On the explosion of a class of continuous-state nonlinear branching processes

In this paper, we consider a class of generalized continuous-state branching processes obtained by Lamperti type time changes of spectrally positive L\'evy processes. When explosion occurs to such a process, we show that the process converges to infinity asymptotically along a deterministic curve, and identify the speed function of explosion. Using generalized scale functions for spectrally negative L\'evy process, we also find an expression of potential measure for such a process when explosion occurs. To show the main theorems, a new asymptotic result is proved for the scale function of spectrally positive L\'evy process.


Introduction
A continuous-state branching process is a nonnegative real-valued Markov process satisfying the additive branching property. It arises as time-space scaling limit of discrete Bienaymé-Galton-Watson processes. On the other hand, it can also be obtained by the Lamperti time change of a spectrally positive Lévy process stopped when hitting 0. We refer to Li (2012) and Chapter 12 of Kyprianou (2014) for nice introductions on continuous-state branching processes.
The classical Bienaymé-Galton-Watson branching processes had been generalized to those with nonadditive branching mechanism; see for example, Sevast'janov and Zubkov (1974), Klebaner (1984), Chen (2002) and Chen et al. (2008). In the same spirit, continuousstate branching processes with nonadditive branching have been proposed in recent years. In particular, the continuous-state polynomial branching process is introduced in Li (2019) as the unique nonnegative solution to a generalized version of the stochastic differential equation in Dawson and Li (2006), which can be identified as a continuous-state branching process with nonadditive, population dependent branching mechanism. The behaviors of extinction, explosion and coming down from infinity for such a process are discussed in Li (2019). A more general class of continuous-state branching processes is proposed in Foucart et al. (2019) via Lamperti type time change of stopped spectrally positive Lévy processes using rate functions R defined on (0, ∞), where the classical continuous-state branching process corresponds to the linear rate function of R(x) = x and the model in Li (2019) corresponds to the rate function of R(x) = x θ . The above continuous-state nonlinear branching processes are further generalized in  as solutions to more general versions of the Dawson-Li equation. For the continuous-state nonlinear branching processes, on one hand, the nonadditive branching mechanism allows richer boundary behaviors such as coming down from infinity; on the other hand, many classical techniques based on the additive branching property fail to work. Criteria for extinction, explosion and coming down from infinity are developed in Li (2019),  and Foucart et al. (2019) for the respective continuousstate nonlinear branching processes via a martingale approach and fluctuation theory for spectrally positive Lévy processes.
The speed of coming down from infinity for such processes is studied in Foucart et al. (2019) by analyzing the asymptotic behaviors of weighted occupation times for the associated spectrally positive Lévy process. Sufficient conditions are found under which the continuous-state nonlinear branching process comes down from infinity along a deterministic curve.
For the continuous-state nonlinear branching processes introduced in Foucart et al. (2019), explosion occurs when the process X has a positive drift and the rate function increases fast enough near infinity. In this paper we study the explosion behaviors for such a continuous-state branching process X. In particular, we identify the speed of explosion that is defined as the asymptotic of X(T + ∞ − t) as t → 0+ for the explosion time T + ∞ . We are not aware of any previous results on the speed of explosion for general Markov processes or for solutions to general stochastic differential equations with jumps. In addition, when explosion happens, using techniques from Li and Palmowski (2018) we also express the potential measure of the process X using the generalized scale functions for the associated spectrally negative Lévy process.
To find the speed of explosion, we treat separately two classes of rate functions, the so called slow regime of rate functions that are perturbations of power functions and the fast regime of rate functions that are perturbations of exponential functions. Our approach relies on analyzing the weighted occupation time for spectrally positive Lévy process. For process X with rate function from the slow regime, given the explosion occurs we can show that the normalization of random variable T + ∞ − T + x converges to 1 in the conditional probability, where T + x denotes the first upcrossing time of level x. Similarly, if the rate function belongs to the fast regime, under the conditional probability of explosion the random variable T + ∞ − T + x , after rescaling, converges in distribution to a random variable whose distribution can be specified using functionals of spectrally positive Lévy process. The convergence results in both cases lead to an asymptotics on the running maximum of the process near the explosion time. By comparing values of the associated spectrally positive Lévy process with its running maximum, we can show that for rate functions in both regimes the explosion occurs in an asymptotically deterministic fashion. In particular, in the fast regime the speed of explosion is asymptotically proportional to − log t as time t → 0+.
