  We derive an invariance principle for the lift to the rough path topology of
stochastic processes with delayed regenerative increments under an optimal
moment condition. An interesting feature of the result is the emergence of area
anomaly, a correction term in the second level of the limiting rough path which
is identified as the average stochastic area on a regeneration interval. A few
applications include random walks in random environment and additive
functionals of recurrent Markov chains. The result is formulated in the
p-variation settings, where a rough Donsker Theorem is available under the
second moment condition. The key renewal theorem is applied to obtain an
optimal moment condition.
