Tail asymptotics of maximums on trees in the critical case

We consider solutions to the maximum recursion on weighted branching trees given by$$X\,{\buildrel d\over=}\,\bigvee_{i=1}^{N}{A_iX_i}\vee B,$$where $N$ is a random natural number, $B$ and $\{A_i\}_{i\in\mathbb{N}}$ are random positive numbers and $X_i$ are independent copies of $X$, also independent of $N$, $B$, $\{A_i\}_{i\in\mathbb{N}}$. Properties of solutions to this equation are governed mainly by the function $m(s)=\mathbb{E}\big[\sum_{i=1}^NA_i^s\big]$. Recently, Jelenkovi\'c and Olvera-Cravioto proved, assuming e.g. $m(s)<1$ for some $s$, that the asymptotic behavior of the endogenous solution $R$ to the above equation is power-law, i.e.$$\mathbb{P}[R>t]\sim Ct^{-\alpha}$$for some $\alpha>0$ and $C>0$. In this paper we assume $m(s)\ge 1$ for all $s$ and prove analogous results.


Introduction
In this paper we study the maximum recursion on trees where N is a random natural number, B and {A i } i∈N are random positive numbers and X i are independent copies of X, which are independent also of N , B, {A i } i∈N . Our main goal is to describe asymptotic properties of the endogenous solution to (1.1) (in the sense of [2]). Observe that for N = 1 a.s. equation (1.1) is just the random extremal equation considered by Goldie [10]. Moreover, in this case, taking logarithm of both sides of the equation, we obtain the classical Lindley's equation related to the reflected random walk. In general, equation (1.1) is called high-order Lindley equation and is a useful tool in studying branching random walks. We refer to Aldous, Bandyopadhyay [2] and Jelenković, Olvera-Cravioto [12] for a more complete bibliography on the subject and description of a class of other related stochastic equations.
We begin with explaining how to construct an endogenous solution to equation (1.1). Let T = k≥0 N k be an infinite Ulam-Harris tree, where N 0 = {∅}. For v = (i 1 , ..., i n ) we define the length |v| = n and by vi we denote the vertex (i 1 , i 2 , ..., i n , i). We write u < v if u is a proper prefix of v, i.e. u = (i 1 , .., i k ) for some k < n. Moreover we write u ≤ v if u < v or u = v. Now we take {(N (v), B(v), A 1 (v), A 2 (v), ...)} v∈T a family of i.i.d. copies of (N, B, A 1 , A 2 , ...) indexed by the vertices of T . Since equation (1.1) depends only on N first values of A i 's, we can assume that A i (v) = 0 for every v ∈ T and i > N (v). For v ∈ T we also define a random variable L(∅) = 1 and L(vi) = L(v)A i (v). We define One can easily deduce that if the maximum above is finite almost surely then the random variable R satisfies (1.1).
The properties of R are governed by the function Jelenković and Olvera-Cravioto [12] recently studied existence and asymptotic properties of R in the case when the equation m(s) = 1 has two solutions α < β and α < 1. They proved, under a number of further assumptions, that R has a power-law distribution of order β, i.e.
for some C > 0. In this paper we consider the critical case, when the equation m(s) = 1 has exactly one solution α and then m (α) = 0. Our main result is the following.
Then the solution R of (1.1) given by (1.2) is well defined and for some constant C > 0.
Equation (1.1) is similar to the linear stochastic equation (called also the smoothing transform) where X i are independent copies of X, which are also independent of a given sequence of nonnegative random variables (N, B, A 1 , A 2 , . . .). This equation was considered in a number of papers, see e.g. [5,6,8,11,13,14]. In these papers existence and some further properties, including asymptotic behavior, of solutions to (1.4) were considered. In particular, the techniques described there can be applied in our settings to study equation (1.1). The details will be given in next sections.
