On the Maximal Displacement of Near-critical Branching Random Walks

We consider a branching random walk on $\mathbb{Z}$ started by $n$ particles at the origin, where each particle disperses according to a mean-zero random walk with bounded support and reproduces with mean number of offspring $1+\theta/n$. For $t\geq 0$, we study $M_{nt}$, the rightmost position reached by the branching random walk up to generation $[nt]$. Under certain moment assumptions on the branching law, we prove that $M_{nt}/\sqrt{n}$ converges weakly to the rightmost support point of the local time of the limiting super-Brownian motion. The convergence result establishes a sharp exponential decay of the tail distribution of $M_{nt}$. We also confirm that when $\theta>0$, the support of the branching random walk grows in a linear speed that is identical to that of the limiting super-Brownian motion which was studied by Pinsky in [28]. The rightmost position over all generations, $M:=\sup_t M_{nt}$, is also shown to converge weakly to that of the limiting super-Brownian motion, whose tail is found to decay like a Gumbel distribution when $\theta<0$.


Introduction and Main Results
The study of extreme values of branching particle systems has attracted a considerable amount of attention during the last few decades. Early works on the tail behavior of branching Brownian motion trace back to Sawyer and Fleischman [12] and Lalley and Sellke [19]. During the same time period the strong law of large numbers for the maxima of branching random walk was studied by Hammersley [13], Kingman [15], Biggins [5] and Bramson [7].
In [20] the authors studied in the critical case the tail distribution of the rightmost position over all generations, namely, (1.1) It was proved in [20] that under some moment assumptions, Here α is a constant that depends on the standard deviations of the jump distribution and the offspring distribution. The asymptotics (1.2) implies that when the critical branching random walk starts from n particles at the origin, the tail distribution of M/ √ n, that is P M ≥ √ nx , decays at rate O(1/x 2 ) for large values of n (see Corollary 2 in [20]). We will show in this paper (see Corollary 1.3(i)) that the corresponding tail distribution of M nt / √ n decays with a rate of exp(−c(t)x 2 ). The difference between the two convergence rates implies that the heavy-tailedness of M in the critical case is due to particles that survive more than O(n) generations.
In the supercritical case, very precise estimates on the tail distribution of the radius of the support of a super-Brownian motion were established by Pinsky in [27] and [28]. Let B r (0) be the ball of radius r centered at the origin. It was proved in [28] (see equation (6) therein) that for a super-Brownian motion X = { X t } t≥0 with drift θ > 0, diffusion coefficient D 2 > 0 and branching coefficient σ 2 , one has P θ δ 0 L t (B r (0) c ) = 0 = e −ur(t,0) , where P θ δ 0 stands for the probability distribution of X with drift θ, X 0 = δ 0 , and L t = t 0 X s ds is the local time process associated with X. As to u r (·), for any r > 0, u r (t, x) is the minimal positive solution to the following nonlinear PDE:  One important implication of Theorem 1 of [27] is the growth rate of the support of X. It was proved in Theorem 1 of [28] that the large time growth rate is linear with rateγ := (2θD 2 ) −1/2 . Specifically, one has lim t→∞ P θ δ 0 L γt (B t (0) c ) = 0 = 1, if γ <γ.
The above convergence in probability is strengthened to be almost sure convergence in [16].
The aforementioned growth rate result brings up the second aim of this paper, namely, to derive the growth rate of near-critical branching random walks. As we mentioned earlier, results on the limiting measure-valued processes in most cases are not precise enough for the research of discrete particle systems. The motivation for this work comes from the study of population and epidemic models, where sharp bounds on the local time are key elements in the proofs of phase transitions. For example, in [18], a phase transition for the spatial measure-valued susceptible-infected-recovered (SIR) epidemic models was established. A key ingredient in the proof is the growth rate of the support of the local time (see the discussion in Section 1.2 of [18]).
Before we state our main results, we define more carefully the branching random walk that we study.
