Maximums on Trees
We study the minimal/endogenous solution to the maximum recursion on weighted branching trees given by
where is a random vector with , and nonnegative weights , and is a sequence of i.i.d. copies of independent of ; denotes equality in distribution. Furthermore, when this recursion can be transformed into its additive equivalent, which corresponds to the maximum of a branching random walk and is also known as a high-order Lindley equation. We show that, under natural conditions, the asymptotic behavior of is power-law, i.e., , for some and . This has direct implications for the tail behavior of other well known branching recursions.
Keywords: High-order Lindley equation, stochastic fixed-point equations, weighted branching processes, branching random walk, power law distributions, large deviations, Cramér-Lundberg approximation, random difference equations, maximum recursion
2000 MSC: 60H25, 60J80, 60F10, 60K05
where is a real-valued random vector with , , and is a sequence of i.i.d. random variables independent of having the same distribution as . Equation (1.1) has applications in a wide variety of fields, including the analysis of divide and conquer algorithms [19, 18], e.g. Quicksort ; the analysis of the PageRank algorithm [20, 13]; and kinetic gas theory . Our work in [14, 15] shows that the so-called endogenous solution, as termed in , of (1.1), under the natural main root condition with positive derivative for some , has the power tail behavior,
where . The main tool used in deriving this result was a generalization of Goldie’s Implicit Renewal Theorem  to weighted branching trees.
Motivated by a different set of applications, we study in this paper the maximum recursion on trees given by
where is a random vector with , nonnegative weights , and , and is a sequence of i.i.d. random variables independent of having the same distribution as . Here and throughout the paper we use and to denote the maximum and the minimum, respectively, of and . We point out that by taking the logarithm in (1.2) when a.s., we obtain the additive equivalent
where , , , and the are i.i.d. copies of , independent of . Note that for and , (1.3) reduces to the classical Lindley’s equation, satisfied by the reflected random walk; and when , the recursion corresponds to a random walk reflected on a random barrier. In general, the preceding additive equation has been studied in the literature of branching random walks since, when , represents the range of the branching random walk (see , §4.2). Recursion (1.3) was termed “high-order Lindley equation” and studied in the context of queues with synchronization in . Unlike the classical Lindley equation, it was shown in  that (1.3) can have multiple solutions. A more complete analysis of the existence and the characterization of the entire family of solutions was carried out in  (e.g., see Theorem 1 in ). In addition, it can be shown that the study of (1.3) arises in the context of today’s massively parallel computing. More specifically, consider a job that is split into smaller pieces which are sent randomly to different processors, and these pieces need to communicate, i.e., need to be synchronized, in order to complete their processing. In the limiting regime as the number of processors goes to infinity, a similar reasoning as in  can be used to show that (1.3) represents the delay for job completion in this massively parallel system. In addition to these applications, a better understanding of (1.2) immediately leads to important insights to other max-plus branching recursions. More precisely, for the case of nonnegative weights, (1.2) is a natural lower bound for many other recursions on trees , e.g., for the same set of weights, elementary arguments show that in (1.1) is stochastically larger than the solution in (1.2).
For all of the reasons described above, we study in this paper the tail behavior of the minimal/endogenous solution to the maximum recursion in (1.2) (or (1.3)). As shown in , equation (1.3) can have multiple solutions. It is worth noting that the minimal/endogenous solution we study here is also central in characterizing all other solutions, as stated in Theorem 1 of  (there is used to denote the minimal/endogenous solution). Furthermore, we would like to point out that under iterations of the fixed-point equation (1.2) (or (1.3)), the minimal/endogenous solution is the primary limiting value, unless one starts with very specific initial distributions (see Theorem 1(ii) in ); we will discuss this in more detail in Section 3. In addition, we emphasize that the tail characterization of the other (non minimal solutions) was given in , but the tail behavior of the minimal one was left open.
Our first main result, stated in Theorem 3.4, describes the tail behavior of the minimal/endogenous solution to the maximum recursion (1.2) (or (1.3)). In this regard, the application of the Implicit Renewal Theorem on Trees (see Theorem 3.4 ), under the natural conditions and for some , readily gives that
where . However, the main difficulty in establishing the power-law behavior lies in proving that . Unlike in the linear case, it is not clear that this constant should be positive at all, since at first glance the expression which determines in Theorem 3.4 of Section 3,
appears just as likely to be negative. Note also that a direct application of a ladder heights argument gives the positivity of the constant for the classical non-branching case (), see Theorem 5.2 in . However, for the branching case no ladder heights equivalent is available. Hence, our first main contribution lies in a new sample-path construction showing that under no additional assumptions (besides those needed for the application of Theorem 3.4 in ). Observe that in the additive case of equation (1.3), our result yields the exponential asymptotics , which is the generalization of the well known Cramér-Lundberg approximation. The latter is widely used in insurance risk theory and queueing.
