Edgeworth expansions for profiles of lattice branching random walks
Consider a branching random walk on in discrete time. Denote by the number of particles at site at time . By the profile of the branching random walk (at time ) we mean the function . We establish the following asymptotic expansion of , as :
where is arbitrary, is the cumulant generating function of the intensity of the branching random walk and
The expansion is valid uniformly in with probability and the ’s are polynomials whose random coefficients can be expressed through the derivatives of and the derivatives of the limit of the Biggins martingale at . Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walk except its extreme values. As an application of this expansion for we recover in a unified way a number of known results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers , where depends on in some regular way. We also prove a.s. limit theorems for the mode and the height of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.
Key words and phrases:Branching random walk, Edgeworth expansion, central limit theorem, profile, Biggins martingale, random analytic function, mod--convergence, height, mode
2010 Mathematics Subject Classification:Primary, 60G50; secondary, 60F05, 60J80, 60F10, 60F15
1.1. Statement of the problem
The branching random walk (BRW) is a model that combines random spatial motion of particles with branching; see  for a historical overview. In this paper, we restrict our attention to the BRW on the integer lattice . The model is defined as follows. At time consider a single ancestor particle located at . At any time every particle is replaced (independently of all other particles and of the past of the process) by a random finite cluster of descendant particles whose displacements w.r.t. the original particle are distributed according to some fixed point process on . The number of particles in may be random. The positions of the particles in need not be independent random variables (nor need they be independent of the number of particles).
Denote by the number of particles at time . Then is a Galton–Watson branching process. We will always assume that this process is supercritical (meaning that ) and with probability never dies out (meaning that a.s.) The latter assumption could be removed, but then all results hold conditionally on non-extinction. Denote the positions of the particles in the BRW at time by
Our main object of interest is the occupation number defined as the number of particles located at site at time :
The random function will be referred to as the profile of the branching random walk at time . The intensity of the branching random walk at time is the measure on defined as the expectation of the profile:
We will study the asymptotic shape of the profile as . More concretely, we will obtain an asymptotic expansion of in powers of which is similar to the classical Chebyshev–Edgeworth–Cramér expansion for sums of independent identically distributed random variables. The latter will be recalled in Section 1.2.
It follows from the definition of the branching random walk that the intensity measure is the -fold convolution of (so that, in particular, ). Throughout the history of the BRW this fact has been used to relate the asymptotic properties of the random measures represented by , as , to the analogous classical properties of convolutions of probability measures. In connection with the central limit theorem, this leads to the Harris conjecture [27, Chapter III, §16] which was proved (in various forms and for various models) in [37, 28, 1, 30, 7, 39, 9, 40]. The work of Biggins [4, 6, 9] refers to this analogy in connection with large deviation principles and local limit theorems.
The present work adds one more example to this list by proving an asymptotic expansion of the profile. Since the asymptotic expansion contains many other classical limit theorems as consequences, we will be able to recover many of the above mentioned results in a unified way (though under sub-optimal moment conditions). Our asymptotic expansion of the profile will be stated in Theorem 2.1. For comparison, an asymptotic expansion of the intensity measure will be given in Proposition 2.6. It turns out that the two expansions do not coincide: while the expansion of the intensity is deterministic, the expansion of the profile contains random terms which can be expressed through the derivatives of a remarkable random analytic function given as the limit of the Biggins martingale; see below.
Our motivation for investigating such asymptotic expansions was to develop a method to study binary search trees, random recursive trees, and some similar types of random trees that appear in the analysis of algorithms. An important characteristic of these trees is their profile, that is the random function counting the number of nodes at a given level , for a tree with nodes. The probabilistic properties of these profiles have been much studied; see, e.g., [18, 19, 20, 14]. Since random trees can be embedded into continuous-time branching random walks, see Chauvin et al.  and Biggins and Grey , it is possible to translate our asymptotic expansion into the setting of random trees. In Sections 2.6 and 2.7, we will answer the BRW analogues of several open questions on the profiles of random trees. For example, we will prove limit theorems on the height and the mode of the branching random walk. While we plan to give applications to random trees in a separate paper, we will point out connections to the existing literature on random trees here.
