A special case of an asymptotic distribution is when the sequence of random variables is always zero or Zi = 0 as i approaches infinity.
However, the most usual sense in which the term asymptotic distribution is used arises where the random variables Zi are modified by two sequences of non-random values.
As an approximation for a finite number of observations, it provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails.
An important example when the local asymptotic normality holds is in the case of independent and identically distributed sampling from a regular parametric model; this is just the central limit theorem.
Barndorff-Nielson & Cox provide a direct definition of asymptotic normality.