  Let $X=(X_1,\ldots,X_n)$ be a vector of i.i.d. random variables where $X_i$'s
take values over $\mathbb{N}$. The purpose of this paper is to study the number
of weakly increasing subsequences of $X$ of a given length $k$, and the number
of all weakly increasing subsequences of $X$. For the former, it is shown that
a central limit theorem holds. Also, the first two moments of each of those two
random variables are analyzed, their asymptotics are investigated, and results
are related to the case of similar statistics in uniformly random permutations.
We conclude the paper with applications on a similarity measure of Steele, and
on increasing subsequences of riffle shuffles.