The paper is organized as follows. In Section 1.2 we recall the classical asymptotic expansion for sums of i.i.d. random variables. In Section 1.3 we fix the notation and state our assumptions on the BRW. In Section 1.4 we briefly comment on our simulations. In Section 2 we state the asymptotic expansion and its numerous consequences. Proofs are given in Section 3.
1.2. Asymptotic expansion for sums of i.i.d. random variables
Let be independent identically distributed (i.i.d.) random variables with and . Define the sequence of their partial sums:
We assume that takes only integer values and that the lattice span of the distribution of is , that is there is no pair , such that all possible values of are contained in the arithmetic progression .
Under these assumptions, the local limit theorem (see, e.g. Theorem 1 in [34, Ch. VII, p. 187]) states that
Under additional moment conditions, there is a complete asymptotic expansion of in powers of , the classical Chebyshev–Edgeworth–Cramér expansion. Various versions of this expansion have been much studied; see the monographs by Petrov , Bhattacharya and Ranga Rao , Hall . To state the expansion relevant to us, suppose additionally that for some . The logarithm of the characteristic function of can then be written in the form
where the numbers are the cumulants of . Note that and . The Chebyshev–Edgeworth–Cramér asymptotic expansion reads as follows, see Theorem 13 in Petrov [34, Ch. VII, p. 205]:
where is a degree polynomial in whose coefficients can be expressed through the cumulants . The first three terms in the expansion are given by
where denotes the -th “probabilist” Hermite polynomial:
The first few Hermite polynomials relevant to us are
The aim of the present work is to obtain asymptotic expansions of this type for the profiles of branching random walks.
1.3. Assumptions on the branching random walk
Consider a branching random walk on the integer lattice , as defined in Section 1.1. We will use the following standing assumptions. Recall that denotes the intensity measure of the BRW at time .
Assumption A: The branching random walk is non-degenerate, that is the set contains at least two elements.
Recall that denotes the number of particles in the BRW at time .
Assumption B: a.s. and .
Assumption A simply excludes a trivial case. As we already mentioned in Section 1.1, the first part of Assumption B could be removed, but then all results hold conditionally on non-extinction.
An important role will be played by the cumulant generating function of the intensity :
Assumption C: The function is finite in some open interval containing .
Let be the maximal open interval on which is finite. It follows that is strictly convex and infinitely differentiable on . A crucial role in the study of the branching random walk is played by the Biggins martingale:
By the martingale convergence theorem for non-negative martingales, we have for all ,
It is, however, possible that a.s. The range of where this does not happen was found by Biggins . Denote by the open interval on which :
Clearly, , so that this interval is non-empty. The endpoints of the intervals and are allowed to be infinite. We also need the following moment condition which supplements Assumption C.
Assumption D: There is a such that for every compact set ,
Under the above assumptions, it follows from Biggins [5, Theorem A] that for all ,
It is a crucial observation due to Biggins [8, 9] and Uchiyama  that the martingale convergence (11) can be extended to a complex neighborhood of the interval . First of all, it is clear that the function is defined as an analytic function of in a sufficiently small open set containing the interval . It follows that for every , the function is well-defined as an analytic function on . It has been shown in  and  that under Assumption D there exist an open set containing the interval and a random analytic function on such that
for every compact set .
Finally, as in the classical Chebyshev–Edgeworth–Cramér expansion, we need to assume that the lattice width associated with the support of the intensity measure equals .
Assumption E: There is no pair , such that is concentrated on the arithmetic progression .
Assumption E is not a restriction of generality because it can always be achieved by a suitable affine transformation; see Example 2.16 below.
where is the Dirac delta-measure at . In our simulations of the BRW we do not keep track of the individual locations as the number of particles grows exponentially. Instead we make use of the following rules to obtain : and
where is the number of particles at time and site that have descendants at time (located at sites and ). Given , , the random variables , , are conditionally independent and .
