Asymptotic behaviour near extinction of continuous state branching processes

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1 Asymptotic behaviour near extinction of continuous state branching processes G. Berzunza and J.C. Pardo August 2, 203 Abstract In this note, we study the asymptotic behaviour near extinction of sub- critical continuous state branching processes. In particular, we establish an analogue of Khintchin s law of the iterated logarithm near extinction time for a continuous state branching process whose branching mechanism satisfies a given condition and its reflected process at its infimum. Key words and phrases: Continuous state branching processes, Lamperti transform, Lévy processes, conditioning to stay positive, rate of growth. MSC 2000 subject classifications: 60G7, 60G5, 60G80. Introduction and main results. A continuous-state branching process or CB-process for short is a non-negative valued strong Markov process with probabilities P x, x 0 such that for any x, y 0, P x+y is equal in law to the convolution of P x and P y. CB-processes may be thought of as the continuous in time and space analogues of classical Bienaymé-Galton-Watson branching processes. Such classes of processes have been introduced by Jirina [7] and studied by many authors included Bingham [2], Grey [5], Grimvall [6], Lamperti [0], to name but a few. More precisely, a CB-process Y = Y t, t 0 is a Markov process taking values in [0, ], where 0 and are two absorbing states, and satisfying the branching property; that is to say, its Laplace transform satisfies E x [ e λyt ] = exp{ xu t λ, for λ 0, Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n Guanajuato, México. gaboh@cimat.mx Centro de Investigación en Matemáticas A.C. Calle Jalisco s/n Guanajuato, México. jcpardo@cimat.mx

2 for some function u t λ. According to Silverstein [4], the function u t λ is determined by the integral equation λ u tλ du = t 2 ψu where ψ satisfies the celebrate Lévy-Khincthine formula ψλ = aλ + βλ 2 + e λx + λx {x< Πdx, 3 0, where a R, β 0 and Π is a σ-finite measure such that 0, x 2 Πdx is finite. The function ψ is known as the branching mechanism of Y. Note that the first moment of Y t can be obtained by differentiating with respect to λ, i.e. E x [Y t ] = xe ψ 0 + t. Hence, in respective order, a CB-process is called supercritical, critical or subcritical depending on ψ 0 + < 0, ψ 0 + = 0 or ψ 0 + > 0. Moreover since P x lim Y t = 0 = e ηx, t where η is the largest root of the branching mechanism ψ, the sign of ψ 0+ yields the criterion for a.s. extinction. More precisely a CB-process Y with branching mechanism ψ has a finite time extinction almost surely if and only if du ψu < and ψ We denote by T 0 the extinction time of the CB-process Y, i.e. T 0 := inf{t 0 : Y t = 0. In this paper, we are interested in a detailed description of how continuous state branching processes become extinct. Hence, in what follows we always assume that assumption 4 is satisfied. One of the starting points of this paper are the results of Kyprianou and Pardo [8] for the CB-process in the self-similar case, i.e. when the branching mechanism is given by ψλ = c + λ α, for < α 2 and c + > 0. Note that such branching mechanism clearly satisfies condition 4. The authors in [8] described the upper and lower envelopes of the self-similar CB-process near extinction via integral test. In particular they obtained the following laws of the iterated logarithm LIL for short for the upper envelope of Y and its version reflected at its running infimum and Y T0 t t /α log log/t = c +α /α, Y Y T0 t t /α log log/t = c +α /α, P x a.s., P x a.s., where Y t denotes the infimum of the CB-process Y, P x over [0, t]. In order to state our main results, we first introduce some basic notation. Let us define the mapping du φt :=, for t > 0, ψu t 2

