Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of ξn:=log((1−An)/An)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\xi _{n}:=\log ((1-A_{n})/A_{n})$\end{document} is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail ℙ(Zn≥m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {P}(Z_{n} \ge m)$\end{document} of the n th population size Zn is asymptotically equivalent to nF¯(logm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\overline F(\log m)$\end{document} as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail ℙ(Zn>m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {P}(Z_{n}>m)$\end{document} which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak.


Introduction and main results
Branching processes considered in this paper are motivated by works of Solomon (1975) and Kesten et al. (1975), who analysed a neighbourhood random walk in random environment (RWRE).This is a random walk (X t , t ∈ Z + ) on Z defined in the following way.Consider a collection (A i , i ∈ Z + ) of i.i.d.(0, 1)-valued random variables.Let A be the σ-algebra generated by (A i , i ∈ Z + ).Let (X k , k ∈ N) be a collection of Z-valued random variables such that X 0 = 0, P X k+1 = X k + 1 A, X 0 = i 0 , . . ., X k = i k = A i k Date: July 28, 2020.2010 Mathematics Subject Classification.60J70,60G55,60J80.The work of S Foss and D Korshunov is partially supported by the RFBR grant 19-51-53010 (2019-2020).The work of Z Palmowski is partially supported by the Polish National Science Centre under the grant 2018/29/B/ST1/00756 (2019)(2020)(2021)(2022). and P X k+1 = X k − 1 A, X 0 = i 0 , . . ., X k = i k = 1 − A i k for all i j ∈ Z, 0 ≤ j ≤ k and k ≥ 0. The collection (A i , i ∈ Z + ) is called a random environment.For this random walk Kesten et al. (1975) studied the appropriately scaled limiting distribution of the hitting time T n = inf{k > 0 ∶ X k = n} of the state n ∈ Z by the walker in the random environment.Their analysis is based on the representation of T n in terms of the total number of particles up to the n-th generation of a certain branching process in random environment (BPRE) with size-1 immigration at each generation step.In this model the offspring distribution in the n-th generation is geometric with a random parameter A n .In other words, let (Z n , n ≥ 0) be a branching process in random environment with one immigrant each time that starts from Z 0 ≡ 0. Then the following representation holds: where, conditioned on A, (B n+1,i , i ≥ 1) are independent copies of a geometric random variable B n+1 with probability mass function Following Kesten et al. (1975), let U n i denote the number of transitions of (X k , k ≥ 0) from i to i − 1 within time interval [0, T n ), i.e., , where Card(C) is the cardinality of the set C. It is easy to derive that (3) Note that U n i = 0 for all i ≥ n and U ∶= ∑ i≤0 U n i < ∞ a.s.if X k → ∞ a.s. as k → ∞.It has been established in Kesten et al. (1975), that (4) Then Kesten et al. (1975) have analysed T n under the so-called "Kesten assumptions" on the environment: and there exists a unique positive solution κ to the equation In particular, the assumption (6) implies that the random variable has an exponentially decaying right tail.It was shown in Kesten et al. (1975) that, under the assumptions ( 5)-( 6), an appropriately scaled T n has asymptotically the same distribution tail as scaled ∑ n−1 k=0 Z k has, that converges to a κ-stable random variable.The aim of our paper is to study the asymptotic behaviour of the branching process Z n under the complementary assumption that the distribution F of the random variable ξ is long-tailed, that is, F (x) > 0 for all x and for some (and therefore for any) fixed y = 0.Here F (x) = 1 − F (x) is the tail distribution function and equivalence (7) means that the ratio of the left-and right-hand sides tends to 1 as x grows.In particular, (7) implies that F is heavy-tailed, i.e.Ee cξ = ∞ for all c > 0. Given (7), the distribution G defined by its tail as G(x) = F (log x), x ≥ 1, is slowly varying at infinity and therefore subexponential, that is, see, e.g.Theorem 3.29 in Foss et al. (2013).
Any strong subexponential distribution F is subexponential, and its integrated tail distribution F I with the tail is subexponential too (see e.g.(Foss et al. , 2013, Theorem 3.27)).In what follows, we write F I (x, y] ∶= F I (x) − F I (y).
We start now with our first main result.
