We consider Galton-Watson branching process with offspring ,
starting with 1 individual in generation 0.
is a non-negative,
integer-valued random variable with mean 1 and variance
.
We note
the offspring
distribution.
We write for the family tree of this branching process and
the probability space of all trees with the law induced by
. We note
the space of size
trees endowed by the
conditional law given
(under this condition, Galton-Watson
trees are simple trees). The law on
will be denoted by
.
Let be a node of
and note
the number of ancestors
of
(
excluded) that have
children. We note
the depth
of the node
(that is
).
We ``show'' strong and uniform properties of the random variables
:
Theorem Let be a positive real number.
For any
To obtain this result (and other ones), we derive properties of ladder variables (the ladder variables are the random variables that intervene in the records of a random walk: time of the records, values of the record, increment that gives a record).
Application to moderate deviations of finite branching random walk are also given.