  F\'elix-Medina and Thompson (2004) proposed a variant of link-tracing
sampling to estimate the size of a hidden population such as drug users, sexual
workers or homeless people. In their variant a sampling frame of sites where
the members of the population tend to gather is constructed. The frame is not
assumed to cover the whole population, but only a portion of it. A simple
random sample of sites is selected; the people in the sampled sites are
identified and are asked to name other members of the population which are
added to the sample. Those authors proposed maximum likelihood estimators of
the population size which derived from a multinomial model for the numbers of
people found in the sampled sites and a model that considers that the
probability that a person is named by any element in a particular sampled site
(link-probability) does not depend on the named person, that is, that the
probabilities are homogeneous. Later, F\'elix-Medina et al. (2015) proposed
unconditional and conditional maximum likelihood estimators of the population
size which derived from a model that takes into account the heterogeneity of
the link-probabilities. In this work we consider this sampling design and set
conditions for a general model for the link-probabilities that guarantee the
consistency and asymptotic normality of the estimators of the population size
and of the estimators of the parameters of the model for the
link-probabilities. In particular we showed that both the unconditional and
conditional maximum likelihood estimators of the population size are consistent
and have asymptotic normal distributions which are different from each other.
