}. In this case many d-periodic behaviours depend on the time scale and are given either through days and nights; in this case d = 2 if the sampling time is 12 hours, d = 2 x 12 if this time one hour and d = 6 x 12 x 60 = 4320 is the sampling time is each minute). If seasons are considered then d = 4 in case of monthly observations; finally if the observed period of activity is that of the of the sun then d = 12 for observations sampled each year. An important question is the type of asymptotics, namely the conference is more oriented in a discrete time asymptotic, for instance a random phenomenon Zt is observed at epochs k∆ for a fixed ∆ > 0 and Xk = Zk∆ for k = 1, … , n; this scheme needs ergodic theorems. The same holds in case ∆ = vn x δ if limn→∞ nvn = ∞.
An alternative asymptotic is known as infi lls statistics and corresponds to the cas ∆ = 1/n x δ; in this case ergodic theorems are replaced by regularity conditions on the trajectories of the process (Zt).

Non-asymptotic results are also essential for real data analysis.

Monotonic trends or locally stationary behaviours can be considered. The use of exogenous data time series is also important: think of nebulosity in the global warming setting.

• Procedures for the estimation are given by many speakers mainly on the rst day of the conference, June 4. They include distributional point-wise or uniform asymptotic behaviours of the corresponding estimations.

• The second day of the conference aims at widening the amount of models through the introduction of continuous time or point process valued models. Limit theorems and speci c dependence properties for a wide variety of such models will also be considered.

• The final day of the conference, June 6, is more dedicated to speci c applications. First the introduction of change point models, with stationary behaviours between unknown periods of time n0 = 1 < n1 ≤ … ≤ nk = n is an important tool for the modeling of non-stationarity. Applications to economy (see e.g. commodities trading such as energy markets, online retail banking…), global warming and coral reefs, or astrophysics will be considered. As stressed in the different talks, it is possible to use a very large variety of techniques for relevant data analyses.

Note also that many further applications to nancial and actuarial data, to astronomical data, or to biological, medical or genomic data sets need further considerations. This means that the conference must be viewed as an exploratory one more than a de nitive consideration of non stationarity issues.

Image parAlexas_Fotos de Pixabay