Abstract:

For many time series, extremes are of practical interest. These may
represent e.g. high temperatures, floods, or stock price crashes. The
statistics of these rare events is given by extreme value theory. For
weakly correlated, stationary time series, extreme value theory
provides a universal description of the size and timings of extreme
events.

Bursts, i.e. a period of activity above some threshold, are similarly
relevant in time series analysis. Typically, one would like to
characterise their duration and size. This will depend on whether the
underlying time series is stationary, and whether the applied
threshold is high.

I will survey results on extremes and bursts in the context of time
series by introducing the theoretical framework and providing some
real-world examples. For non-stationary series, extremes and bursts
depend more specifically on the underlying process, and I will
indicate what can be learned from toy models such as (fractional)
Brownian motions and branching processes.