The model: For any fixed n ∈ N and a constant η ∈ R, P η n stands for the probability distribution of a discrete time branching random walk X = (X k (x)) k≥0,x∈Z initiated by n particles at the origin and with the following properties. In each generation, particles first jump (independently from each other) according a distribution with a finite range, F RW = {a k } k∈[−R,R] , which has mean 0 and variance σ 2 R , and then each particle branches independently according to an offspring distribution F η B = {p η i } i≥0 , which has p η 0 > 0, expectation 1 + η, variance σ 2 (η) and third moment γ(η). The σ(η) and γ(η) satisfy that for some δ > 0, Notation. We often use the abbreviated notation P n = P 0 Observe that under this model, particles jump first and then reproduce, just as in [20]. This does not change the limiting tail behavior of the maximal displacement as explained in Remark 3 therein and noting the Taylor expansion of function Q given by (2.3) below.
We study the tail behavior of the maximal displacement of X up to generation [nt] for t ≥ 0, that is, Let θ ∈ R and n ∈ N. Our first main result establishes the weak convergence of M nt / √ n under P θ/n n , to the rightmost point in the support of the local time L t of the limiting super-Brownian motion.
Remark 1.2. The convergence in Theorem 1.1 is new even when θ = 0, i.e., the critical case. When θ = 0 and F θ/n BR is Poisson or binomial, the convergence of u θ/n n· ( √ n·) follows from the convergence in the critical case and the convergence of the likelihood ratio between the near-critical and critical systems. In general, such an argument may fail because there may be no likelihood ratio not to mention its convergence.
We now describe a corollary to Theorem 1.1. Consider the following PDE (1.6) The existence and uniqueness of positive solutions to (1.6) will be proved in Section 4. We will also need the following nonlinear ODE. From Proposition 2 in [27] we have that for each ρ ∈ (0, √ 2), there exists a unique positive increasing solution f = f ρ ∈ C 2 ([0, ∞)) to the equation (1.7) Moreover, one has In the following corollary we derive sharp bounds on u θ/n n· ( √ n·).
where the following bounds on φ hold: (i) when θ ≥ 0, for every ε > 0, there exists c ε > 0 and M ε > 0 such that for all x > M ε , In the next main result we derive the large time growth rate of the support of X θ/n : (ii) for any γ >γ, there exists N (γ) > 0 such that for all n > N (γ), P θ/n n X nγt (( √ nt, ∞)) > 0 for all t large enough | X survives = 1.
Remark 1.5. In [28] the author established the linear growth of the support of supercritical super-Brownian motions by observing that the bounds on (φ(t, x)) given in Corollary 1.3 implies that (1.9) It follows from Corollary 1.3 that for any ε > 0, there exist T 0 > 0 such that for any t > T 0 , we can find an N 0 = N 0 (t) such that for all n ≥ N 0 , (1.10) This result, however, is not enough to establish the linear growth of the support of branching random walk because we would need (1.10) to hold for all t large enough. Such a uniform convergence seems difficult to prove given the discontinuity of the limit in (1.9) as a function of γ. We prove the linear growth result using another argument.
Finally, analogous to [20], we derive the tail distribution of the maximal displacement over all generations, namely, M = sup k M k . Theorem 1.6. When θ = 0, for every x 0 > 0, uniformly over x ∈ [x 0 , ∞), we have lim (1.12) Moreover, where θ + = max(θ, 0). Remark 1.7. When θ = 0, the convergence (1.11) and PDE (1.12) are also true; see Corollary 2 and Proposition 23 (and its proof ) in [20]. The tail behavior of M , however, is completely different according to whether θ = 0 or not. When θ = 0, by Corollary 2 in [20], the tail distribution of M/ √ n decays at a rate of 1/x 2 . In contrast, in the sub-near-critical case (or the super-near-critical case and conditioned on extinction), the tail distribution of M/ √ n is similar to a Gumbel distribution.
Remark 1.8. In [20], the convergence (1.11) was established by first proving the convergence of a complicated object lim x→∞ w ∞ (x + y/w ∞ (x))/w ∞ (x), where w ∞ (x) = P 1 (M > x) for any x; see equation (23) therein for the precise statement. In this paper, we prove the convergence of P θ/n n (M ≥ √ nx) directly.
Organization of the paper: The rest of this paper is organized as follows. In Section 2 we establish a discrete Feynman-Kac formula for the tail distribution of the maximal displacement, which will be used in Section 3 to establish the tightness of (nw for each k and x. In Section 4 we identify the limit as a unique solution to a nonlinear parabolic PDE with infinite boundary condition, based on which we establish Theorem 1.1 and Corollary 1.3. In Section 5 we prove Theorem 1.4, and in Section 6 we prove Theorem 1.6.

