  We present a technique to study normalizing strategies when termination is
asymptotic, that is, it appears as a limit, as opposite to reaching a normal
form in a finite number of steps. Asymptotic termination occurs in several
settings, such as effectful, and in particular probabilistic computation --
where the limits are distributions over the possible outputs -- or infinitary
lambda-calculi -- where the limits are infinitary normal forms such as Boehm
trees.
  As a concrete application, we obtain a result which is of independent
interest: a normalization theorem for Call-by-Value (and -- in a uniform way --
for Call-by-Name) probabilistic lambda-calculus.
