There is a vast literature on branching processes. Here we cite the monographs [1, 3, 12]; moreover, we also cite the monographs [18] for the multitype case, [10], which focuses on statistical inference, and [13] and [15] for applications in biology.

The simplest example of a branching process is the Galton–Watson process. We consider the case of a population that has a unique individual at the beginning and all the individuals (of all generations) live for a unitary time; moreover, at the end of their lifetimes, every individual of the population (of every generation) produces a random number of new individuals acting independently of all the rest, according to a specific fixed distribution. So, if we consider a sequence of random variables {Vn:n≥0} such that Vn is the population size at time n (for all n≥0 ), we have V0=1 and Vn:= ∑k=1Vn−1Xn,k(forn≥1), where {Xn,i:n,i≥1} is a family of nonnegative integer-valued i.i.d. random variables. In other words, Xn,1,…,Xn,Vn−1 represent the offspring generated at time n by each of Vn−1 individuals that live at time n−1 . We recall some other preliminaries on the Galton–Watson process in Section 2, where, in particular, we consider a slightly different notation to allow the case with a random initial population (instead of the case with a unitary initial population cited before).

In this paper, we present large deviation results. The theory of large deviations is a collection of techniques that gives an asymptotic estimate of small probabilities in an exponential scale (see, e.g., [6] as a reference). We recall some preliminaries in Section 2. The literature on large deviations for branching processes is large. Here we essentially recall some references with results concerning the Galton–Watson process.

In several references, the large-time behavior for the supercritical case is studied, namely the case where the offspring mean μ is strictly larger than one (in such a case, the extinction probability is strictly less than one). Here we recall [2] (see also [4] for the multitype case), [5], where the main object is the study of the tails of W:=limn→∞Vn/μn , [19] with a careful analysis based on harmonic moments of {Vn:n≥0} , [20] (and [21]) with some conditional large deviation results based on some local limit theorems, [8] where the central role of some “lower deviation probabilities” is highlighted for the study of the asymptotic behavior of the Lotka–Nagaev estimator Vn+1/Vn of μ.

Other references study the most likely paths to extinction at some time n0 when the initial population k is large. The idea is to consider the representation of a branching process with initial population equal to k as a sum of k i.i.d. replications of the process with a unitary initial population; in this case, Cramér’s theorem for empirical means of i.i.d. random variables (on Rn0 ) plays a crucial role. A most likely path to extinction in [16] (see also [17]) is a trajectory that minimizes the rate function among the paths that reach the level 0 at time n0 . A generalization of this concept for the most likely paths to reach a level b≥0 can be found in [11].

In this paper, we are interested in a different direction. Namely, we are interested in the empirical means of i.i.d. replications of the total progeny of a Galton–Watson process. The total progenies of branching processes are studied in several references: here we cite the old references [7, 14, 22] for a Galton–Watson process, and [9] (see Section 2.2) among the references concerning different branching processes. The total progeny of a Galton–Watson process is an almost surely finite random variable when the extinction occurs almost surely, and therefore the supercritical case will not be considered. Some relationships between the offspring distribution and the total progeny distribution of a Galton–Watson process are well known (see (3) for the probability mass functions and (4) for the probability generating functions).

A new relationship is provided by Proposition 1, where we illustrate how the rate function for the empirical means of total progenies can be expressed in terms of the analogous rate function for the empirical means of a single progeny. This is a quite natural problem to investigate large deviations, and, as we can expect, (4) has an important role in the proof; in fact, the large deviation rate function for empirical means of i.i.d. random variables (provided by Cramér’s theorem recalled below; see Theorem 1) is given by the Legendre transform of the logarithm of the (common) moment generating function of the random variables. Moreover, the relationship provided by Proposition 1 can have interest in information theory because the involved rate functions can be expressed in terms of suitable relative entropies (or Kullback–Leibler divergences); see, for example, [23] for a discussion on the rate function expressions in terms of the relative entropy.