A Discrete Feynman-Kac Formula
Recall that F θ/n B stands for an offspring distribution with mean 1+ θ/n. Denote by f θ/n the probability generating function of F θ/n B . Define (2.1) We also define w θ/n k (y) = P θ/n The derivation of the Feynman-Kac formula uses ideas from Section 2.2 in [20].
The following lemma (see Lemma 4.1 in [24]) gives a convolution equation for w θ/n k (·) based on Q θ/n (·) and the random walk distribution Recall that F θ/n B has a finite third moment, hence by the Taylor expansion of Q θ/n (·) at s = 0 we have and Then Lemma 2.1 can be rewritten as the following, which is more convenient for our purpose.
We will also need the following result on the boundedness and monotonicity of H θ/n (see Lemma 4.3 in [24]): We denote by {W k } n≥0 a random walk on Z with the following law: in other words, {W k } k≥0 is a reflection of W , the random walk associated with our branching system. We use P x and E x to denote the probability measure and expectation of {W k } k≥0 with W 0 = x, and omit the subscript when x = 0 (and when there is no confusion). Moreover, to improve readability, we often Define the stopping times where we use the convention that for any k ≤ 0, k j=1 = 0 and k j=1 = 1. In particular, Y x,y =τ x ∧ τ y . The following lemma gives a discrete Feynman-Kac formula.
Proof. Note thatτ y,z ≤τ for every 0 ≤ y < z ≤ ∞. The conclusions follow by taking the stopping timesτ y,z ∧ (m − k) andτ y,z ∧ m and applying the optional stopping theorem to the martingale {Y

Tightness
In this section we prove that the function sequence nw We start with some exponential bounds on the distribution of the maximum of W .
Proof. (i) is a special case of as Corollary A.2.7 in [22].
In the following lemma we compute the probability that X (under P θ/n n ) dies out as n → ∞. Proof. Recall that f θ/n is the probability generating function of F θ/n B . Let q n be the smallest non-negative root of the equation f θ/n (q) = q. By Theorem 2 in Chapter I.A.5 of [3], P θ/n 1 X dies out = q n , and therefore P θ/n n X dies out = q n n . (3.1) Therefore, Before stating our next lemma, we recall the duality principle which states that a supercritical branching process conditional on extinction has the same distribution as its dual subcritical process; see, for example, Theorem 3 in Chapter I.D.12 in [3]. Specifically, let Z = {Z n } n≥0 be a Galton-Watson process with Z 0 = 1 and an offspring distribution F B = {p i } i≥0 that has mean m > 1 and p 0 > 0. Define to be the event of extinction, and let q = P (B) ∈ (0, 1). Then the duality principle says that the process {Z n } n≥0 conditional on event B has the same distribution as a subcritical Galton-Watson branching process { Z n } n≥0 with Z 0 = 1 and where f denotes the probability generating function of F B . In the following lemma we derive an exponential bound on sup n≥1 nw Proof. (i) We shall only prove the result when θ ≥ 0. Recall that M = sup n≥0 M n stands for the maximum displacement over all generations. From Theorem 1 in [20], which applies to critical branching random walks, we have Therefore, if we denote by B the event of extinction of the branching random walk X, then by the duality principle we get Moreover, by (3.3), there exist positive constants c 1 and c 2 independent of n and x such that for all x ≥ δ, n ≥ 0, and we get (i).
(ii) By Lemma 2.4(ii) and (2.7), we have Because w θ/n 0 (y) = 0 for y ≥ 1, we have for all x > 0 and n ≥ 1, Recall that W has a range R. From the monotonicity of w θ/n nt (x) in x and part (i), we get that there exists a positive constant C such that where we used Lemma 3.1(i) in the last inequality. The conclusion follows.
The following lemmas are key ingredients in proving the tightness of of (nw θ/n nt ( √ nx)).
Proof. We only need to prove for the case when y > x because otherwise the LHS equals 0. We will also only prove for the case when θ ≥ 0; the case when θ < 0 can be proved similarly.
Note that For I 1 (n, y − x, t), we have Because e x is a Lipschitz function, we get About term I 2 (n, y − x), using Lemma 3.1(ii) we get I 2 (n, y − x) ≤ e θt P √ nx τ √ ny > n(y − x)) ≤ C(T )e −1/(y−x) , and the result follows.