Some parts of our approach resemble those in Foucart et al. (2019) and in Bansaye et al. (2016) for studying the coming down from infinity behaviors of the respective processes. But an additional difficulty emerges in our work due to the overshoot when the nonlinear branching process first upcrosses a level x at time T + x . To overcome this difficulty, for the associated spectrally positive Lévy process we identify the Laplace transform of its stationary overshoot distribution, and we obtain a new asymptotic result on the corresponding scale function. For the case of fast regime, instead of showing the convergence of Laplace transform for the weighted occupation time as in Foucart et al. (2019), we apply the occupation density theorem to the weighted occupation time and the properties of regularly varying functions to show the almost sure convergence that eventually leads to the desired convergence in law.
The rest of the paper is arranged as follows. In Section 2 we first introduce some preliminary results on spectrally positive Lévy processes and the associated scale functions together with the exit problems and the weighted occupation times. The continuous-state nonlinear branching processes are also defined via the Lamperti type transforms in this section. The main results are presented in Section 3. Section 4 contains several examples. All the proofs are deferred to Section 5. Several intermediate results are also proved in Section 5.
Denoted by P x the probability law of ξ for ξ 0 = x, and write P when x = 0. We denote throughout this paper that p := Φ(0) and γ := E(ξ 1 ) = −ψ ′ (0). (1) If p > 0, then the process ξ is transient and goes to ∞ as t → ∞, and the following result holds.
The following limiting result on the resolvent density in (3) is useful in this paper, and we refer to Theorem I.21 of Bertoin (1996) for a similar result called "renewal theorem".
Remark 2. By change of measure, we obtain the following general result where a lighttailed condition on Π is required. For q ≥ 0, let α be the left-root of t → ψ(t) − q with −ψ ′ (α) < ∞ that is α < Φ(q) with ψ(α) = q, then α ≤ 0 and where −α is also known as the unique nonnegative root of the Cramér-Lundberg equation ψ(−t) = q in risk theory.
The proof is based on the following result used in Döring and Kyprianou (2015), see also Theorem 5.7 of Kyprianou (2014) and Bertoin et al. (1999). If γ ∈ (0, ∞) then for some non-degenerate weak limit ρ on [0, ∞), called the stationary overshoot distribution in Döring and Kyprianou (2015), which is characterized in the following lemma. .
Define the first passage times of X by The following identities on the first passage times follow immediately from the Lamperti type transform. For any x > 0 we have At the absorbing time η(τ − 0 ), process X becomes extinct at the finite time We first characterize the extinction and explosion behaviors for process X using integral tests. Note that similar results are obtained in Li (2019) Proposition 1. Extinction occurs for process X with a positive probability, that is , process X explodes with a positive probability, that is Note that the second statement in Proposition 1 is an immediate consequence of the following result from Döring and Kyprianou (2015): if E(ξ 1 ) ∈ (0, ∞) and f is a positive locally integrable function, then P( Even for γ = ∞, one can find from the proof for sufficiency in Döring and Kyprianou (2015) that the identity on the right hand side of (9) is still a sufficient condition for that on the left hand side to hold. Therefore, for the main results on explosion we always assume that R satisfies the following explosion condition H 0 : Two more conditions are needed for the main results in this paper: For which, although stronger than the explosion condition H 0 , allows to find explicit expressions for general ω for further analysis; c.f. Remarks 3 and 4.
Write ω(·) := 1/R(·). Process η(·) is the weighted occupation time process for the process ξ considered in Li and Palmowski (2018), where fluctuation theory of the ω-killed spectrally one-sided Lévy processes is studied, and where ω is positive and locally bounded on (0, ∞). The following ω-scale function W (ω) is useful and is defined as the unique locally bounded function satisfying for x, y > 0 where the second equation is proved in Lemma 4.2 of . The following result that extends the classical result of (3) can be derived from Theorem 2.2 of Li and Palmowski (2018) and we leave the proof to interested readers; see also Remark 4 in Li and Palmowski (2018).
In addition, the ω-resolvent measure of ξ is given by The following additional limiting properties of W (ω) are also presented in this section.