Another result presented in this paper is Proposition 2, that is a version of Proposition 1, where the initial population V0 is a random variable with a suitable distribution. Finally, in Propositions 3 and 4, we prove large deviation results for some estimators of the offspring mean μ in terms of i.i.d. replications of the total progeny and of the initial population (we are considering the case where the initial population V0 is a random variable as in Proposition 2).

We conclude with the outline of the paper. We start with some preliminaries in Section 2. In Section 3, we prove the results concerning the large deviation rate functions related to Cramér’s theorem. Finally, in Section 4, we prove the large deviation results for the estimators of the offspring mean μ.

We start with some preliminaries on the Galton–Watson process. In the second part, we recall some preliminaries on large deviations.

The aim of this section is to prove Propositions 1 and 2. In view of this, we recall Lemmas 1 and 2, which give two immediate applications of Cramér’s theorem (Theorem 1) with d=1 ; in Lemma 2, we consider the case with a unitary initial population almost surely (thus, as stated Remark 1, the case with q1=1 or, equivalently, g=id ).

Now we can prove our main results. We start with Proposition 1, which provides an expression for IGf,id in terms of If .

Now we present Proposition 2, which concerns the LDP for the empirical means of i.i.d. bivariate random variables {(Yn,Zn):n≥1} distributed as (Yf,g,V0f,g) . In particular, we obtain an expression for the rate function IGf,g,g in terms of If in Lemma 1 and Ig defined by (5) Ig(z):=supγ∈R{γz−logg(eγ)}. Proposition 2.

Let {(Yn,Zn):n≥1} be i.i.d. random variables distributed as (Yf,g,V0f,g) . Assume that E[eβYf,g+γV0f,g] is finite in a neighborhood of (β,γ)=(0,0) . Let {(Y¯n,Z¯n):n≥1} be the sequence of empirical means defined by (Y¯n,Z¯n):=(1n∑k=1nYk,1n∑k=1nZk) (for all n≥1 ). Then {(Y¯n,Z¯n):n≥1} satisfies the LDP with good rate function IGf,g,g defined by IGf,g,g(y,z)=yIf(y−zy)+Ig(z)ify≥z>0,Ig(0)ify=z=0,∞otherwise. 0,\\{} I_{g}(0)& \hspace{2.5pt}\textit{if}\hspace{2.5pt}y=z=0,\\{} \infty & \hspace{2.5pt}\textit{otherwise}.\end{array}\right.\]]]>

In this section, we prove two LDPs for two sequences of estimators of the offspring mean μf . Namely, if {(Y¯n,Z¯n):n≥1} is the sequence in Proposition 2 (see also the precise assumptions in Remark 3; in particular, we have μf<1 ), then we consider: 1.

{Y¯n−Z¯nY¯n:n≥1} ;

The LDPs of these two sequences are proved in Propositions 3 and 4. Moreover, Corollary 1 and Remark 4 concern the comparison between the convergence of the first sequence {Y¯n−Z¯nY¯n:n≥1} and its analogue when the initial population is deterministic (equal to the mean). Propositions 3 and 4 are proved by combining the contraction principle (see e.g. Theorem 4.2.1 in [6]) and Proposition 2 (note that the rate function IGf,g,g in Proposition 2 is good, as it is required to apply the contraction principle). We remark that, in the proofs of Propositions 3 and 4, we take into account that IGf,g,g(0,0)=∞ by Proposition 2 and Ig(0)=∞ . At the end of this section, we present some remarks on the comparison between the rate functions in Propositions 3 and 4 (Remarks 5 and 6).

We start with the LDP of the first sequence of estimators.

We have an immediate consequence of this proposition that concerns the case with a deterministic initial population equal to μg (almost surely). Namely, if we consider the probability generating function g◇ defined by g◇(s):=sμg (for all s), then we mean the case g=g◇ , and therefore:

Now we present the LDP for the second sequence of estimators.