**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 Probabilistic, Analytical and Statistical Models of Heavy
Tailed Phenomena in one or more dimensions (**Sydney
Resnick)** 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.
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.
********************************************** The tail empirical process for extremally independent heavy
tailed time series (**Philippe Soulier)** - 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.
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. |