  Let $X,X_1,X_2,\cdots$ be independent real valued random variables with a
common distribution function $F$, and consider $\{X_1,\cdots,X_N \}$, possibly
a big concrete data set, or an imaginary random sample of size $N\geq 1$ on
$X$. In the latter case, or when a concrete data set in hand is too big to be
entirely processed, then the sample distribution function $F_N$ and the the
population distribution function $F$ are both to be estimated. This, in this
paper, is achieved via viewing $\{X_1,\cdots,X_N \}$ as above, as a finite
population of real valued random variables with $N$ labeled units, and sampling
its indices $\{1,\cdots,N \}$ with replacement $m_N:= \sum_{i=1}^N w_{i}^{(N)}$
times so that for each $1\leq i \leq N$, $w_{i}^{(N)}$ is the count of number
of times the index $i$ of $X_i$ is chosen in this virtual resampling process.
This exposition extends the Doob-Donsker classical theory of weak convergence
of empirical processes to that of the thus created randomly weighted empirical
processes when $N, m_N \rightarrow \infty$ so that $m_N=o(N^2)$.
