UPPER DEVIATIONS FOR SPLIT TIMES OF BRANCHING PROCESSES

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1 Applied Probability Trust 7 May 22 UPPER DEVIATIONS FOR SPLIT TIMES OF BRANCHING PROCESSES HAMED AMINI, AND MARC LELARGE, ENS-INRIA Abstract Upper deviation results are obtained for the split time of a supercritical continuous-time Marov branching process. More precisely, we establish the existence of logarithmic limits for the lielihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times which shows that the scaling is completely different from the upper deviations. Keywords: continuous-time branching process; split time; large deviation 2 Mathematics Subject Classification: Primary 6J8 Secondary 6F. Introduction We consider a one dimensional continuous time Marov branching process {Zt; t } with infinitesimal generating function fs s, where fs = i p is i with p i for all i and i p i = see e.g., [3]. We recall the construction of the process {Zt} and introduce some required notations. Let {ξ i, i } be a sequence of i.i.d. random variables with generating function f and let η i = ξ i. We define S = Z = for, and S n = + n i= η i for n. Let I := inf{n N; S n = }. If S n for all n N, then I =. Given the sequence {ξ i }, let τ,..., τ I independent exponential random variables with means E [ τj {ξ i } ] = Sj. be mutually Postal address: Projet TREC, INRIA-École Normale Supérieure, Paris, France address: hamed.amini@ens.fr - marc.lelarge@ens.fr

2 2 H. Amini and M. Lelarge We define the sequence of split times by T = and T n = τ + + τ n for n I, and let Sn for Tn t < Tn, n I, Zt = for TI t. On the event I <, we tae the convention T n = + for any n I +. This event corresponds to the extinction of the branching process. Let λ denote its probability which is the smallest root in [, ] to s = fs. We also define λ = f λ. We can now state our main theorem. Theorem. In previous setting with < f <, we denote ξ min = min{i, p i > }. For any x > and, we have > Tn with lim n log n log P x + f log n = xgξ min,, gξ min, = ξ min 2 + p ξ min = + λ ξ min =. Although split times have been studied in [2] in the case where ξ min 2 and very precise large deviations are nown for supercritical Galton-Watson processes [4, 5, 8,, ], to the best of our nowledge, cannot be obtained directly from results in the literature and we provide a complete proof in the next section. To get some insights about, we interpret the process Zt as a population process describing living particles at time t. Note that if ξ min 2, then T n is exactly the first time when the population increased by an amount of n starting with an initial population of. Thans to [2], the growth rate of T n is of order log n f. Hence the upper deviation for T n analysed in corresponds to an event where the population remains small for a large amount of time. Since each initial particle gives rise to a population which cannot become extinct, each of this sub-population have to remain small and the linearity of g in is then easy to understand. When ξ min =, note that some split times are not real split times for the population since a particle can replicate itself at a split. An easy change of time argument allows to reduce this case to previous case and introduces a factor p. Finally when ξ min =, the situation is radically different as extinction is possible. In this case, the function g does not depend

3 Upper deviations for split times 3 on. Indeed the typical event for population starting with particles to remain small but positive for a long time is that of these particles become extinct while only one particle replicates itself for a very long time. This can be seen by the following argument giving an interpretation of λ. Let X p be a Galton-Watson tree with offspring distribution p = p i i, i.e., a random tree where the root and all successive descendants have a random number of children independent from the rest and with distribution p. Let X + p X p be the set of particles of X p that survive, i.e., have descendants in all future generations. Then X + p contains the root of the original branching process with probability λ, and is empty otherwise. We denote by ξ the random variable with distribution Pξ = = p. For η [, ], we let ξ η be the thinning of ξ obtained by taing ξ points and then randomly and independently eeping each of them with probability η, i.e., r Pξ η = = p r η η r. r= Note that the number of surviving children has the distribution ξ λ. Let ξ + denote the offspring distribution in X + p. Conditioning on a particle being in X + p same as conditioning on at least one of its children surviving, so that Pξ + = = Pξ λ = ξ λ = p λ λ λ = f λ = λ. is exactly the Hence, the factor λ in is obtained by the same ind of change of time that led to the factor p in the case ξ min =. Our Theorem will be used to establish the asymptotic of the diameter in random graphs with exponential edge weights see [6] for regular graphs corresponding to p r = fro some r 2, which is the subject of our forthcoming paper []. We thin that it is of independent interest as it gives some insights on how slow the growth of a continuous time branching process can be. Remar. We describe here how to heuristically derive our main result using nown properties of continuous-time Marov branching processes. It is well nown see e.g., [3] that Zte f t has an almost sure limit W. The limiting random variable W