Our model is closely related to the branching random walks, which can be defined as follows. An initial ancestor is located at the origin. Its N children, the first generation, are placed in R according to the distribution of the point process Θ (1.1). Each of the particles produces its own children which are positioned (with respect to their parent) according to the same distribution of Θ and they form the second generation. And so on. The resulting system is called a branching random walk.
Notice that if B = 1, then M = − log R describes the global minimum of the branching random walk, that is the leftmost position of all the particles in the system. Thus our Theorem 1.3 implies that lim t→∞ e αt P[M < −t] = C and C > 0. The same result, however under much weaker hypotheses and using different techniques based on the spinal decomposition, was recently proved by Madaule [15].

Upper and lower estimates of R
The aim of this section is to provide upper and lower estimates for R defined in (1.2), that is to prove under assumptions given in Theorem 1.3: for some constant C > 0 and large t. Notice that the upper estimate implies in particular that R given by (1.2) is finite a.s. Before giving proofs we recall a useful tool, called the many-to-one formula. First, let us introduce a random variable Y with distribution given by for any positive Borel function f . By (A2) the right hand side of the above defines a probability measure. Moreover (A3), (A5) and (A6) imply that the random variable Y is centered, nonarithmetic and has finite exponential moments, i.e.
E e ±δY < ∞, for some δ > 0. Now, let {Y i } be a sequence of independent copies of Y defined by (2.1) and let S n be the sequence of their partial sums, S n = n k=1 Y k . For a fixed n and any test function f : R n → R, the following many-to-one formula holds see e.g. Theorem 1.1 in Shi [17].
Proof. We apply here estimates proved by Buraczewski and Kolesko [6] in a slightly different settings. The authors considered tails of fixed points of the inhomogeneous smoothing transform, that is solutions to the stochastic equation (1.4). 'Inhomogeneous' means that the B-term do not reduce to 0. All solutions to this equation were described by Alsmeyer and Meiners [3]. In particular the endogenous solution is given by assuming that the above series is finite a.s. It is known that if α < 1, then under hypotheses of Theorem 1.3, the random variable R is finite a.s. (see [6], Proposition 2.1) and moreover for some C > 0, see [6], Theorem 1.1.
We will also need that for any strictly positive constant δ, the function is bounded, see [6], Lemma 2.2.
Since the results in [6] are proved only for α < 1 we split the proof into two cases. First we assume that α < 1 and we apply directly the results stated above. Next we reduce the general situation to this case. Case 1. Assume that α < 1 and choose δ < 1 − α.
For the upper bound simply note that and the desired estimates on right hands side come from (2.4).
Lower estimates are more difficult to prove and usually require some tricky arguments. Below we present a proof based on the result by Buraczewski and Kolesko [6], who studied the linear stochastic equation (1.4). However, for reader's convenience in Appendix A we present a complete proof, borrowed from Aïdékon [1], based on the second moment method. To simplify the arguments we write it for a very particular case when N is constant and B = 1 a.s. (for a proof when N is random see Madaule [15]).
Here, we proceed in two steps.
For large M > 0, whose precise value will be specified below, we write Using the many-to-one formula (2.2) we obtain for the bounded function W defined in (2.5). The above implies On the other hand, by (2.4), we have the lower estimate for some C 2 > 0 and sufficiently large t. Therefore, taking M big enough, we can find C > 0 such that Step (ii). We now consider general B. For this purpose we define R = v∈T L(v). For any We apply here similar arguments as in [6] (Proposition 2.1). Putting f (x 1 , ..., x n ) = 1(x 1 ≥ − log t, ..., x n−1 ≥ − log t, x n < − log t) in the many-to-one formula (2.2) we obtain From the discussion in the first step there is C > 0 such that and the right hand side of the above is properly bounded by arguments given in the first case to the random variable R.

Asymptotics of R
To prove the precise asymptotic of R we adopt to our settings the arguments presented by Durrett and Liggett [8] (see also [4,6]), where the problem was reduced to study asymptotic properties of solutions to a Poisson equation. The details are as follows. We define φ(x) = P[R > x] and D(x) = e αx φ(e x ). Our aim is to prove and Y is the random variable defined in (2.1).
