NSCPHD1282 Mathematical Foundations of Heavy Tailed Analysis

Course description:

The course consists on two parts that will be tough consecutively: the first part is devoted to the fundamental concepts of heavy tails and regular varying distributions whereas the second part develops these concepts for time series. The two parts are presented separately below. Their common subject area is Probability and mathematical statistics

• Part 1 (Teacher)

Probabilistic, Analytical and Statistical Models of Heavy Tailed Phenomena in one or more dimensions (Sydney Resnick)

• Scientific content

Regularly varying functions and regularly varying measures.

• Regular variation of measures on complete separable metric spaces such as R, R^p, R^\infty, D[0,1].
• Convergence concepts and mapping theorems; the power of continuity applied to limit theorems.
• TABOF spaces and forbidden zones defining regions of the state space that are "tail regions"?
• Heaviest tail wins: Products, Breiman's theorem, Tauberian theorems as exercises in continuity.

Hidden regular variation, asymptotic independence, asymptotic dependence and thin supports of limit measures.

• Problems where there is more than one regular variation defined.
• Problems where there are infinitely many regular variation properties defined.
• Where did the jumps go asymptotically in a Levy process with regularly varying Levy measure.

• Learning outcome

How to statistically detect a multivariate heavy tail

• The Hillish and Pickandsish plots as diagnostics.
• Connection to the conditional extreme value model.
• Other traditional statistical techniques: QQ plotting, Hill plots and variants, peaks over threshold methods, extreme quantiles and VaR, multivariate methods including rank based methods. Examples from exchange rates, social networks such slashdot, netflow data, packet level data, response data.

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• Part 2 (Teacher)

The tail empirical process for extremally independent heavy tailed time series (Philippe Soulier)

• Scientific content

• Heavy tailed time series; extremal dependence and independence: the tail process; conditional scaling exponent and limiting conditional distributions.
• Conditions for the functional central limit theorem for the tail empirical process of an extremally independent time series. Estimation of the conditional scaling functions and exponents and of the conditional limiting distributions.
• Checking conditions.

• Learning outcome

The tail empirical process is the main tool to investigate the tail behaviour of a stationary regularly varying time series. Its properties are well understood when the series is extremally dependent, that is when extreme values can occur in clusters. In the

seemingly simpler situation of extremal independence where the extreme values can not cluster, hence closer to the i.i.d. situation. However, some finer properties can be investigated. In particular we may be interested in finding different normalizations for which non degenerate limiting conditional distributions given one extreme observation exist and to estimate these normalizations and distributions. To achieve this, a modified version of the tail empirical process must be introduced.