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.
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.
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