Theorem 1.1.Under the assumption (7), If, in addition, the distribution F is subexponential, then, for any fixed n ≥ 2, Theorem 1.1 shows that the tail of Z 1 is surprisingly heavy and is getting heavier in each next generation.What should be underlined, this type of behaviour is a consequence of the environment only, and not of the branching mechanism which is of geometric type.In contrast to a series of papers Seneta (1973), Darlin (1970), Schuh and Barbour (1977), Hong and Zhang (2019), we do not analyse the convergence results, with focusing on the tail behaviour of Z n for each n.Consider now a branching process with state-independent immigration satisfying the stability condition The classical Foster criterion implies that the distribution of Z n stabilises in time, i.e. the distribution of the Markov chain Z n converges to it unique limiting/stationary distribution as n grows.It follows from Theorem 1.1 that, for any n, the tail of the stationary distribution must be asymptotically heavier than nF (log m), i.e.P(Z > m) F (log m) → ∞ as m → ∞, where Z is sampled from the stationary distribution.The distribution tail asymptotics of Z n and Z are specified in the following two results.The first result provides two asymptotic lower bounds, for finite and infinite time horizons, where the first bound is uniform for all generations.
Theorem 1.2.Assume that A ≤ Â a.s.for some constant Â < 1.Then the following lower bounds hold.(i) If the distribution F is long-tailed, then (ii) If the integrated tail distribution F I is subexponential and the stability condition (10) holds, then The next result presents conditions for existence of upper bounds that match the lower bounds of Theorem 1.2.
Theorem 1.3.Let the stability condition (10) hold and the distribution F be such that Then the following upper bounds hold.(i) If the distribution F is strong subexponential, then Distributions satisfying the first condition in (13) are called square-root insensitive, see e.g.(Foss et al. , 2013, Sect. 2.8).Typical examples of distributions satisfying (13) are: any regularly varying distribution, a log-normal distribution and a Weibull distribution with parameter less than 1 2. We do not know, whether the square-root insensitivity condition is essential or not for the upper bounds in Theorem 1.3.In the literature, there are various scenarios where extra randomness leads to appearance of further terms in the tail asymptotics due to the effects caused by the central limit theorem, namely, for the Weibull distribution F (x) = exp(−x β ) with parameter β ∈ [1 2, 1), the number of extra terms depends on the interval [n (n + 1), (n + 1) (n + 2)), n = 1, 2, . . . the parameter β belongs to; see e.g.Assmusen et al. (1998) and Foss and Korshunov (2000) for the tail asymptotics of the stationary queue length in a single-server queue or Denisov et al. (2020) for the stationary tail asymptotics for Markov chains with asymptotically zero drift.However, we are not certain that similar arguments may be relevant to the model considered in the present paper.If the distribution F satisfies all the conditions of Theorems 1.2 and 1.3, then the corresponding lower and upper bounds match each other and we conclude the following tail asymptotics: 16) These asymptotics may be intuitively interpreted as follows: Z n is taking a large value if one of the ξ's is sufficiently large, i.e. one of the success probabilities A's is small.This phenomenon may be named as the principle of a single atypical environment and formulated as follows.
For any c > 1 and ε > 0 let us introduce events where S j,n−1 ∶= ξ j + . . .+ ξ n−1 .Roughly speaking, the event n (m) describes a trajectory such that, for large m, the value of Z k is relatively not big, while the success probability A k is close to zero and, as a result, a single atypical environment occurs, and after time k the environment follows the strong law of large numbers with drift −a.As stated in the next theorem, the union of all these events provides the most probable way by which the large deviations of Z n do occur.
Theorem 1.4.Assume that conditions of Theorems 1.2 and 1.3 hold.Then, for any fixed ε > 0, Let us highlight a deep link of BPRE to stochastic difference equations.It follows from the recurrence equality and its limit E(Z A) are distributed as a finite time horizon perpetuity and the solution to the stochastic fixed point equation, respectively.Their tail asymptotic behaviour in the heavy-tailed case is the same as given in ( 16)-( 17), that is, see Dyszewski (2016) for ( 21) and Korshunov (2020) for general case.The remainder of the paper is dedicated to the proofs of the results above.We close our paper by Section 6 which contains some discussion and possible extensions.
2. Finite time horizon tail asymptotics, proof of Theorem 1.1 We start with some useful representations.Firstly, Secondly let us observe that the k-fold convolution of geometric distribution is known in closed form, and its probability mass function is hypergeometric: Therefore, for k ≥ 2, which yields the following binomial representation that is convenient for further analysis, The above representations allow us to prove two auxiliary results.
