  Transiting exoplanetary systems are surpassingly important among the
planetary systems since they provide the widest spectrum of information for
both the planet and the host star. If a transiting planet is on an eccentric
orbit, the duration of transits T_D is sensitive to the orientation of the
orbital ellipse relative to the line of sight. The precession of the orbit
results in a systematic variation in both the duration of individual transit
events and the observed period between successive transits, P_obs. The
periastron of the ellipse slowly precesses due to general relativity and
possibly the presence of other planets in the system. This secular precession
can be detected through the long-term change in P_obs (transit timing
variations, TTV) or in T_D (transit duration variations, TDV). We estimate the
corresponding precession measurement precision for repeated future observations
of the known eccentric transiting exoplanetary systems (XO-3b, HD 147506b, GJ
436b and HD 17156b) using existing or planned space-borne instruments. The TDV
measurement improves the precession detection sensitivity by orders of
magnitude over the TTV measurement. We find that TDV measurements over a ~4
year period can typically detect the precession rate to a precision well
exceeding the level predicted by general relativity.
