  In earlier stages in the introduction to asymptotic methods in probability
theory, the weak convergence of sequences $(X_n)_{n\geq 1}$ of Binomial of
random variables (\textit{rv}'s) to a Poisson law is classical and easy-to
prove. A version of such a result concerning sequences $(Y_n)_{n\geq 1}$ of
negative binomial \textit{rv}'s also exists. In both cases, $X_n$ and $Y_n-n$
are by-row sums $S_n[X]$ and $S_n[Y]$ of arrays of Bernoulli \textit{rv}'s and
corrected geometric \textit{rv}'s respectively. When considered in the general
frame of asymptotic theorems of by-row sums of \textit{rv}'s of arrays, these
two simple results in the independent and identically distributed scheme can be
generalized to non-stationary data and beyond to non-stationary and dependent
data. Further generalizations give interesting results that would not be found
by direct methods. In this paper, we focus on generalizations to the
non-stationary independent data. Extensions to dependent data will addressed
later.
