  In the regularly varying time series setting, a cluster of exceedances is a
short period for which the supremum norm exceeds a high threshold. We propose
to study a generalization of this notion considering short periods, or blocks,
with lp-norm above a high threshold. Our main result derives new large
deviation principles of extremal lp-blocks, which guide us to define and
characterize spectral cluster processes in lp. We then study cluster inference
in lp to motivate our results. We design consistent disjoint block estimators
to infer features of cluster processes. Our estimators promote the use of large
empirical quantiles from the lp-norm of blocks as threshold levels which eases
implementation and also facilitates comparison for different p>0. Our approach
highlights the advantages of cluster inference based on extremal
l$\alpha$--blocks, where $\alpha$>0 is the index of regular variation of the
series. We focus on inferring important indices in extreme value theory, e.g.,
the extremal index.