3 and note that φ : 0, 0, is a bijection and thus its inverse exist, here denoted by ϕ. From 2, it is straightforward to get u t λ = ϕt + φλ λ, t > 0. Since φ = 0, we clearly have u t = ϕt, for t > 0. Hence we deduce that for every x, t > 0, [ ] P x T 0 t = P x Y t = 0 = lim E x e λyt = e xϕt. λ Finally, let us introduce the lower and upper exponents of ψ at infinity, { { ψλ ψλ γ := sup c 0 : lim = and η := inf c 0 : lim = 0. λ λ c λ λ c Since ψ satisfies 3, we necessarily have γ η 2. Now, we introduce the following exponent, δ := sup { c 0 : Q 0, s.t. Qψuu c ψvv c, u v. 5 Therefore, we necesarilly have δ γ η 2. Note that in the case where ψ is regularly varying at with index α >, then δ = γ = η = α. Our first result consist in a law of the iterated logarithm LIL for short at 0 for the upper envelope of the time-reversal process Y T0 t, 0 t T 0, under P x. Theorem. Assume that δ >, then Y T0 t ϕt log log ϕt = P x-a.s., for every x > 0. Recall that Y and Y Y T0 t, 0 t < T 0 denote the running infimum of Y and the time-reversal process of Y reflected at its running minimum, respectively. We also introduce the so-called scale function W : [0, [0, which is the unique absolutely continuous increasing function whose Laplace transform is /ψ. Let us suppose that for all β <, the scale function W satisfies the following hypothesis H x 0 W βx W x <. We remark that the above hypothesis is satisfied, in particular, when ψ is regularly varying at. Theorem 2. Suppose that δ >, then under the hypothesis H, we have Y Y T0 t ϕt log log ϕt = P x -a.s., for every x > 0. 3

4 It is important to note that in the self-similar case, i.e. when ψλ = c + λ α for < α 2 and c + > 0, we have log log ϕt ϕt = log logc +α t c + α t α α for t > 0. We can replace the above function by log log t ϕt = c + α t α log log t for t > 0, since they are asymptotically equivalent at 0. Similarly, we can take the previous function in the regularly varying case. 2 Proofs Let P x, x R be a family of probability measures on the space of càdlàg mappings from [0, to R, denoted by D, such that for each x R, the canonical process X is a Lévy process with no negative jumps issued from x. Set P := P 0, so P x is the law of X +x under P. The Laplace exponent ψ : [0,, of X is specified by E[e λxt ] = e tψλ, for λ R, and can be expressed in terms of the Lévy-Khintchine formula 3. Henceforth, we shall assume that X, P is not a subordinator recall that a subordinator is a Lévy process with increasing sample paths. In that case, it is known that the Laplace exponent ψ is strictly convex and tends to as λ goes to. In this case, we define for q 0 Φq = inf { λ 0 : ψλ > q the right-continuous inverse of ψ and then Φ0 is the largest root of the equation ψλ = 0. Theorem VII. in [] implies that condition Φ0 > 0 holds if and only if the process drifts to. Moreover, almost surely, the paths of X drift to, oscillate or drift to accordingly as ψ 0+ < 0, ψ 0+ = 0 or ψ 0+ > 0. Lamperti [0] observed that continuous state branching processes are connected to Lévy processes with no negative jumps by a simple time-change. More precisely, consider a spectrally positive Lévy process X, P x started at x > 0 and with Laplace exponent ψ. Now, we introduce the clock A t = t 0 ds X s t [0, τ 0, where τ 0 = inf{t 0 : X t 0, and its right-continuous inverse θt = inf{s 0 : A s > t. Then, the time changed process Y = X θt, t 0, under P x, is a continuous state branching process with initial population of size x. The transformation described above will henceforth be referred to as the CB-Lamperti representation. 4

5 Now, define X := X, the dual process of X. Denote by P x the law of X when issued from x so that X, P x = X, P x. The dual process conditioned to stay positive is a Doob h-transform of X, P x killed when it first exists 0, with the harmonic function W. In this case, assuming that ψ 0+ 0, one has P xx t dy = W y W x P x X t dy, t < τ 0, t 0, x, y > 0. Under P x, X is a process taking values in 0,. It will be referred to as the dual Lévy process started at x and conditioned to stay positive. The measure P x is always a probability measure and there is always weak convergence as x 0 to a probability measure which we denote by P. Let σ x = sup{t > 0 : X t x, be the last passage time of X below x R. The next proposition, whose proof may be found in [8], gives us a time-reversal property from extinction for the C.B. process see Theorem. Recall that, under P, the canonical process X drifts towards and also that X t > 0 for t > 0. Proposition. If condition 4 holds, then for each x > 0 { Y T0 t : 0 t < T d 0, P x = {X θt, 0 t < A σx, P. Recall that ψ is a branching mechanism satisfying conditions 3 and 4; and whose exponent δ defined by 5 is strictly larger than. Therefore, since δ >, there exist c, and C 0, such that ψλ Cψλuu c, 6 for any u, λ [,. Now, we introduce the so-called first and last passage times of the process {X θt, t 0, P by { { S y = inf t 0 : X θt y and U y = sup t 0 : X θt y, for y 0. Note that the processes S y, y 0 and U y, y 0 are increasing with independent increments since the process {X θt, t 0, P has no positive jumps. Observe that A σx and U x are equal and that the latter, under P, has the same law as T 0 under P x. This clearly implies that the distribution of U x, under P, satisfies P U x t = e xϕt, 7 for every x, t > 0. Let J t, t 0 be the future infimum process of X θt, t 0 which is defined as follows J t = inf s t X θs, for t 0. 5