Lemma 2.1.Under the assumption (7), Lemma 2.2.Under the assumption (7), there exist γ < ∞ and ε > 0 such that In particular, for any fixed j ≥ 1, Proof of Lemma 2.1.Since, for any fixed ε > 0, is exponentially decreasing as m → ∞, the asymptotic behaviour of the right-hand side in ( 24) is determined by the tail behavior of A near 0. Notice that, for 0 < a < b < 1, Hence, for any fixed c > 0, we have It follows from the long-tailedness of the distribution F of ξ that the right-hand side of above equation is asymptotically equivalent to e −c F (log m) as m → ∞.Letting c ↓ 0 we complete the proof of the lower bound To obtain the matching upper bound, let us consider the following decomposition which is valid for all integer Let us show that the series in the middle term in the last line is negligible for large values of K. Indeed, firstly, and hence Since the distribution F is assumed long-tailed, there exists a constant γ < ∞ such that F (x − y) ≤ γe y F (x) for all x, y > 0. Therefore, Hence we conclude that Due to the long-tailedness of F this implies that, for any fixed K, Proof of Lemma 2.2.There exist K ∈ N and ε 1 > 0 such that the following inequalities hold Similar to the case j = 0 considered in the proof of Lemma 2.1, we make use of the following decomposition: The maximum of the function x j (1− x) m over the interval [0, 1] is attained at point j (m + j) and is equal to j j m m (m + j) m+j .Therefore, for some ε owing to the long-tailedness of F .Further, the series on the right hand side of (30) possesses the following upper bound because (1 − kj 3m) m ≤ e −kj 3 .Let us now bound the latter series.It follows from the inequality (28) that (k + 1) j e −kj 3 = e j(log(k+1)−k 3) ≤ e −jk 6 for all k ≥ K.
Then, using arguments similar to those in ( 27), which implies the result due to the inequalities (31) and which is guarantied by (29).
Proof of Theorem 1.1.Let us prove the statement by induction in n ≥ 1.The assertion for n = 1 follows from the representation (24) and Lemma 2.1.Assume that the assertion of Theorem 1.1 is valid for some n ≥ 1.Let us show that then it follows for n + 1 ≥ 2. Our aim is to obtain the tail asymptotics of the distribution of where (B n+1,i , i ≥ 1) are independent copies of a geometric random variable B n+1 with success probability A n (its probability mass function is specified in (2)) and independent of Z n conditioned on A. Then the following representation holds where we have conditioned on A and used the fact that Z n and (B n+1,i , i ≥ 1) are independent conditioned on A. We start with proving an upper bound.For that, let us split the sum in (33) into three parts, from 0 to K, from K + 1 to εm − 1 and from εm to ∞ where an integer K is chosen large enough and real ε > 0 is small enough.This splitting together with non-negativity of B's implies that By the induction hypothesis and long-tailedness of F , for any fixed ε, So it is left to show that, for any fixed K, and, for any δ > 0, there exist a sufficiently large K and a sufficiently small ε > 0 such that We start with proving (35).Let ξ(A) be a Bernoulli random variable with success probability A and S m+k (A) be the sum of m + k independent copies of ξ(A).It follows from the representation (23) that , for all β > 0.
The minimal value of the right hand side is attained for β such that e −β = A(m+1) , hence This allows us to conclude from Lemma 2.2 that, for k ≤ εm, Therefore, Representing P(Z n = k) as a probability difference P(Z n > k − 1) − P(Z n > k) and rearranging the sum on the right hand side we conclude that this sum is not greater than Then the induction hypothesis yields an upper bound, for some γ 1 < ∞, Due to the long-tailedness of F , for any δ > 0 there exists a sufficiently large K such that the first term on the right hand side is not greater than δF (log m) for all sufficiently large m.After rearranging we conclude that the sum on the right hand side is not greater than F (log(m + 1) − log(εm))F (log(εm − 1)) Since F is long-tailed, the first term here is asymptotically equivalent to where the distribution G is defined via its tail as G(x) = F (log x), and can be approximated by an integral, Since the distribution F is assumed subexponential, we can choose a sufficiently large K and a sufficiently small ε > 0 such that the last probability is not greater than δF (log m) for all sufficiently large m, see (Foss et al. , 2013, Theorem 3.6), which completes the proof of (35).
To complete the proof of the upper bound it now suffices to show (34).This follows immediately from the representation (23), the asymptotics (25) and Lemma 2.1.
We will obtain now the matching lower bound.For that, let us split the sum in (33) into two parts, from 0 to cm and from cm + 1 to ∞ where c is a large number sent to infinity later on.This splitting implies that since all B's are non-negative.By Lemma 2.1, Further, by the law of large numbers, Hence, the dominated convergence theorem allows us to conclude that Finally, by the induction hypothesis and long-tailedness of F , for any fixed c, Substituting ( 38)-( 40) into (37) and letting c → ∞ we conclude the induction step for the lower bound.
