Recurrence and transience of contractive autoregressive processes and related Markov chains

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.


Introduction
The classification of irreducible Markov chains as recurrent or transient is one of the fundamental objectives in the study of Markov chains. Scalar nonnegative autoregressive processes (X n ) n∈N 0 of the form X n = aX n−1 + Y n , where 0 < a < 1 and (Y n ) n∈N is i.i.d., and, closely related, subcritical Galton-Watson processes (Z n ) n∈N 0 with immigration (Y n ) n∈N and average offspring a ∈ (0, 1) are classical Markov chains. The study of these processes has a rich history which started more than half a century ago. However, most of the literature on these processes deals only with the positive recurrent case, i.e. the case where there exists a stationary probability distribution.
To the best of our knowledge there is at present no complete classification in simple terms of recurrence versus transience of these processes although this problem has been investigated for several decades, see [Pak75], [Pak79], [Kel92,Part I], [GM00, p. 1196], [ZG04], [Bau13] and the review below.
In the present article we characterize recurrence and transience of these processes in terms of a and the cumulative distribution function of Y 1 . More precisely, we show that either process is for some y ∈ (0, ∞), see Theorem 3. (For the branching process we need to assume a certain moment condition on the offspring distribution.) Note that the right hand side of (1) can often be easily checked by ratio tests.
We also extend this result to certain multidimensional cases in random environment by classifying nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix and subcritical multitype Galton-Watson processes in random environment with immigration. The criterion also applies to two other related processes, sometimes called max-autoregressive process and general random exchange process.
We first introduce these four processes and review the existing literature on the subject. (The precise definition of recurrence and transience is given in the next section.) Autoregressive processes. Autoregressive models are among the most widely used stochastic models, see e.g. [Kes73], [BD91], [MT93], [BDM16]. We consider nonnegative multidimensional autoregressive processes X = (X n ) n≥0 of order one (AR(1) processes) with random coefficient matrix, defined as follows. Fix a dimension d ∈ N. Let Y = (Y n ) n≥0 be a sequence of [0, ∞) d -valued random vectors, called innovations, and let (A n ) n≥1 be a sequence of [0, ∞) d×d -valued random matrices. Assume that (A n , Y n ) n≥1 is i.i.d. and independent of Y 0 . To avoid cases which are not interesting in the present context we suppose that the support of the law of Y 1 is unbounded. Set X 0 := Y 0 and (2) X n := A n X n−1 + Y n for n ≥ 1.
Relation (2) is sometimes called a random difference equation or random affine recursion, see also [BDM16,p. 1]. Solving this recursion we obtain the explicit expression We only consider the subcritical (contractive) case where the maximal Lyapunov exponent of A n is strictly less than 0. For convenience we phrase our statements in terms of the negative λ of the Lyapunov exponent, defined as the a.s. limit (4) λ := lim n→∞ S n n , where S n := − ln A 1 . . . A n .
(Here S 0 := 0.) This limit is known to exist if E[log + A 1 ] < ∞, see [FK60,Theorem 2]. For bounds and efficient methods for the computation of λ see e.g. [Pol10]. It has been shown for the subcritical case (λ > 0) that under various conditions X is positive recurrent iff and constant coefficient matrix A = A n . For a discussion of the multidimensional case with random A n see [Erh14]. (Note that the equivalence of positive recurrence and the existence of an invariant probability measure, which is well-known for countable state spaces, also holds in this setting, see e.g. [Kel06,Section 6].) The case where (5) fails is sometimes referred to as super-heavy tailed, see [ZG04]. Among the few works which deal with recurrence versus transience of AR(1) processes are the unpublished preprint [Kel92, Part I] and [ZG04]. (For some recent work with deals with super-heavy tailed innovations see [BI15].) Both papers treat one-dimensional AR(1) processes with constant coefficient A 1 = a ∈ (0, 1). In [Kel92, Part I, Theorem (3.1)] Kellerer shows that X is In [ZG04, Theorem 1] Zeevi and Glynn consider log-Pareto distributed innovations Y n , whose common distribution is given by P [ln(1 + Y 1 ) > t] = (1 + βt) −p for some β > 0 and p > 0. For this case they characterize recurrence and transience by showing that X is positive recurrent if p > 1, null recurrent if p = 1 and β ln(1/a) ≥ 1, and transient otherwise. Note that in this example the distinction between recurrence and transience easily follows from (1) and Raabe's test.
Branching processes with immigration. The classical Galton-Watson model as a basic model for branching populations, see e.g. [Har63] and [AN72], has been extended in various directions, for example by allowing finitely many different types of individuals with different offspring distributions [AN72, Chapter V], by letting the offspring distribution depend on time in a random way [AN72, Chapter VI.5], or by allowing immigration [AN72, Chapter VI.7]. Following e.g. [Key87], [Roi07], and [Vat11], we consider a combination of these three generalizations, namely multitype Galton-Watson branching processes Z = (Z n ) n≥0 in random environment with immigration.
We postpone the precise definition of Z to the next section and first give an informal description of the model. Let d ∈ N and (Y n ) n≥0 be as above and let us assume for the moment that all Y n are N d 0 -valued. There are d different types of individuals, enumerated by 1, . . . , d. The i-th component of the N d 0 -valued random variable Z n is the number of individuals of type i present in generation n. Given Z n−1 , the n-th generation Z n is obtained as follows. Each member of generation n−1 gets independently of the other members of that generation a random number of children of the d different types. The distribution of the number of children of a certain type may depend on the type of the parent. It may also depend in an i.i.d. way, called the random environment, on the number n − 1 of the generation. The n-th generation consists of the children of the individuals of the previous generation and additional immigrants of type 1, . . . , d, whose numbers are given by Y n .