Furthermore, as an immediate corollary one obtains the strict positivity of in the linear case with nonnegative . In this setting, the work in  used a straightforward convexity argument to show that for , but the corresponding question for was left open. The strict positivity of for was recently resolved in  as part of the more general real-valued case, but under additional assumptions that include . Note that in the additive equation (1.3), this extra moment assumption corresponds to the finiteness of exponential moments of the . Since the new results on the maximum hold without such additional assumptions, Theorem 3.4 fully completes the prior work for nonnegative . In addition, as already mentioned, the maximum is a natural lower bound for other max-plus recursions, and therefore this result can potentially be used to prove the power-tail asymptotics of the endogenous solutions to other recursions, e.g., the discounted tree sums considered in .
We now go back to the linear recursion (1.1) with real-valued weights , which has recently been considered in [4, 15, 2] (see also  for the multivariate case). The characterization of all the solutions of (1.1) when is real-valued, and a.s. was given in . The work in  establishes the Implicit Renewal Theorem on Trees for the real-valued case and shows that, under the usual conditions and , the endogenous solution to (1.1) has a power tail behavior of the form , . In that paper the strict positivity of in its full generality remained open. It was this open problem that motivated the work in , where it was shown, using complex analysis and analytical functions, that under the additional assumptions a.s., and .
In this paper, we revisit the problem of the strict positivity of for the general real-valued case using our result on the maximum equation (1.2) (with nonnegative weights ), under no additional assumptions on the vector besides those needed for Theorem 3.4 in . However, we do require that does not reduce to a constant given . Our main set of arguments is based on Lévy’s symmetrization approach. We would like to mention that although the proof of Theorem 4.1 in  (the part that establishes the positivity of for the case ) also relies on symmetrization, our proof is completely different. While in the one-dimensional case it was enough to center the weights around their median, in the branching case we need complete symmetry. More precisely, the proof consists in first showing the positivity of the constant for symmetric trees, and then extending it to the general case through a coupling argument; see Corollary 4.4. In general, as previously stated, we expect that Theorem 3.4 for the maximum, coupled with the Implicit Renewal Theorem on Trees (Theorem 3.4 in ), can be used to derive the exact power law asymptotics of the solutions to other branching recursions .
The paper is organized as follows. Section 2 includes a brief description of the weighted branching process. Section 3 contains our first main result about the asymptotic behavior of the minimal/endogenous solution to the maximum recursion (1.2), including the strict positivity of . Section 4 presents our proof of the positivity of the constant for the general mixed-sign linear recursion (1.1).
2 Model description
We use the model from  for defining a weighted branching tree. First we construct a random tree . We use the notation to denote the root node of , and , , to denote the set of all individuals in the th generation of , . Let be the number of individuals in the th generation, that is, , where denotes the cardinality of a set; in particular, .
Next, let be the set of positive integers and let be the set of all finite sequences , where by convention contains the null sequence . To ease the exposition, for a sequence we write , provided , and to denote the index truncation at level , . Also, for we simply use the notation , that is, without the parenthesis. Similarly, for we will use to denote the index concatenation operation, if , then .
We iteratively construct the tree as follows. Let be the number of individuals born to the root node , , and let be i.i.d. copies of . Define now
It follows that the number of individuals in the th generation, , satisfies the branching recursion
Now, we construct the weighted branching tree as follows. Let be a sequence of i.i.d. copies of . determines the number of nodes in the first generation of according to (2.1), and each node in the first generation is then assigned its corresponding vector from the i.i.d. sequence defined above. In general, for , to each node we assign its corresponding from the sequence and construct . For each node in we also define the weight via the recursion
where is the weight of the root node. Note that the weight is equal to the product of all the weights along the branch leading to node , as depicted in Figure 1.
3 The maximum recursion:
In this section, we study the maximum fixed-point equation given by
where is a random vector with , and , and is a sequence of i.i.d. random variables independent of having the same distribution as . As already mentioned, the additive version of (3.1), given in (1.3), was termed “high-order Lindley equation” and studied in the context of queues with synchronization in . The full characterization of its multiple solutions was given in . More recently, a related recursion where , , and the are real valued deterministic constants, has been analyzed in . The more closely related case of and being random was studied earlier in . For this and other max-plus equations appearing in a variety of applications see the survey by .
Using standard arguments, we start by constructing an endogenous solution to (3.1) on a tree and then we show that this solution is finite a.s. and unique under iterations provided that the initial values and the weights satisfy appropriate moment conditions.
Following the notation of Section 2, define the process
on the weighted branching tree . Recall that the convention is that denotes the random vector corresponding to the root node. Next, define the process according to
It is not hard to see that satisfies the recursion
where are independent copies of corresponding to the tree starting with individual in the first generation and ending on the th generation. One can also verify that
where is a sequence of i.i.d. random variables independent of and having the same distribution as .
