  This paper introduces an analytical method to calculate segment-level
wavefront error tolerances in order to enable the detection of faint
extra-solar planets using segmented telescopes in space. This study provides a
full treatment of spatially uncorrelated segment phasing errors for segmented
telescope coronagraphy, which has so far only been approached using ad hoc
Monte-Carlo simulations. Instead of describing the wavefront tolerance globally
for all segments, our method produces spatially dependent requirements. We
relate the statistical mean contrast in the coronagraph dark hole to the
standard deviation of the wavefront error of each individual segment on the
primary mirror. This statistical framework for segment-level tolerancing
extends the Pair-based Analytical model for Segmented Telescope Imaging from
Space (PASTIS), which is based uniquely on a matrix multiplication for the
optical propagation. We confirm our analytical results with Monte-Carlo
simulations of E2E optical propagations through a coronagraph. Comparing our
results for the Apodized Pupil Lyot Coronagraph designs for the Large
UltraViolet Optical InfraRed (LUVOIR) telescope to previous studies, we show
general agreement but provide a relaxation of the requirements for a
significant subset of segments. These requirement maps are unique to any given
telescope geometry and coronagraph design. The spatially uncorrelated segment
tolerances we calculate are a key element of a complete error budget that will
also need to include allocations for correlated segment contributions. We
discuss how the PASTIS formalism can be extended to the spatially correlated
case by deriving the statistical mean contrast and its variance for a
non-diagonal aberration covariance matrix. The PASTIS tolerancing framework
therefore brings a new capability that is necessary for the global tolerancing
of future segmented space observatories.