2.1. Asymptotic expansion of the profile
Consider a branching random walk on which satisfies Assumptions A–E. Recall that denotes the number of particles of the branching random walk that are located at site at time . Take some . Our limit theorems will be stated in terms of the “tilted” profile
As known from large deviations theory, the tilting operation allows to better access the properties of the profile for . Define the corresponding tilted cumulant
as the -th derivative of at , . In particular, we need the notation
Introduce the “standardized coordinate”
Note that for ease of reading we omit the dependence on in .
Our first theorem gives an asymptotic expansion of the tilted occupation number in powers of .
Consider a branching random walk satisfying Assumptions A–E. Fix and a compact interval . Then we have
where is a degree polynomial in . The coefficients of are random and can be expressed through
Stating the complete formula for requires introducing complicated notation and is therefore postponed to the proof in Section 3.3. Here, we provide only the first three terms of the expansion:
Theorem 2.1 remains valid if in the formula for we replace the derivatives of by the corresponding derivatives of . The reason is that w.p. , converges to exponentially fast together with all its derivatives (see Lemma 3.3), while the error term in Theorem 2.1 is of polynomial order only. This fact is used in our simulations where we replace by . It was shown in , see also , that in a suitable range of the asymptotic distribution of the appropriately normalized difference is a mixture of centered normals. A functional limit theorem for this difference was obtained in .
2.2. Local and global central limit theorems
Taking only the first term in the expansion given in Theorem 2.1 (meaning that ) we obtain
The most interesting case is when there is no tilting and we obtain the following local limit theorem for the BRW:
Roughly speaking, (24) says that the profile has an approximately Gaussian shape with mean , standard deviation , and the total mass of the profile is ; see the left part of Figure 1. In Figures 1 and 2 we use vertical bars for the discrete profiles and continuous lines for the approximating functions.
Let us now look at the next term in the expansion. Theorem 2.1 with and yields that
where the -term is uniform in . There are two correction terms on the right-hand side of (25). The term involving is the “shape correction” to the Gaussian profile. The same term appears in the expansion of the expected profile ; see Section 2.3. The term involving can be thought of as a random “location correction”. Indeed, this term says that in order to obtain a better approximation to the BRW profile we have to take a Gaussian profile centered at rather than at ; see the left and right parts of Figure 1.
Note that the random variable appearing above is the a.s. limit of the martingale
Since the total number of particles at time is , we can view as an estimate for the “shift” of the profile w.r.t. its “expectation” . This explains the appearance of as a “location correction” in a quite natural way. Similarly, the variable which will appear frequently below is the limit of the martingale
There is also a global central limit theorem for the BRW, originally known as the Harris conjecture [27, Chapter III, §16]. It states that for all ,
Various forms of (26) and (24) have been obtained in [37, 28, 1, 30, 7, 39, 9, 40, 24]. In fact, we can obtain a full asymptotic expansion in (26). To this end, one takes sums in Theorem 2.1 and uses the Euler–MacLaurin formula to approximate sums by integrals. We will record here only the first non-trivial term of this expansion.
Consider a branching random walk satisfying Assumptions A–E. Then,
where the -term is uniform over .
2.3. Comparing the profile and the expected profile
It is interesting to compare the expansion of stated in Theorem 2.1 with the expansion of . Since the intensity measure of the branching random walk at time is just the -fold convolution of , we can apply the classical expansion (5) to the expected profile . The proof of the following proposition is standard and will be given in Section 3.2. Formally, it can be obtained from Theorem 2.1 by taking the expectation and noting that for all which implies that all higher derivatives of have zero expectation.
Consider a branching random walk satisfying Assumptions A, C, E. Fix and a compact interval . Then we have
where is the same as in (5) but with replaced by . The -term is uniform in and .
We can considerably simplify the expansion given in Theorem 2.1 if instead of we consider . This normalization removes all terms involving and we obtain
Consider a branching random walk satisfying Assumptions A–E. Fix and a compact interval . Then,
where is the same as but without the term involving , that is
The -term in (29) is uniform in and .