We now define the random variable according to
Note that is monotone increasing sample-pathwise, so is well defined. Also, by monotonicity of and (3.3), we obtain that solves
where are i.i.d. copies of , independent of , see also Section 2 in . Clearly this implies that , as defined by (3.4), is a solution in distribution to (3.1). However, this solution might be . Next, we establish in the following lemma the finiteness of the moments of , and in particular that a.s.; its proof uses standard contraction arguments but is included for completeness; e.g. see Theorem 6 (i) in . Conditions under which is infinite a.s. can be found in Corollary 4 in .
Assume that and for some . Then, for all , and in particular, a.s. Moreover, if , , where stands for convergence in norm.
Proof. Note that
It follows that
That whenever follows from noting that and the same arguments used above.
Although this paper focuses only on the solution defined by (3.4), it is important to mention that equation (3.1) can have multiple solutions, as the work in  describes. The solution receives the name “endogenous” since it is constructed explicitly from the weighted branching tree, and the name “minimal” since it is the stochastically smallest solution, in the sense that any other solution to (3.1) satisfies for all . For the case when and there exists a unique such that and (referred to as the “regular case”), Theorem 1 (ii) and (iii) of  characterizes the entire family of solutions to (3.1). Moreover, under some additional technical conditions, all other solutions to (3.1) are given in terms of ( in ) and the limit of the martingale . To better understand the nature of these other solutions, as well as to highlight the importance of the endogenous/minimal solution , we will next define the process that is obtained from iterating equation (3.1) starting from an initial value .
and are i.i.d. copies of an initial value , independent of the entire weighted tree . is referred to as the “terminal” value in  (, ) since it corresponds to the value of the leaves in the weighted branching tree with finitely many generations. It follows from (3.3) and (3.5) that
where are independent copies of corresponding to the tree starting with individual in the first generation and ending on the th generation, and is the set of all nodes in the th generation that are descendants of individual in the first generation. Moreover, are i.i.d. copies of , and thus, is equal in distribution to the process obtained by iterating (3.1) with an initial condition . This process can be shown to converge in distribution to for any initial condition satisfying the following moment condition (see also Theorem 9 in ).
Suppose and for some , then
with . Furthermore, under these assumptions, the distribution of is the unique solution with finite -moment to recursion (3.1).
Proof. The result will follow from Slutsky’s Theorem (see Theorem 25.4, p. 332 in ) once we show that . To this end, recall that is the same as if we substitute the by the . Then, for every we have that
Since by assumption the right-hand side converges to zero as , then . Furthermore, by Lemma 3.1. Clearly, under the assumptions, the distribution of represents the unique solution to (3.1), since any other possible solution with finite -moment would have to converge to the same limit.
(a) Lemma 3.2 establishes a certain type of uniqueness of the solution to (3.1), in the sense that is the only possible limit for the iterative process for any initial value possessing finite moment. It is therefore to be expected that all other solutions to the maximum recursion must arise from violating this assumption. (b) Theorem 1 (ii) of  states that in the regular case (see the comments after Lemma 3.1), if and (), then , where
Moreover, solves (3.1) provided a.s. and .
Now we are ready to state the main result of this section, which characterizes the asymptotic behavior of .
(a) The condition is only needed to obtain the result about the negative tail. The result about the positive tail only requires . (b) The equivalent result for the lattice case can be obtained by using the corresponding Implicit Renewal Theorem on Trees in . (c) Corollary 5 in  provides upper bounds for the tail behavior of any finite solution to the maximum equation (3.1).
Proof. The first part of the proof about the right tail, , will follow from an application of the Implicit Renewal Theorem on Trees, Theorem 3.4 in , once we verify the finiteness of
Also, since , then
Combining these two observations gives that (3.6) is finite, and by Theorem 3.4 (a) in  we obtain the result with the integral representation of . To derive the second expression for follow the same steps used at the end of the proof of Theorem 4.1 in .
For the negative tail, , simply note that
where in the last step we used Markov’s inequality. Since , then as , proving the result.
The rest of the proof is devoted to showing that the constant if and only if . Note that if a.s. then and therefore , so it only remains to show that whenever . Hence, assume from now on that .
The main idea of the proof is to construct a minorizing random variable for for which we can directly estimate the expectation appearing in the numerator of . We start by fixing and choosing and such that , , and ; define . Note that such always exists under the assumptions of the theorem, since when we have that for any ,
and when we have that for any ,
Now let be nonnegative i.i.d. random variables, independent of , having the same distribution as , where satisfies
(e.g., take to have density ). For each define the random variable
where is such that
and note that for any ,
We now apply the first part of this theorem to the new random variable to obtain
as . The positivity of will then follow once we show
We start by writing as
To analyze note that