Proposition 3. For any x, y > 0, and Moreover, the function H (ω) defined above satisfies for y > 0, Moreover, under this condition, we can always express the function H (ω) in (14) in terms of sum of a sequence of integrals, that is, can also be expressed as a sum of integrals.

Main results
With the notations introduced in the previous sections, we are ready to present our main results whose proofs are deferred to Section 5. Denote by (Q t ) t≥0 the semigroup of X before absorption, i.e., Theorem 1. Suppose that the condition H 1 holds for function ω. For q ≥ 0, let W (qω) and H (qω) be the functions defined in (11) and (13) with respect to q · ω, respectively. We have for every is finite and can be obtained recursively by Remark 4. On the other hand, if the condition H 1 fails to hold, we can also find expressions for the resolvent measure of X on (0, ∞) by introducing functions other than H (qω) .
where H (ω) (y) is the function defined in Remark 3.
For the behavior of process X near time T + ∞ under the condition H 0 , we further define the probability law of X conditioned on explosion, that is, for any x > 0 and any x denotes the probability law of ξ conditioned to stay positive. We first have the following limiting result concerning the explosion time. Recall that function ϕ is defined in (10).
where ̺ is a random variable with its probability law ρ specified in (6), andξ is an independent copy of ξ that is also independent of ̺.
A sufficient condition for the condition H 2 is that function x → ω(log x) varies regularly with index −λ, which holds by applying Karamata's theorem, c.f. Theorem 1.5.11 and Proposition 1.5.9.b of Bingham et al. (1987). For the case λ > 0, it further follows from the theorem that In particular, if R(x)e λx varies regularly with index α for some λ ≥ 0 and α ∈ R, then x → ω(log x) varies regularly with index −λ. An interesting example for R is a power-like function satisfying the condition H 0 and for some constants α ≥ β with α − β < 1. Then H 2 holds with λ = 0. Actually, by the condition H 0 , we have β < −1, thus for some constant c > 0 and x large enough, We also have the following main result concerning the speed of explosion.

Examples
In this section, examples are provided to find the functions with explicit expressions, where the idea of contour integral from Jacobsen and Jensen (2007) is implemented in Examples 1 and 3. for s > 0, we have .
Taking Laplace transforms on both sides of (11) gives for s > p, For R(x) = e λx for some λ > 0, which is also an example considered in Li and Palmowski (2018), the generalized scale function W (ω) can be expressed by Bessel function for the case of linear Brownian motion.

Plugging into the second equation for H (ω) in Proposition 3 gives
Noticing that s + t > p in the third line of the above expression, ν(·) is thus a measure on [p, ∞) satisfying the equation with ν({p}) = 1. In particular, if we further assume that µ(dt) = h(t) dt for some measurable h ≥ 0 on (0, ∞), then ν({p}) = 1 and ν(ds) = k(s) ds for some locally integrable function k(·) defined on (p, ∞) satisfying equation which can be expressed as a summand of integrals. Consequently, for y > 0 H (ω) (y) = e −py + ∞ p e −ys k(s) ds.
Example 2. For the asymptotic results in Theorem 2 and 3, • if R(x) = (c + x) θ for θ > 1, then Example 3. If ω(z) = z −1 for z > 0, then X reduces to the linear branching process. Condition H 0 fails to hold, we have from Remark 3 that H (ω) satisfies Similar to the first example, we look for a solution of the form h( for some kernel function k(·). Plugging it into the equation above and using that fact Comparing the two sides of equation above, with (s + t) > p, we have k(t) = 0 for t < p and for t > p k(t)ψ(t) = t p k(s) ds, which gives for y > 0 where δ > 0 is a constant such that H (ω) (1) = 1, which coincides with Theorem 1 of Duhalde et al. (2014) without immigration, and where we need the fact that

Proofs
This section is dedicated to the proofs of the main results. Lemmas 1, 2 and 3 for SPLP are of independent interest and are proved first. They will be applied in the proofs of the main theorems. For the proofs of Theorems 1, 2 and 3, since processes X and ξ are connected via the Lamperti type time transform, we focus ourself on the study of ξ and its weighted occupation times.