6 Proposition 2. Assume that δ >, then J t ϕt log log ϕt =, P -a.s. Proof: For simplicity, we let ft = log log ϕt ϕt for t > 0. In order to prove this result, we need the following two technical lemmas. Recall that φ is the inverse function of ϕ. Lemma. For every integer n and r >, put i The sequence t n : n decreases. t n = φr n and a n = ft n. ii The series n P J tn > ra n converges. Proof of Lemma : The first assertion follows readily from the fact that φ is decreasing. In order to prove ii, we note that for n, ϕt n = ϕφr n = r n. This entails log log ϕt n = log log r n. 8 Now, since the last passage times process is the right continuos inverse of the future infimum of X θt, t 0, we have P J tn > ra n = P U ran < t n. Hence 7 and 8 imply n P U ran < t n n r, n log r which converges, and our statement follows. Lemma 2. For every integer n 2 and r >, put s n = φe nr and b n = fs n. We have that the series P n U bn/r s n diverges. Proof of Lemma 2: First we note that for n, ϕs n = ϕφe nr = e nr. This entails log log ϕs n = log n r. Hence, the identity 7 implies n P U bn/r s n 6 = n n,

7 which diverges, and our claim follows. We are now able to establish the law of the iterated logarithm. In order to prove the upper bound, we use Lemma. Take any t [t n+, t n ], so, provided that n is large enough ft log log ϕt n, ϕt n+ since ϕ decreases. Note that the denumerator is equal to r n+ and the numerator is equal to log log r n. We thus have ft n r. 9 ft On the other hand, an application of the Borel-Cantelli Lemma to Lemma shows that and we deduce that n J t ft J tn ft n r, J tn ft n P -a.s., ft n r 2, ft P -a.s. To prove the lower bound, we use Lemma 2 and observe that the sequence b n, n 2 decreases. First, from Lemma 2, we have P U bn/r U bn+ /r s n P U bn/r s n =, n so by the Borel-Cantelli Lemma for independent events, we obtain If we admit for a while that lim inf U bn/r U bn+ /r s n, P -a.s. we can conclude lim inf U bn+ /r s n = 0, P -a.s., 0 U bn/r s n, P -a.s. This implies that the set {s : U fs/r s is unbounded P -a.s. Plainly, the same then holds for {s : J s fs/r, and as a consequence J t ft r, P -a.s. Next we establish the behaviour in 0. First, since δ > we observed from inequality 6 that there exist c, and C 0, such that εφt = ε t du ψu C c ε t ψ C ε du 7 c u = C ε c t du ψu = φ C/ε c t, 2

8 for any t and 0 < ε < minc,. On the other hand, we see that for n large enough and 0 < ε < minc, { P U bn+ /r > εs n = exp b n+ r ϕεs n b n+ 2 ϕεs n. Hence from inequality 2, we have C P U bn+ /r > εs n ε c exp {n r n + r logn + C c e r n r +, ε where the las identity follows since n + r n r rn r. In conclusion, we have that the series n P U bn+ /r > s n converges, and according to the Borel-Cantelli lemma, U bn+ /r s n ε, P -a.s., which establishes 0 since 0 < ε < minc, can be chosen arbitrarily small. The proof of is now complete. The two preceding bounds show that r J t ft r2, P -a.s. Hence the result follows taking r close enough to. Proof of Theorem : In order to establish our result, we first prove the following law of the iterated logarithm holds X θt ft =, P a.s., 3 and then use Proposition. The lower bound of 3 is easy to deduce from Proposition 2. More precisely, = J t ft X θt ft P a.s. Now, we prove the upper bound. Let r > and denote by X θ for the supremum process of {X θs, s 0, P, which is defined by X θt = sup 0 s t X θs for t 0. Lemma 3. Let c >, 0 < ɛ < /c and r >. For t n = φr n, n, then there exist a positive real number K such that P J tn > ɛc ft n ɛk 2 P X θtn > c ft n. Proof: From the Markov property, we have P J tn > ɛc ft n P X θtn > c ft n, J tn > ɛc ft n tn = P S c ft n dt P c ft n J tn > ɛc ft n 0 P X θtn > c ft n P c ft n inf X θs > ɛc ft n. 0 s 8