Proof. We start with finding a recursive formula for φ. For this purpose we denote by µ the distribution of (N, B, A 1 , A 2 , ...) and write By the definition of Y and the many-to-one formula (2.2) we obtain We now show some properties of function G. Proof. We decompose G as a sum of two functions Notice that f 1 is positive. Indeed, it is sufficient to apply the following inequality, valid for 0 ≤ u i ≤ v i ≤ 1 (see [8], p. 283): We first show that f 1 (x) tends to 0. For this purpose recall an easy inequality u ≤ e −(1−u) , valid for any real u and write where F (u) = e −u − 1 + u. Observe that the function F is increasing on [0, ∞), therefore by Lemma 2.3.
Note that H(u) = F (u) u is bounded and tends to 0 as u → 0. These observations and the dominated convergence theorem give us lim sup To bound f 2 we use Chebyshev's inequality with α < β < α + δ Lemma 3.5. Assume (A6). There is > 0 such that e |x| G(x) ∈ L 1 (R).
Proof. Once more we use decomposition (3.4). Take any 0 < < min(α/2, δ). Let us first consider function e |x| f 1 (x). To show integrability on (∞, 0], recall that f 1 is positive and use Chebyshev's inequality with To deal with the right tail we use the fact that F is an increasing function on [0, ∞) and F (u) ≤ u. Choose β such that 3 4 α < β < α. Again using Chebyshev's inequality we write where the last equality holds by Fubini's theorem. We now use a substitution u = E[R β ]e −βx N i=1 A β i and again by Fubini's theorem we obtain To show that the above is finite, we write To estimate the first integral we only need to bound integrand near zero. To obtains this, it is sufficient to observe that lim u→∞ F (u) u 2 = 1 2 and our assumptions on β and imply α+ β < 2. For the second integral notice that F (u) ≤ u for any u ≥ 0, therefore For the expectation factor we use the inequality valid, under assumption E[N 1+δ ] < ∞ for any sequence of positive random variables {X i }, r > 1 and p ∈ (1, r(1+δ) r+δ ) (see [6], Lemma 3.4). Plugging r = α+ α and X i = A rβ i we obtain Integrability of e |x| f 2 (x) comes easily from Chebyshev's inequality. Indeed, once more take Our aim is to deduce some asymptotic properties of the function D, knowing that it is a solution to the Poisson equation (3.2) for some well behaved function G. A typical argument reduces the problem to the key renewal theorem, which in turn requires G to be directly Riemann integrable (see [9] for the precise definition). For this purpose we need to prove some local properties of G and this cannot be done directly. To avoid this problem we proceed as in Goldie's paper [10] and for an integrable function f we define the smoothing operatoȓ

Moreover,f is always continuous function and if f is integrable, then
f is directly Riemann integrable (dRi) (see Goldie [10], Lemma 9.2).
Smoothing both sides of equation (3.2) we obtain Notice thatG has now better properties than G. Below we will describe asymptotic behavior of D and finally deduce the main result. Define The following lemma holds.
Lemma 3.7. For a given function f suppose that there is a positive such that e |x| f (x) ∈ L 1 (R) .
Proof. We only show the first property, since the second one is a consequence of the integration by parts. One has Recall that Rf (s)ds = R f (s)ds. Similarly to the above, for x ≥ 0 one has hence f is dRi as a function bounded by dRi functions. Proof. Take K big enough to ensure that for any x ≥ K we have D(x) > 0. We define a family {h x } x≥K of continuous functions by Such a family and its properties was already considered e.g. in [6,8,13], with a slightly different definition of function D, though. Notice that the family {h x } x≥K is uniformly bounded and equicontinuous. Indeed, boundedness is straightforward since by (2.6) and Lemma 2.3 one has D(x) ≤ C and D(x) > 1 C > 0 for x sufficiently large and hence the same holds forD. To obtain equicontinuity, for h > 0, we write and the very last expression does not depend on x (it does not depend on y as well, so we obtain even uniform equicontinuity).