Lemma 3.5. For any 0 < t 0 < T < ∞ and x 0 > 0, there exist N 0 > 0 and C = C(t 0 , T, x 0 ) > 0 such that for all n ≥ N 0 and t ∈ [t 0 , T ] we have Proof. Let x 0 > 0, x 0 < x < y < ∞ and t ≥ t 0 . It follows from Lemma 2.4(ii) with m = nt and (2.7) that θ/n nt (Wτ√ nx/2 ∧(nt) )1 {τ √ ny >(nt)∧τ √ nx/2 } =: I 1 (n, x, y, t) + I 2 (n, x, y, t), where in the second inequality we used the monotonicity of w θ/n nt (x) in x. We first handle I 1 (n, x, y, t). By Lemma 3.4, for all 0 ≤ y − x ≤ 1, which, by Lemma 3.3(i), is bounded by C(y − x)/n. Next we bound I 2 (n, x, y, t). Using the monotonicity of w θ/n nt (x) in x again we have where in the last inequality we used Lemma 3.1(ii) and the gambler's ruin result. By Lemma 3.3(i) again, we get The conclusion follows.
Lemma 3.6. For any 0 < t 0 < T < ∞ and x 0 > 0, there exist δ > 0, N 0 > 0 and C(t 0 , T, x 0 ) > 0 such that for all n > N 0 , t ∈ [t 0 , T ], nt ≤ m ≤ n(t + δ) and x 0 < x < ∞, we have Proof. From monotonicity it is enough to prove the lemma when m = [n(t + δ)]. Let x 0 < x < ∞ and let ξ > 0 be a small number to be chosen later. From the bound (2.7) and Lemma 2.4(i) with y = x − ξ, z = ∞ and k = [nt] we get (3.9) Note that m − [nt] ≤ δn + 1. Using the monotonicity of w θ/n k (x) in x we get that By Lemma 3.5 and Lemma 3.3(i), if ξ and δ are small enough then (3.10) As to J 2 (m, n, x), using the monotonicity of w θ/n k (x) in k and x, the finite range of W and noting that m − [nt] ≤ δn + 1, we have By Lemma 3.1(i), there exist C 2 > 0 and β > 0 such that for all x ≥ x 0 , By choosing δ to be small enough and using Lemma 3.3(i) we get J 2 (m, n, x) ≤ ε n . (3.11) The conclusion follows.

Scaling Limit
In this section we prove that the limiting function φ from Proposition 3.8 satisfies a nonlinear PDE with an infinite boundary condition at x = 0. Then we prove uniqueness of solutions to the PDE. For any continuous process {Y t } t≥0 , definē The following proposition gives a Feynman-Kac representation of φ(t, x).
Proposition 4.1. Let (φ(t, x)) t>0,x>0 be any sub-sequential limiting function from Proposition 3.8. Then for any x 0 > 0 and for all x > x 0 , t > 0, we have Proof. For any x > x 0 and t > 0, by Lemma 2.4(ii), we have Using (2.6) and that log(
Therefore, by our assumption on the convergence of nw Furthermore, from (1.4) and (2.4) we get σ 2 (θ/n) → σ 2 . By the same reasoning as for (4.4) we get Finally, we have , as n → ∞.
Plugging the above limits into (4.3), together with bounded convergence theorem we get the conclusion.
Corollary 4.2. Suppose that (φ(t, x)) t>0,x>0 is a sub-sequential limiting function from Proposition 3.8. Then it satisfies the following PDE: Proof. This can be proved similarly to Theorem 8.2.1(a) in Section 8.2 of [25].
In the following lemma we derive the initial and boundary conditions of φ from Proposition 3.8.  Proof. (i) It is sufficient to show that for any x > 0 and ε > 0, there exists t 0 > 0 such that This follows from the bounds (3.5) and (3.6).