9 Now from the Lamperti transform and Lemma VII.2 in [], we have P c ft n inf X θs > ɛc ft n = P 0 s c ft n τ [0, ɛc ft n = = W c ɛft n W c ft n, where τ [0,z = inf{t 0 : X t [0, z. On the other hand an application of Proposition III. in [] gives that there exist a positive real number K such that then it is clear K xψ/x W x K, for all x > 0, 4 xψ/x W c ɛft n W c ft n K2 ɛ ψ /c ft n ψ ɛ /c ft n. From the above inequality and Lemma 3 in [3], we deduce P c ft n inf X θs > ɛc ft n ɛk 2, 0 s which clearly implies our result. Now, we prove the upper bound for the LIL of X θt, t 0, P. Let c > and fix 0 < ɛ < /c. Recall from 7 that P J tn > ɛc ft n = P U ɛc ftn < t n = n log r ɛc. Hence from Lemma 3, we deduce P S tn > c ft n K 2 ɛ n n n log r ɛc <, since ɛc >. Therefore an application of the Borel-Cantelli Lemma shows X θtn ft n c, P -a.s. Using 9, we deduce X θt ft X θtn ft n ft n rc, ft P -a.s. Hence from the first part of the proof and taking r close enought to above, we deduce X θt ft [, c ], P -a.s., By the Blumenthal zero-one law, it must be a constant number k, P -a.s. 9

10 Now we prove that the constant k equals. Fix ɛ 0, /2 and define { R n = inf n s : J s ɛ. kfs Note that for n sufficiently large /n < R n < and that R n converge to 0, P -a.s., as n goes to. From Lemma VII.2 in [], the strong Markov property and since the process {X θt, t 0, P has no positive jumps, we have P JRn kfr n 2ɛ = P J Rn 2ɛX θr n ɛ = Ê P J Rn 2ɛX θr n ɛ = Ê W lɛxθrn W X θrn where lɛ = ɛ/ ɛ. Applying 4 and Lemma 3 in [3], give us W lɛxθrn Ê K 2 lɛ, W X θrn, X θrn wich implies Since R n /n, P Jt kft lim P JRn kfr n 2ɛ, for some t /n 2ɛ > 0. P JRn kfr n 2ɛ. Therefore, for all ɛ 0, /2 P Jt 2ɛ, i.o., as t 0 kft JRn lim P kfr n 2ɛ > 0. The event on the left hand side is in the lower-tail sigma-field of X, P which is trivial from Bertoin s contruction see for instance Section in [4]. Hence J t ft k 2ɛ, P -a.s., and since ɛ can be chosen arbitraily small, we deduce that k. Proof of Theorem 2: Here we follow similar arguments as those used in the last part of the previous result. Assume that the hypothesis H is satisfied. From Theorem, it is clear that X θt J t X θt ft ft =, P -a.s. 0

11 Fix ε 0, /2 and define R n = inf { n s : X θs fs ɛ. First note that for n sufficiently large /n < R n < P -a.s. Moreover, from Theorem we have that R n converge to 0 as n goes to, P -a.s. From Lemma VII.2 in [], then strong Markov property and since {X θt, t 0, P has no positive jumps, we have P XθRn J Rn fr n 2ɛ = P J Rn = Ê P J Rn ɛ ɛ X θr n ɛ ɛ X θr n, = Ê W lɛxθrn W X θrn X θrn where lɛ = 2ɛ/ ɛ. Since the hypothesis H is satisfied, an application of Fatou-Lebesgue Theorem shows that W lɛxθrn Ê W lɛx θrn n + W X Ê <, θrn W X θrn which implies that Next, we note P XθRp J Rp fr p lim P XθRn J Rn fr n 2ɛ > 0. 2ɛ, for some p n P XθRn J Rn fr n 2ɛ. Since R n converges to 0, P -a.s. as n goes to, it is enough to take limits in both sides. Therefore for all ɛ 0, /2 P Xθt J t 2ɛ, i.0., as t 0 lim P XθRn J Rn 2ɛ =. ft n + fr n Then, X θt J t ft 2ɛ, P a.s., and since ɛ can be chosen arbitrarily small, we get the result.