2.4. Uniform expansions
For every point Theorem 2.1 yields a family of asymptotic expansions of parametrized by . It is natural to choose as the solution to because then . The information function is defined by
Consider a branching random walk satisfying Assumptions A–E. Fix . Then we have
where the -term is uniform over all for which exists and stays in a fixed compact set . Similarly,
where the -term is uniform over all such that exists and stays in a fixed compact set .
The expansion of is well known because it is a classical formula for the precise large deviations of sums of i.i.d. random variables; see [12, 2, 33]. Note that the terms with odd in (33) and (34) vanish; see Remark 3.6 below.
Taking in the above theorem yields the following known result [6, 9]; see Figure 2. In the context of random trees, results similar to Corollary 2.10 have been obtained by Chauvin et al. , Drmota et al. , Chauvin et al. [14, Theorem 3.1] and Sulzbach .
Equation (33) of Corollary 2.8 holds uniformly in as long as stays in the interval and remains bounded away from its ends. Here, both derivatives are understood as the corresponding one-sided limits and are allowed to be infinite. One may ask whether the whole range of the BRW is covered or just some part of it. A basic result due to Biggins , see also , describes the asymptotic range of the branching random walk:
If we additionally assume that is finite everywhere on (rather than on some interval) or that the support of the intensity measure is finite, then it is easy to prove (see, e.g., ) that and . In this case, Equation (33) of Corollary 2.8 covers the whole range of the “central order statistics” of the BRW. The “extremal” and “intermediate” order statistics at are not covered. However, in some (rather exotic) examples it is possible that is a strict subinterval of .
Example 2.11 (see ).
Consider a BRW for which for and otherwise. Here, is a parameter. Then is finite for and moreover, the left derivative is also finite, whereas for one has .
If is sufficiently large, then the BRW is supercritical and we have . This means that , whereas . Hence is strictly larger than .
2.5. Continuous-time branching random walks
All results of this paper apply to continuous-time branching random walks on which are defined as follows. At time one particle appears at position . After an exponential time with parameter , the particle disappears and at the same moment of time it is replaced by a random cluster of particles whose displacements w.r.t. the original particle are distributed according to some fixed point process on . The new-born particles behave in the same way as the original particle. All the random mechanisms involved are assumed to be independent. Denote the number of particles at time by and note that is a branching (Markov) process in continuous time; see [27, Chapter V]. Note that the law of the continuous-time BRW is uniquely determined by the following two parameters: the intensity and the law of the point process . The occupation number is defined as the number of particles located at site at time . If we restrict the time to integer values only, we obtain a discrete-time BRW called the “discrete skeleton” of the original continuous-time BRW. If Assumptions A–E of Section 1.3 hold for the discrete skeleton, then all the results of the present paper can be translated in an evident way to the continuous-time setting by replacing by . Note, however, that one has to be careful whenever the arithmetic properties of are involved; see Sections 2.6, 2.7. The proofs require only straightforward modifications.
2.6. Strong limit theorems for the occupation numbers
Recall that denotes the number of particles located at time at . Let us take an integer sequence which behaves in some regular way. We ask whether the random variables have a non-degenerate a.s. limit, after an appropriate affine normalization. The next proposition is known and follows immediately from (37) and (38).
Consider a branching random walk satisfying Assumptions A–E. Let be an integer sequence such that for some . Then we have
Next we ask whether we can obtain more refined limit theorems for the “centered” variables
This question is especially natural if and any particle in the BRW generates the same number of descendants. Then and hence the limit random variable provided by Proposition 2.12 is a.s. constant. The same phenomenon occurs in the setting of random trees, where the natural analogue of is equal to .
We consider an integer sequence which, for some , is represented in the form . The result will depend on the asymptotic behavior of . Recall that .
Consider a branching random walk satisfying Assumptions A–E. Let be an integer sequence such that
for some and some .