4 4 H. Amini and M. Lelarge has an atom at zero of size λ. Furthermore, the random variable W, conditioned to be positive, admits a continuous density on R +, and we denote by W λ d = W W >. The random variable W λ is given by see e.g., [3, 9, 2] W λ = W i e f E i, i= where the E i s are i.i.d. Exp, independent of the W i which are i.i.d. with Laplace transform φt = E e t W whose inverse function is given by x φ f x = x exp fs s + s ds, λ < x. 2 Note that P > Tn = P x + f log n e f x+ f log n Z = P x + f log n < n, I < Z x + f log n < n xf, I <. Since the left-hand term in the above probability converges to W as n goes to infinity, it is reasonable to analyze PW λ < n xf. Note that [ ] ψs := Ee sw λ = E exp s W e f E, where E is an Exp random variable and ψ. is the Laplace transform of W λ. Moreover, E exp s W e f E = e x φ se f x dx. Combining Equation 2 together with the Tauberian theorem [7, Section XIII.5], it is easy to infer that PW λ < n xf n xgξmin,. In the next section, we present the proof of Theorem. For completeness, we give in Section 3, an estimation for the lower deviation probability of the split times which shows that the scaling is completely different from the upper deviations. 2. Proof of Theorem Let us define α n := log 3 n. In the following, we will use the following property of the exponential random variables, without sometimes mentioning. If Y is an exponential random variable of rate γ, then for any θ < γ, we have E [ e θy ] = γ γ θ.

5 Upper deviations for split times 5 We first assume that p = p = so that ξ min 2. Thus η i for all i, so that S n is increasing in n and I =. In this case, all the elements of the sequence {τ n} are finite and this sequence has been studied in [2]. We now prove the upper bound in for this specific case. Note that since S i + i for all i, we have for any θ < + : E [ expθτi ] [ ] S = E i Si θ + i + i θ. 3 We have for any ɛ >, P Tn x + f log n where xɛ = x ɛ f. For the first term, we have + P iα n τ i n i=α n+ xɛ log n 4 τi + ɛ f log n, iα n τ i xɛ log n = xɛ log n 2iα n τ i xɛ log n y e y dy, since τ is an exponential random variable with mean independent of the τi s with i 2. We need to bound the right-hand term and we proceed as follows: 2iα n τ i xɛ log n y n xɛ e y E exp α n n xɛ e y i=2 n xɛ e y exp + i i αn i=2 2iα n τ i i By Chernoff By Equation 3 Since + x e x < n xɛ e y α n log n, for n sufficiently large. Hence we get for n sufficiently large xɛ log n < xɛn xɛ αn log 2 n < n xɛ α + iα n τ i n.

6 6 H. Amini and M. Lelarge We now give an upper bound for the second term in 4. We first recall a basic result of probability: for any ɛ >, there is a constant γ > such that for n large enough, we have: We define the event E n = P Sn + n f e γn. + ɛ { S i } + i f +ɛ/3, α n i n, so that by the union bound we have PE n on log n. Using the fact that α n = oα n, we have for n sufficiently large, E [ exp αn n i=α n+ τ i E n ] n i=α n n + + αn + i f +ɛ/3 αn + ɛ/2 α n if i=α n αn + ɛ/2 exp f log n. Now we have by Marov s inequality n P τi + ɛ f log n E n i=α n+ E [ exp αn n i=α n+ τ i E n ] αn + ɛ exp f log n αn ɛ exp 2f log n o n log n. Hence we get n P τi + ɛ f log n PE n + o n log n = o n log n. 5 i=α n+ Note that in order to get 5, we only used the fact that p = p = to ensure that T n < for all n. In particular, the argument is still valid if p =. x To summarize in the case p = p =, we obtain for any ɛ > and with xɛ = ɛ f that P Tn x + f log n n xɛ αn + + on log n,

