  If $G : \mathbb{R}_+ \to \mathbb{R}_+$, the $G$-moment of a vector
$\mathbf{x}\in\mathbb{R}_+^n$ is $G(\mathbf{x}) = \sum_{v\in[n]}
G(\mathbf{x}(v))$ and the $G$-sampling problem is to select an index $v_*\in
[n]$ according to its contribution to the $G$-moment, i.e., such that
$\Pr(v_*=v) = G(\mathbf{x}(v))/G(\mathbf{x})$. Approximate $G$-samplers may
introduce multiplicative and/or additive errors to this probability, and some
have a non-trivial probability of failure.
  In this paper we focus on the exact $G$-sampling problem, where $G$ is
selected from the class $\mathcal{G}$ of Laplace exponents of non-negative,
one-dimensional L\'evy processes, which includes several well studied classes
such as $p$th moments $G(z)=z^p$, $p\in[0,1]$, logarithms $G(z)=\log(1+z)$,
Cohen and Geri's soft concave sublinear functions, which are used to
approximate concave sublinear functions, including cap statistics. We develop
$G$-samplers for a vector $\mathbf{x} \in \mathbb{R}_+^n$ that is presented as
an incremental stream of positive updates. In particular:
  * For any $G\in\mathcal{G}$, we give a very simple $G$-sampler that uses 2
words of memory and stores at all times a $v_*\in [n]$, such that $\Pr(v_*=v)$
is exactly $G(\mathbf{x}(v))/G(\mathbf{x})$.
  * We give a ``universal'' $\mathcal{G}$-sampler that uses $O(\log n)$ words
of memory w.h.p., and given any $G\in \mathcal{G}$ at query time, produces an
exact $G$-sample.
  With an overhead of a factor of $k$, both samplers can be used to $G$-sample
a sequence of $k$ indices with or without replacement. Our sampling framework
is simple and versatile, and can easily be generalized to sampling from more
complex objects like graphs and hypergraphs.