Proofs of Lemmas 1, 2 and 3
Our proof of Lemma 1 is based on the Itô excursion theory, where the compensation formula and the exponential formula for Poisson point process are applied; c.f. Chapter O of Bertoin (1996). Here we use the standard notions in the fluctuation theory of Lévy process from Bertoin (1996). Let χ :=ξ − ξ be the Lévy process reflected at its running maximum, whereξ t = sup s≤t ξ s is the running maximum of ξ. Let l be a local time process of χ at 0 and l −1 be its right inverse. Since lim t→∞ ξ(t) = ∞, χ is a recurrent Markov process.
Proof of Lemma 1. Fix 1 > ε > 0. Let t > 0 be such that s = l(t) > 0 andξ(t) − ξ(t) > ε ·ξ(t). Then t ∈ (l −1 (s−), l −1 (s)),ǭ s > ε ·ξ(t) andξ(t) =ξ(l −1 s− ). Sinceξ(t) → ∞ as t → ∞, we only have to look at those s such thatǭ s > 1 in the following, that is, On the other hand, since χ is absent of positive jumps, the law ofǭ givenǭ > 1 under n(·) is identical to the law of | inf t<τ + 0 ξ(t)| under P −1 , that is, for y > 1 n(ǭ > y|ǭ > 1) = P −1 inf Notice that similar to Lemma VI.2 of Bertoin (1996), the heights of excursion process χ are independent of ξ (l −1 s− ), s > 0 . Therefore, we have that conditional on (ξ(l −1 s− ), s > 0), Since W (x) = e px W p (x) ≥ W p (1)e px for x ≥ 1, andξ(l −1 s− ) =ξ(l −1 s ) = ξ(l −1 s ) for almost every s > 0 P-almost surely, by the right-continuity of ξ and the definition of l −1 , we have from Fubini's theorem that where for the equality above we use the fact that ξ(l −1 s ) is a subordinator with Laplace exponent −ψ(β) p−β . Therefore, P almost surely, we have and finish the proof of the first assertion. For the second limit, from the previous result and the identity inf it is sufficient to check that 1 ξ t inf s>t (ξ s − ξ t ) converge to 0 in probability. From the Markov property and the fact thatξ t → ∞ as t → ∞, the desired conclusion follows.
Proof of Lemma 3. The proof is based on the observation that ξ and its Ladder height process have the same overshoot when first up-crossing a level. Thus, the stationary overshoot is identical in law to the limit of the overshoot of the ladder process. More specifically, consider a Ladder height process of ξ, which is a subordinator with a version of Laplace exponent κ(β) = ψ(β) β−p , c.f. Theorem VII.4 of Bertoin (1996). Let δ and ν(dz) be the associated drift parameter and jump measure, respectively. Then we have from Theorem 5.7 of Kyprianou (2014 where µ = κ ′ (0) = ψ ′ (0) −p ∈ (0, ∞), which finishes the proof. We are now ready to prove Lemma 2 by applying Lemma 3.
Proof of Lemma 2. For x > 0, define the hitting time of ξ by Since process X is absent of negative jumps, then τ {x} = τ + x + τ − x • θ τ + x , and by (2) and Lemma 3 we have It is proved in Lemma 3.1 of Li and Zhou (2019) that for x, y ∈ (c, b). Letting b → ∞, it follows from (4) that On the other hand, applying the strong Markov property, we have where we use the fact that ξ is spatially homogenous. Therefore, for x, y > 0, Applying (22) the proof is completed.

Proof of Theorem 1
Proposition 1 is proved by making use of Proposition 3 on the asymptotic of function (x, y) → W (ω) (x, y). Recall that p, γ are constants defined in (1), respectively, and the Lamperti type identities between the first passage times for X and ξ in (8).
On the other hand, since W (ω) (x, y) ≥ W (x − y), we have from (12) that, which proves the "only if" part in the assertions. H (ω) (0+) < ∞ under H 1 also follows. This completes the proof.
We only need to prove the result of Proposition 1 on extinction.
Proof of Proposition 1. Letting c → 0+ in Proposition 2, we have for every b > x > 0, By Proposition 3, On the other hand, if 1 0+ ω(z)W p (z) dz < ∞, we also have for every q > 0, where W (qω) is the generalized scale function with respect to qω(·). By the scale function identity, for every x, y, q, r > 0, (2019), we have that q → W (qω) (x, y) is increasing. It is not hard to find that W (qω) (x, y) → W (x − y) as q → 0+, which shows that

see Lemma 4.3 of Li and Zhou
and the second assertion is proved.