7 Upper deviations for split times 7 and the upper bound for follows. We now prove a lower bound for. We start with for any ɛ > P Tn x + f log n P τ xɛ log n n P τi ɛ f log n, ɛ where xɛ = x + f. Since P τ xɛ log n = n xɛ, we need to show that n lim inf P τi ɛ n f log n >. i=2 This follows from the almost sure convergence of log n n i=2 τ i to f see e.g., Corollary in Section 2 of [2]. We now consider the case where p > while p =. We start with the upper bound and the decomposition 4. Note that f > implies p <. Let τ i i=2 be the real split times of the process Zt, i.e., the times where Zt increases. Let N be a geometric random variable with parameter p, i.e., PN = j = p j p, independent of everything else. Then τ has the same law as a sum of N mutually independent exponential random variables with mean. Hence it is distributed as an exponential random variable with mean p. For the first term, note that τ i get iα n τ i xɛ log n τ i xɛ log n. iα n τi, so that we It is shown in [3] see Section III.9 that the branching process Zt associated to the split times τ i s is still a Marov branching process with the same infinitesimal generating function but with p = and with [ ] E τ j { ξ i } = p S, j where { ξ i } is a sequence of i.i.d. random variables with generating function fs = fs p s = p j s j, p and with S j = + j i= ξ i j. Hence thans to previous analysis, we get τ i xɛ log n n pxɛ α p+ iα n j=2 n.

8 8 H. Amini and M. Lelarge Since 5 is still valid, the upper bound follows from 4. The lower bound follows also from a simple adaptation of the argument above: P Tn n x + f log n P τ xɛ log n, τi ɛ f log n. At time τ we have Z τ = + η, where the random variable η is distributed as η conditioned on being greater than. Let j be such that P η j /2. The process {Zt; t τ } has the same law as the original process starting with + η particles and is independent of τ. Since τi is stochastically decreasing in, we have P Tn x + f log n P τ xɛ log n, N α n P 2 P i= i=n n α n η j, τ +j i ɛ f log n n p xɛ p αn n αn i=. τ +j i ɛ f log n n αn Since we still have the almost sure convergence of log n i= τ +j i to f, we obtain the lower bound. We now consider the case where p > so that the probability of extinction PI < = λ is positive and strictly less than one because f >. Following [3], we define if I < ; Zt = the number of particles among Z t which have infinite line of descent if I =. We have Z = Bin, λ and for j : P Z = j I = = λ j λ j j λ. 6 By Theoreme I.2. [3], the process { Zt; t } conditioned on the event I = is a Marov branching process with infinitesimal generating function fs s where. fs = i p i s i = f λs + λ λ λ for s.

9 Upper deviations for split times 9 Clearly f = p = so that this process survives and we define the corresponding split times T n = τ + + τ n for all n as we did for the original process Zt but now with a random number of initial particles given by 6. Note that we have f = f and p = f λ = λ,. On the event I =, we clearly have τ n τ n for all n, hence thans to previous analysis, we have for any j, lim sup n log n log P Tn x + f log n I =, Z = j xj λ, and the upper bound follows in the case I =. We now consider the case I <. First note that T n T n, hence we need to consider only the case =. It follows from Theorem I.2.3 in [3] that conditioned on the event I <, the branching process has the same law as a Marov branching process Zt with infinitesimal generating function fs s where fs = λ fλ. The total progeny is finite and f = f λ = λ <. Moreover, we have P Tn x + f log n I < P Z E [ Z x + f x + f λ x λ = n f, log n ] log n > and the upper bound follows. We now derive a lower bound. First note that using the Marov property, given Zt = j, for j, the random variable Zt is distributed as a binomial random variable with j trials and probability of success λ. Let {X i ; i } be a sequence of independent Bernoulli random variables with E[X ] = λ. To ease notation, let x n = x + f log n and for ɛ >, let E be the event E = { m i= X i λ ɛm, m }. We have P > T n x n = P Zx n n, I = Z = P Zx n n, Zx n = Zx n i= X i, I =, E Z =