(ii) We need to prove that for any t > 0, Note that for any x ≤ t we have where we used Lemma 3.1(ii) in the last inequality. To bound the probability in the last term, note that by the weak convergence of X to X we have  Proof. Without loss of generality, we set σ R = σ = 1 in (1.6). Define  Assume that (4.8) has two positive solutions u 1 and u 2 such that u 1 (t 0 , x 0 ) = u 2 (t 0 , x 0 ) for some x 0 > 1 and t 0 ∈ (0, T ). Without loss of generality, assume that u 1 (t 0 , x 0 ) < u 2 (t 0 , x 0 ). Then by continuity, for c > 1 that is close enough to 1 we have Then We have lim x↓1 f (t, x) = ∞, for all t > 0. It follows that the infimum of f (·) = f (·; c) must be attained at some point (t * , x * ) = ((t * (c), x * (c))) ∈ (1, M ) × (0, T ], and we have However, by (4.8) and (4.10) we have (4.14) When θ ≥ 0, the last term is negative due to that f (t * , x * ) = u 1 (t * , x * ) − v 2 (t * , x * ) < 0. This contradicts (4.13), and we conclude the proof. Consider now the case when θ < 0. Take any sequence (c n ) such that (4.9) holds for all c n and c n ↓ 1. Note that by continuity we can choose M independent of c n such that for all n large enough, sup x>M, 0<t≤T (u 1 (t, x) + v 2 (t, x; c n )) < (u 2 (t 0 , x 0 ) − u 1 (t 0 , x 0 ))/4.
It follows that (t o , x o ) must be also inside (1, M ) × (0, T ], and by (4.14), for all n large enough, and we again get contradiction with (4.13).
We are now ready to prove Theorem 1.1. To show the convergence of u θ/n · (·), by the independence among branching trees generated by different initial particles, we have Therefore, To finish the proof of Theorem 1.1, in the below we analyze the exit probability of the limiting super-Brownian motion X with drift θ, diffusion coefficient σ 2 R and branching coefficient σ 2 .
For any r > 0, define the following sequence of functions {ψ r,m (x)} ∞ r,m=1 ∈ C ∞ (R): Let v r,m be the solution to with the initial condition v r,m (0, x) ≡ 0. By the same argument as for the convergence (5) in [28], v r,m is increasing in m, and we can define v r (t, x) = lim m→∞ v r,m (t, x), for every x ∈ R, t > 0. (4.17) Moreover, by repeating the argument for equation (6) in [28], with δ x as the initial measure, we have We now analyze v r (t, x). By (4.16), (4.17) and the monotone convergence theorem and using further (4.18), we get that that v r is a weak solution to We want to strengthen the conclusion to be that v r is a classical solution to (4.19). To see this, recall that u r is the minimal positive solution to (1.3). From Theorem A and Proposition A in [28] we have and u r satisfies that for every ε > 0, there exists c ε > 0 and M ε > 0 such that for all r > M ε , for all t ≥ 0 and x ∈ [0, r), (4.21) Noting that v r (t, x) ≤ u r (t, x), using the bound (4.21) and regularity of weak solutions to parabolic PDE (see Chapter 7.1.3 of [11]), we see that v r is a positive classical solution to (4.19).
Next, we prove Corollary 1.3 Proof of Corollary 1.3. First note that (1.8) has been derived in the proof of Theorem 1.1.
Next we prove (i). By Theorem 1.1 and (4.20) we have The bound in (i) then follows from (4.21).
To prove (ii), note that w(t, Since lim x↓0 w(t, x) < ∞ for any t > 0, we can apply the same argument as in the proof of Proposition 4.4 to get that w(t, x) ≤ φ(t, x) for all x > 0 and t ≥ 0. The conclusion follows.

Proof of Theorem 1.4
We first prove Part (i).
Proof of Theorem 1.4 (i). Recall that {L x k } k≥0 stands for the local time process of X. By Corollary 12.2.7 in [22], there exists C > 0 such that for all v > 0 and k ∈ N, Because (L · (·)) is an integer-valued process, by the Borel-Cantelli lemma, we get that P θ/n n (L n(k+1) (B c √ nvk ) = 0 for all k large enough) = 1.
Note that for all t > 0, The conclusion follows.
Next we prove Part (ii). The proof uses ideas from the proof of Theorem 2.1 in [16]. Recall that X k (x) is the number of particles at site x at generation k. For any γ ∈ R, define and Then {W γ k } k≥0 is a martingale; see Chapter VI.4 of [3] or Theorem 1 in [15]. Because {W γ k } k≥0 is nonnegative, the limit W γ := lim k→∞ W β k almost surely exists, and by Fatou's lemma, E(W γ ) ≤ 1. The following lemma characterizes when E(W γ ) = 1 and provides the key ingredient in proving Part (ii) of Theorem 1.4.