An application of Proposition 1 shows that, under the condition H 0 , the moment function m n defined in Theorem 1 can also be written in terms of ξ as The following proposition on m n is frequently used in our proofs. A result similar to the following Proposition 4 can be found in Lemma 8.11.1 of Bingham et al. (1987), and here we provide a proof for readers' convenience. U(x, dy)ω(y)m n−1 (y) for x ≥ 0 and n ≥ 1.
Using the idea similar to Proposition 4 in the following, we have for x > 0, Then Lemma 2 shows that if γ ∈ (0, ∞), We refer to Li and Zhou (2018) for more detailed discussions on the related results. Notice that the 0-1 law in the first part of Proposition 1 can also be proved by showing that We are now ready to prove Theorem 1. Notice that ω in Theorem 1 is assumed to satisfy H 1 , which fulfills the condition of Proposition 3, and under which for any x > 0 as shown in Proposition 1.
Proof of Theorem 1. Let f ≥ 0 be a continuous function on (0, ∞) with compact support. Noticing that dη(t) = ω(ξ t ) dt, we have by change of variable that The q-resolvent of X follows from Proposition 2 and 3, by letting b go to ∞ and c go to 0+.
The moment generating function of T + ∞ is obtained from Proposition 4. From (23), we know that the density of U is bounded by Therefore, with m 0 (x) = m 0 (x) = 1 − e −px ≤ 1, we have n for all n ≥ 1.
Since Carleman's condition on the moments is satisfied, the distribution of ∞} under the condition H 1 is uniquely determined by its moments (m n ) n≥0 , and the desired conclusion follows.

Proofs of Theorems 2 and 3
For a positive function f satisfying the condition H 0 , its tail integral is defined as ∞ x f (y) dy, and we need the following asymptotic results for its integrals, c.f. Theorem 1.6.5 of Bingham et al. (1987). We write as usual Denote by and as x → ∞, Proof of Proposition 5. Put g(u) := ∞ log u f (z) dz. Then u λ g(u) is slowly varying because the tail integral of f satisfies the condition H 2 .
With Lemma 2 and Proposition 5 above, we first have the following asymptotic result.
Lemma 4. Suppose that γ ∈ (0, ∞) and f is an integrable function on (0, ∞) with Proof of Lemma 4. Notice that W (y − x) = 0 for y < x, we have from (3) that for x > 0, Since W (y)e −py = W p (y) ↑ Φ ′ (0) < ∞ as y → ∞, we have from Proposition 5 that On the other hand, for every ε > 0, applying Lemma 2, for some k > 0, (23), then for x > k where we used the assumption that the function of tail integral x → ∞ log x f (y) dy is slowly varying. This finishes the proof.
We are now ready to prove part A of Theorem 2. After investigating the asymptotic behaviors of the first two moments of η(∞) at infinity, the 1st and the 2nd moments of J(τ + x ) are estimated by the Markov property of ξ. Proof of part A of Theorem 2. In the following moment argument we assume that under which we have m 2 (x) < ∞ by Theorem 1. In case the above assumption does not hold, we can first prove the convergence result under Q x · T + ∞ < T − c = ∞ = P x · τ − c = ∞ for c > 0, and then let c → 0+ to obtain the desired result.
We first claim that, as x → ∞, and in addition, for h ∼ g for some decreasing function g such that g(log x) varies slowly at ∞. Noting that given (29) and (30), since J(τ + x ) = η(∞) • θ τ + x , we have for x > 1, which implies that and the desired weak convergence follows.
To prove (29), we apply Lemma 4 to function f 1 (y) = ω(y)(1 − e −py ). It is not hard to see that Thus, by the assumptions of Theorem 2, f 1 fulfills the condition in Lemma 4. It follows from (27) that Then, we take f 2 (y) = ω(y)m 1 (y). From the result above, for any ε ∈ (0, 1) let k 1 > 0 satisfy ( It follows that for x > k 1 , and f 2 satisfies the condition of Lemma 4. Applying (28) and Lemma 4 we have To prove (30), let k 2 > 0 satisfy where ρ is the stationary overshoot distribution defined in (6). Then for x > k 2 , where the monotonicity of g is applied to the first and the last inequality. Thus, Lastly, applying the strong Markov property for ξ we further have This finishes the proof.
Remark 9. In the proof of case A, we have However, with the presence of positive jumps, m 1 (x) and E ↑ x η(∞) may fail to be monotone in x in general.