10 H. Amini and M. Lelarge On the event E, we have Zx n i= X i λ ɛzx n. Thus, P > T n x n P Zx n P Zx n P n λ ɛ, I =, E Z = n λ ɛ Z = Z = Z = PE Z =. We have P Z = Z = PE Z = >, and the process { Zt; t } conditioned on the event Z = is a Marov branching process with infinitesimal generating function fs s with p = λ. Hence the lower bound follows from previous analysis. 3. On the lower deviation probability In this section, we consider the lower deviation probability for the split time of a branching process. Proposition. In previous setting with < f <, for any x > and, and for a sufficiently large constant C, we have x P Tn < f log n = o n C e nx. 7 Proof. We need the following lemma. Lemma. Let X,..., X t be a random process adapted to a filtration F = σ[ø], F,..., F t, and let µ i = EX i, Σ i = X X i, Λ i = µ µ i. Let Y i ExpΣ i, and Z i ExpΛ i, where all exponential variables are independent. Then we have Y Y t st Z Z t. Proof. By Jensen s inequality, it is easy to see that for positive random variable X, we have ExpX st ExpEX. Then by induction, it suffices to prove that for a pair of random variables X, X 2 we

11 Upper deviations for split times have Y + Y 2 st Z + Z 2. We have PY + Y 2 > s = E X [PY + Y 2 > s X ] E X [PExpX + ExpX + µ 2 > s] PZ + Z 2 > s. We infer by Lemma, T n st n i= where all exponential variables are independent. Thus, we have PT n t = e n xit Exp + f i, n i= f i+x i dx...dx n f i + e f n y y nt i= n i= yi e yn dy...dy n f i +, where y = i= x n i. Letting y play the role of y n, and accounting for all permutations over y,..., y n giving each such variable the range [, y], we obtain n PTn t f n i= i + f n! t. e f +y e f n i= yi dy...dy n dy [,y] n = n. = n. n i= i + f n! t e f +y n + i= t f n y i= f i f i= f e f y i dy i dy e f +y e f y n dy cn f + f where c > is an absolute constant. Now we use the fact that t e f y n e n x, e f +y e f y n dy,

12 2 H. Amini and M. Lelarge for all y x f x P Tn f log n log n =: t n. We infer for C > cf n tn f + Acnowledgements f + e nx dy = o n C e nx. The authors than an anonymous referee for the suggestions which improved the presentation of the paper. References [] Amini, H. and Lelarge, M. The diameter of weighted random graphs in preparation. [2] Athreya, K. B. and Karlin, S Limit theorems for the split times of branching processes. Journal of Mathematics and Mechanics 7, [3] Athreya, K. B. and Ney, P. E. 24. Branching processes. Dover Publications. [4] Biggins, J. D. and Bingham, N. H Large deviations in the supercritical branching process. Advances in Applied Probability 25, [5] Bingham, N. H On the limit of a supercritical branching process. Journal of Applied Probability 25, [6] Ding, J., Kim, J. H., Lubetzy, E. and Peres, Y. 2. Diameters in supercritical random graphs via first passage percolation. Combinatorics, Probability and Computing 9, [7] Feller, W. 99. An Introduction to Probability Theory and Its Applications, 2nd Edition. Vol. 2. Wiley. [8] Fleischmann, K. and Wachtel, V. 27. Lower deviation probabilities for supercritical Galton-Watson processes. Annales de l Institut Henri Poincaré. Probabilités et Statistiques 43, [9] Harris, T. E. 95. Some mathematical models for branching processes. 2nd Bereley Symposium on Mathematical Statistics and Probability [] Liu, Q Asymptotic properties of supercritical age-dependent branching processes and homogeneous branching random wals. Stochastic Processes and their Applications 82, [] Ney, P. E. and Vidyashanar, A. N. 24. Local limit theory and large deviations for supercritical branching processes. The Annals of Probability 4, [2] Sevastyanov, B. A. 95. The theory of branching random processesruss. Usephi Mat. Nau. 6,