12 3 Concluding remarks on quasi-stationarity We conclude this paper a brief remark about a kind of conditioning of CB-process which result in a so-called quasi-stationary distribution. Specifically we are interested in establishing the existence of a normalization constant {c t, t 0 such that the weak limit lim P xy t /c t dz T 0 > t, t exist for x > 0 and z 0. Results of this kind have been established for CB-processes for which the underlying spectrally positive Le vy process has a second moment in [9]; see also [], and for the α-stable CB-process with α, 2] in [8]. In the more general setting, [2] formulates conditions for the existence of such a limit and characterizes the resulting quasi-stationary distribution. The result below shows, when the branching mechanism is regularly varying at, an explicit formulation of the normalization sequence {c t : t 0 and the limiting distribution is possible. Lemma 4. Suppose that the branching mechanism ψ is regularly varying at with index α, 2]. Then, for all x 0, with c t = /ϕt [ ] lim E x e λyt/ct T0 > t = t [ + λ α ]. /α Proof: The proof pursues a similar line of reasoning to the the aforementioned references [8, 9,, 2]. From it is straightforward to deduce that [ ] lim E x e λyt/ct T0 u t λ/c t > t = lim t t u t ϕt + φλ/c t = lim, t ϕt if the limit on the right hand side exists. However, since ψ is regularly varying at with index α, we have that /ψ is regularly varying at with index α. On the other hand an application of Karamata s Theorem see for instance Bingham et al [3] gives and therefore, φt t α ψt as t, φλt lim t φt = λ α for λ > 0. In other words φ is regularly varying at with index α and then its inverse ϕ is regularly varying at with index, thus for all λ, ɛ > 0, and t sufficiently large, we α have α α λ λ t φλϕt t. ɛ + ɛ Therefore since ϕ is decreasing, we deduce ϕ + λ/ + ɛ α t ϕt + φλ/c t ϕt ϕt 2 ϕ + λ/ ɛ α t, ϕt

13 for t sufficiently large. On the other hand, since ϕ is regular varying we deduce + λ α α +ɛ ϕt + φλ/c t ϕt +, λ α α ɛ for t sufficiently large. Hence the result follows taking the limit in the above inequality as ɛ goes to 0. References [] J. Bertoin: Lévy Processes. Cambridge University Press, Cambridge, 996. [2] N. Bingham: Continuous branching processes and spectral positivity. Stochastic Process. Appl. 4, , 976. [3] N.H. Bingham, C.M. Goldie and J.L. Teugels: Regular variation. Cambridge University Press, Cambridge [4] R.A. Doney: Fluctuation theory for Lévy processes. Ecole d eté de Probabilités de Saint- Flour, Lecture Notes in Mathematics No Springer, [5] D.R. Grey: Asymptotic behaviour of continuous-time continuous state-space branching processes. J. Appl. Probab.,, [6] A. Grimvall: On the convergence of sequences of branching processes. Ann. Probab., 2, [7] M. Jirina: Stochastic branching processes with continuous state-space. Czech. Math. J. 8, , 958. [8] A. Kyprianou and J.C. Pardo: Continuos-state branching processes and selfsimilarity.j. Appl. Probab. 45, [9] A. Lambert: Quasi-stationary distributions and the continuous-state branching process conditioned to be never extinct. Elect. J. Probab. 2, , [0] J. Lamperti: Continuous-state branching processes. Bull. Amer. Math. Soc., 73, , 967. [] Z-H. Li: Asymptotic behaviour of continuous time and continuous state branching processes. J. Austr. Math. Soc. Series A. 68, [2] A. Pakes: Conditional limit theorems for continuous time and state branching processes. Research Report no. 9/99 Department of Mathematics and Statistics, The University of Western Australia

14 [3] J.C. Pardo: On the rate of growth of Lvy processes with no positive jumps conditioned to stay positive. Elect. Comm. in Probab. 3, , [4] M.L. Silverstein: A new approach to local time. J. Math. Mech. 7, ,