In view of the Arzelà-Ascoli theorem, the family {h x } x≥K is relatively compact in the topology induced by the uniform norm on compact sets. We conclude that there is a sequence x n → ∞ and h such that h xn → h uniformly.
Using (3.6) we writeD which we divide byD(x), to obtain By Lemma 2.3 we have D(x) > C and alsoD(x) > 0 for any sufficiently large x. Thus, by Lemma 3.3 we see that (3.11) lim Passing with x n → ∞ in (3.10), using (3.11) and dominated convergence theorem yield As a consequence of Choquet-Deny theorem (see e.g. [16], Theorem 1.3 in Chapter 5), any positive Y -harmonic function is constant, thus we see that h(y) = h(0) = 1. It implies that h is the unique accumulation point and (3.9) holds. Now we are ready to prove main results.
Proof of Theorem 1.3. Let Y i be an i.i.d sequence with distribution defined by (2.1). Denote S n = n i=1 Y i (we assume S 0 = 0). Define also stopping times L = inf{n ≥ 0 : S n < 0} and T k = inf{n > T k−1 : S n ≥ S T k−1 }, T 0 = 0. Equation (3.6) implies that for any x the process forms a martingale with respect to the natural filtration generated by Y i . By the optional stopping theorem or in another words (3.12) E By the duality principle (see [9], Chap. XII) |G(x + S T i )| and the right hand side series is finite sinceG is dRi. From Proposition 3.8 we concludȇ D(x) ≤ Ce |x| and since E e S L < ∞ (see [9], Chap. XII) we can pass with n to infinity in (3.12) to get Again by the duality principle we have The key renewal theorem now yields .
Integrating (3.13) one has The same argument as before allows us to pass with x to the infinity under the integral sign, hence we have Now we divide (3.14) by x and pass to the limit, which implies The upper bound onD(x) show that above limit equals 0, hence This procedure may be repeated using integrating (3.13) with different limits, i.e. we write or equivalently We pass with x to the infinity and again by renewal theorem obtain .
Finally, observe that where the last equality is a consequence of Lemma 9.3 in [10]. Proof. The arguments presented below base on [1]. For v ∈ T we define τ t (v) = inf{1 ≤ k ≤ |v| : L(v k ) > t} and for S n as in the proof of Lemma 2.3 denote τ t = inf{n : S n < − log t}. Moreover denote Z t = {v ∈ T : τ t (v) = |v|} and let N t = v∈T 1 (v ∈ Z t ) = n |v|=n 1 (v ∈ Z t ) be the number of elements of Z t . Note that an event {N t > 0} is contained in { v∈T L(v) > t} hence it is sufficient to prove the lower bound for P[N t > 0]. For this purpose we use the so-called second moment method, i.e. by the Cauchy-Schwarz inequality one has .
For x ≥ 1 we call P x the distribution such that P x [L(∅) = x] = 1 (hence P = P 1 ) and E x the corresponding expectation. First, we will show that there is a positive constant C such that for sufficiently large t the following holds We define for any s ≥ 1 the undershoot L s = − log s − S τs . Using the many-to-one formula (2.2) we obtain Obviously, for any s one has E e −αLs ≤ 1, since L s > 0. For the lower bound note that for any positive M E e −αLs ≥ e −αM P [L s ≤ M ] . Choosing appropriately large M , there is a constant C such that P [L s ≤ M ] > C > 0, uniformly with respect to s (see [7], Proposition 4.2) which shows (A.2).
Let us define N t (n) = |v|=n 1(v ∈ Z t ) and N t (≤ n) = k≤n |v|=k 1(v ∈ Z t ). We have To estimate E [1(v ∈ Z t )N t (≤ n)] we decompose N t (≤ n) along the vertex v, i.e. we write where N v k t is the number of descendants u of v k at the level at most n which are not descendants of v k+1 and such that u is an element of Z t . Denote by F n the σ-algebra generated by the tree up to the level n. The above implies that