For the proof of case B of Theorem 2, we make use of the local time for the process ξ, c.f. Chapter V of Bertoin (1996) for more detailed discussion. Given a SPLP ξ, its local time is well-defined and defined as the density of occupation measure by, P-a.s., L(y, t) := lim ε→0+ 1 2ε t 0 1(|ξ s − y| < ε) ds, for y ∈ R, t > 0. and the following occupation density formula holds for all measurable bounded function For the proof of part B we need the following lemmas on the regularly varying functions and the local time.
Lemma 5. Let ω be the function in the case B in Theorem 2. Let f ≥ 0 be a measurable function locally integrable such that the set {x ∈ R, f (x) > 0} is bounded from below, R e −λy f (y) dy < ∞ and R e −2αy f 2 (y) dy < ∞ for some 2α ∈ (0, λ).
Remark 10. Lemma 5 appears similar to the Abelian theorem, c.f. Theorem 4.1.3 of Bingham et al. (1987) where ω(log ·) is assumed to be regularly varying, and also similar to Theorem 1.7.5 in Bingham et al. (1987), where conditions related to slow decrease is imposed. The condition here can be replaced by other, possibly weaker, conditions. For example, if f has bounded variation and is bounded, right-continuous, and {x ∈ R, f (x) > 0} is bounded from below, an application of the uniform converge theorem could give the same result.
Proof of Lemma 5. Since the set {x ∈ R, f (x) > 0} is bounded from below, it is sufficient to prove (31) for f vanishing on (−∞, 0), and we only focus on integrals on (0, ∞). Firstly, it is straightforward that (31) holds for f (y) = 1(y > c) for every c > 0. By the assumption in case B, for some K, M > 0, we have ∞ x ω 2 (y) dy ≤ M · ϕ 2 (x) for all x > K.
Applying Fubini's theorem, for x > K we have by applying Karamata's theorem to the last identity since ϕ 2 (log x) is regular varying with index −2λ. Thus, the Hölder inequality yields where the second term is dominated by some constant. The desired result then follows from the monotone class theorem for functions and localization.
Lemma 6 is proved following the argument used in Theorem V.1 of Bertoin (1996), where Plancherel's theorem is applied.
Thus, e −αξt , e −2αξt and g(y) are all integrable. The Fourier transform of g gives for every u ∈ R, Under the new measure P (α) the above quantity equals to where Ψ α (s) = t −1 log E (α) e isξt = −ψ α (−is) is the characteristic exponent of ξ under P (α) . Noticing that −ψ(α) > 0, we have where Theorem II.16 in Bertoin (1996) is applied. The proof is finished by applying Plancherel's theorem. Now, we are ready to prove the result of part B.
Proof of part B of Theorem 2. Let f and g be bounded continuous and nonnegative functions.
Applying the strong Markov property of ξ at τ + x , we have Denote byξ an independent copy of ξ with probability law ofP and define for z > 0 LetL(y, t) be the local time ofξ at level y and time t. Sinceξ is transient, applying Theorem I.20 of Bertoin (1996), Lemma 6 and the fact that inf t>0ξ t < ∞, one can check that, for 2α < λ ∧ p,L(y, ∞) fulfills the conditions of Lemma 5P-a.s.. Therefore, On the other hand, by the uniform converge theorem for ϕ, see Theorem 1.5.2 of Bingham et al. (1987), we have ϕ(x + y)/ϕ(x) → e −λy as x → ∞, uniformly for y ∈ [0, ∞).
Applying (6) and the facts above to (32), we complete the proof.
For the proof of Theorem 3, we follow the same idea from Foucart et al. (2019) and Bansaye et al. (2016).
Proof of Theorem 3. The theorem is proved by first claiming that under P ↑ 1 , The desired conclusion then follows from Lemma 1.
Then we further have from β(t) > k 3 that the following inequalities hold.
Thus, we always have lim inf y→∞ ϕ(y) ϕ(hy) = ∞ for all h > 1 in this case. Since J(τ + x )/ϕ(x) converges in law to a random variable on (0, ∞), there exist M > 1 and k 5 > 0 such that for all x > k 5 , which can be compared with (34). The same argument as in the previous case can be applied to prove (33). Applying the result of (C) in Proposition 5, we finish the proof.