Lemma 5.1. If |β| < 2θ/σ 2 R , then for all sufficiently large n, Proof. The proof is based on Lemma 5 in [6]. From our assumptions on the step distribution and branching law, by Lemma 5 in [6], we need to verify that if |β| < 2θ/σ 2 R , then for all sufficiently large n, By the Taylor expansion and using the fact that z∈Z za z = 0 we get It follows that and we verify (5.3).
We are now ready to prove Theorem 1.4 (ii).
Proof of Theorem 1.4 (ii). By a similar argument to the proof of Lemma 2.2 in Chapter 2.1 of [29], for any γ ∈ R, P θ/n 1 (W γ > 0) is either 0 or equal to the survival probability of X . Therefore by Lemma 5.1, if |β| < 2θ/σ 2 R , then for all sufficiently large n, P θ/n Now pick any 0 < β < 2θ/σ 2 R , and let ε ∈ (0, β) be an arbitrarily small number. Note that It follows from the Taylor expansion in the proof of Lemma 5.1 that for all n large enough, converges almost surely, the last term converges to 0 as t → ∞, and we obtain that It follows from (5.7) and (5.6) that P θ/n 1 (X nt (( √ n(β − ε)σ 2 R t, ∞)) > 0 for all t large enough | X survives) = 1.
Because β can be arbitrarily close to 2θ/σ 2 R and ε can be arbitrarily small, we get the desired conclusion.
6 Proof of Theorem 1.6 In this section we study M , the maximum displacement throughout the whole process. Denote w θ/n ∞ (y) = P θ/n 1 (M ≥ y), y ≥ 0. In the following lemma we show that for any θ ∈ R and x 0 > 0, the function sequence (nw Proof. By duality between the supercritical and subcritical branching processes (see discussion before Lemma 3.3) and equation ( In the following proposition we describe the limiting function ψ(·). For any θ > 0, define By duality between the supercritical and subcritical branching processes we have (n w θ/n ∞ ( √ nx)) converges to ( ψ(x)), which, by (3.3), satisfies the following simple relationship with ψ: ψ(x) = ψ(x) + 2θ/σ 2 , for all x > 0.
Recall thatτ Y x was defined in (4.1) for any continuous process {Y t } t≥0 and x ∈ R. We further define for x < y ≤ ∞, Proposition 6.2. The limiting function ψ(·) in Lemma 6.1 satisfies that Proof. The fact that ψ satisfies the boundary conditions in (6.3) follows from (6.1), (6.2) and Lemma 4.3. It remains to show that ψ satisfies the ODE in (6.3). By (6.2) again, it suffices to show the case when θ < 0.
We only give the sketch of proof because it uses similar techniques to proving the convergence of (nw θ/n nt ( √ nx)). A simple modification of the proof of Lemma 4.5 in [24] yields the following discrete Feynman-Kac formula for w It follows by a similar argument to the proofs of Proposition 4.1 and Corollary 4.2 that the limiting function ψ(·) satisfies that for all 0 < x 0 < y ≤ ∞, x 0 ,y 0 ψ(σ R B s )ds · ψ(σ R Bτσ R B x 0 ,y ) . (6.4) Based on this expression, we want to show that ψ satisfies σ 2 R 2 ∂ 2 ψ ∂x 2 = −θψ + σ 2 2 ψ 2 , x 0 < x < y.
By Theorem 3.1 in Chapter 3 of [10], we only need to verify that ψ is Lipschitz.
To do so, we take y = ∞ in (6.4), which yields By the strong Markov property, for any δ ≥ 0 and x > x 0 , It follows that ψ(x) is decreasing in x, and we have Therefore ψ is Lipschitz and we complete the proof.
In the rest of this section we assume that θ > 0.
We are interested in the asymptotic behavior of ψ(x) as x → ∞. By (6.2), it suffices to study the asymptotic behavior of ψ(x). For notational ease, denote  In the following lemma we establish uniqueness of solutions to (6.6) (note that the uniqueness does not follow from Theorem 3.1 in [10], which applies to the case with bounded domain and given boundary condition). Lemma 6.3. For any a > 0 and b > 0, there exists at most one solution to (6.6).
We are now ready to prove Theorem 1.6.
Plugging a and b in (6.5) yields x , as x → ∞.