3. Proof of the lower bound, Theorem 1.2 Note that, by the strong law of large numbers, for any fixed ε > 0, where We show that the most probable way for a big value of Z n to occur under the long-tailedness condition ( 7) is due to atypical random environment when one of the following events occurs, k ≤ n − 1: where , c 2 , ε > 0 are fixed, c 2 will be sent to infinity later on, while c 1 and ε will be sent to 0. Since A is bounded by Â < 1, a A is bounded away from 0 by 1 Â − 1.
Let us bound from below the probability of the union of events C A (k, n).We start with the following lower bound and As follows from (26), Therefore, and as m → ∞ uniformly for all n ≥ 1. Substituting these bounds into (42) and applying (41), for any fixed ε > 0 we can conclude the following lower bound, as m → ∞ uniformly for all n ≥ 1, where g(c 2 ) → 1 as c 2 → ∞.As above, conditioning on A yields where Then, owing to (44), to prove (11) it suffices to show that lim inf Hence we are left with proving (46).Observe that the event C A (n) implies that C A (k, n) occurs for some k ≤ n − 1.Then the probability of the event conditionally on C A (n), possesses the following asymptotic lower bound Therefore, it only remains to show that inf as m → ∞ uniformly for all k ≤ n − 1 and n ≥ 1.To prove this convergence, let us note that, conditioned on A, Applying exponential inequality, we obtain the following upper bound, for all λ > 0, A ≤ 1 for all A ∈ (0, Â). Therefore, which due to monotonicity property of the branching process Z n implies that Then induction arguments lead to the following upper bound we conclude that we get for any sequence of ξ's such that which is the case on C S (c 2 , ε, k, n − 1) and hence on C A (k, n), as follows from the left hand side inequality in (43) for all ε ∈ (0, − E ξ).So, we have shown (47), and the proof of the first lower bound in Theorem 1.2 is complete.The lower limit for the stationary distribution follows similar arguments if we start with an analogue of (45), Then, similar to (44), using the fact that F I is long-tailed we conclude that which together with (46) justifies the lower bound for the stationary tail distribution.which implies that Substituting ( 51) and ( 52) into (50) we deduce that, uniformly for all n ≥ 1, By the condition (13), uniformly for all n ≥ 1. Due to the arbitrary choice of ε > 0 the proof of the upper bound ( 14) is complete.
The above arguments can be streamlined if we made use of the link ( 19) to stochastic difference equations.Indeed, conditioning on the environment we get that For the first term on the right hand side we apply asymptotics (20).For the estimate of the second term we can apply Markov's inequality to get and that the union of events on the left hand side implies Z n > m with high probability, that is, n (m) → 1 as m → ∞ uniformly for all n.
Then it follows from the equality Letting ε ↓ 0 concludes the proof.

Related models
The techniques developed in this paper may be applied for analysing similar models.We mention here a few of them.
Random-size immigration.One may replace size-1 immigration by a random-size-immigration where random sizes are i.i.d. and independent of everything else, with a common light-tailed distribution (or, more generally, the sizes may be stochastically bounded by a random variable with a light-tailed distribution).
A branching process { Ẑn , n ≥ 0} with state-dependent size-1 immigration is a particular case here: an immigrant arrives only when the previous generation produces no offspring: Clearly, Ẑn ≤ Z n a.s., for any n.Moreover, one can show that, for each n, the low bounds for P(Z n > m) and P( Ẑn > m) are asymptotically equivalent.Then, in particular, the statement of Theorem 1.1 stays valid with Ẑn in place of Z n .Continuous-space analogue.Instead of the recursion (1), one may consider a "continuousspace" recursion of the form where B n are subordinators with a light-tailed distribution of the Levy measure (that depends on random parameters) and {Y n } are i.i.d."innovations" with a light-tailed distribution.A similar problem for a branching process with immigration, but without random environment has been studied in a recent paper by Foss and Miyazawa (2020).

B
n+1,j > m A P(Z n = k) ≤ δF (log m) for all sufficiently large m.(35) + 1 and A is bounded away from 1, there exists a sufficiently small λ 0 > 0 such that E e λ 0 e −ε 2 − B a A m on the event E(Z n A) ≤ me − √ log m which completes the proof.The proof of the stationary upper bound (15) follows similar arguments with initial upper boundP(Z > m) n > m A ; E(Z n A) ≤ me − √ log m .andfurther use of the asymptotics (21) instead of (20) which is valid due to subexponentiality of the integrated tail distribution F I .The proof of Theorem 1.3 is complete.5. Proof of the principle of a single atypical environment, Theorem 1.4As follows from the arguments presented in Section 3, for any fixed c and ε > 0,