This process Z is closely related to AR(1) processes in the following way. Define the (i, j)-th entry of the matrix A n as the conditional expectation of the number of children of type i in generation n of a parent of type j given the random environment. Then the conditional expected value of Z given the random environment and given the numbers of immigrants satisfies the recursion (2) and is therefore an AR(1) process, see (17) below.
In the one-dimensional case the results in the literature concerning the distinction between recurrence and transience of Z are more complete than the corresponding results for autoregressive processes. Pakes considers in [Pak75] and [Pak79] subcritical single-type processes with immigration in an environment which is constant in time. He gives several sufficient conditions for recurrence or transience in terms of generating functions and provides several examples.
For subcritical single-type branching processes in random environment Bauernschubert derives in [Bau13, Theorems 2.2, 2.3] conditions similar to (6) and (7). She shows that under suitable assumptions Z is For a different but similar model in continuous time, Li, Chen, and Pakes [LCP12, Theorem 3.3 (ii)] give a necessary and sufficient criterion for recurrence and transience, also in terms of generating functions. Unfortunately, "it is not easily applicable in specific cases" [LCP12,p. 136]. This raises the question whether a modification of our criterion (1) also holds for that model.
Max-autoregressive processes. By replacing the sum in (2) with the maximum we obtain the process M = (M n ) n≥0 defined by M 0 := Y 0 and (10) M n := max{A n M n−1 , Y n }, n ≥ 1.
Here the maximum is taken for each coordinate of R d separately. Such processes have been studied e.g. in [Gol91] and [RS95] and are sometimes called maxautoregressive. They appear naturally in our proof. If d = 1 then similarly to (3), M n = max n m=0 A n . . . A m+1 Y m for all n ≥ 0. For general dimension d ≥ 1 we have where the inequalities hold for each of the d components. We are not aware of any results in the literature on the classification of recurrence versus transience of max-autoregressive processes.
General random exchange processes. These are one-dimensional processes R = (R n ) n≥0 which have been studied e.g. in [HN76] and are defined as follows. Let (W n ) n≥0 be a sequence of nonnegative random variables with unbounded support and let (T n ) n≥1 be a sequence of real-valued random variables such that (T n , W n ) n≥1 is i.i.d. and independent of W 0 . Set R 0 := W 0 and (12) R n := max{R n−1 − T n , W n }, n ≥ 1.
The starting point of our investigation was the following classification of recurrence and transience of R in the special case where T n is a positive constant (random exchange process). To the best of our knowledge this observation was first made by Kesten in the appendix to [Lam70], where it was phrased in terms of long range percolation.
The claim follows.
The significance of Proposition A in the present context is that on a heuristic level one can easily deduce from it in several steps the recurrence/transience criterion for our processes X, Z and M introduced above. However, our actual proof will not follow these steps.
Step 1. If R satisfies only the more general recursion R n = max{R n−1 − c, W n } for some constant c ∈ N then the event {W 1 ≤ m} in (13) has to be replaced by {W 1 ≤ mc}.
Step 2. If we do not require the minimum y of the support of W 1 to be 0 then the event {W 1 ≤ mc} in Step 1 has to be replaced by {W 1 ≤ y + mc}. The same holds for any y satisfying P [W 1 ≤ y] > 0.
Step 3. It is easy to guess but much harder to prove that for the general random exchange process satisfying (12) and E[T 1 ] > 0 the event {W 1 ≤ y + mc} from Step 2 has to be replaced by {W 1 ≤ y + mE[T 1 ]}, see Corollary 7 below.
Step 4. The process e R is a one-dimensional max-autoregressive process M which satisfies the recursion M n = max{A n M n−1 , Y n } with A n = e −Tn and Y n = e Wn . It follows from Step 3 that if y is such that P [Y 1 ≤ y] > 0 then M should be recurrent iff n≥0 Step 5. By the strong law of large numbers, λ = −E[ln A 1 ] if d = 1. Thus for multi-dimensional max-autoregressive processes one should replace −E[ln A 1 ] in Step 4 by λ and get that for all y satisfying P [ Y 1 ≤ y] > 0, M is recurrent iff Step 6. It is a well-known phenomenon (max-sum-equivalence) that the sum of heavy tailed random variables tends to be comparable to the largest summand. Thus one might expect that the recurrence criterion for M derived in Step 5 also applies to X and, due to the relation between X and Z described above, to Z as well.
That the conclusion of this heuristics is indeed true under suitable conditions is the content of our main results, Theorem 3 and Theorem 5.
Let us now describe how the present article is organized. In the next section we provide additional definitions and collect some elementary statements. In Section 3 we first treat the special case of constant, deterministic environments because in this case our proof is shorter and requires weaker assumptions than in the genuinely random case. We also provide an application to so-called frog processes with geometric lifetimes. The general case of random environments is dealt with in Section 4, where we also give an application to random walks in random environments perturbed by cookies. In the appendix we collect some general bounds which we need in Section 4 but were not able to find in the literature.

Preliminaries
2.1. Notation. The ℓ p -vector norms (1 ≤ p ≤ ∞) on R d and their associated matrix norms are denoted by · p . We abbreviate · ∞ by · . Recall that for a matrix A, A is the maximum row sum and A 1 is the maximum column sum. The i-th coordinate of a vector x is denoted by [x] i and the (i, j)-th entry of a matrix A by By c 1 , c 2 , . . . we mean suitable strictly positive and finite constants which may depend on other constants.

2.2.
Branching processes. While branching processes are most often defined and studied in terms of generating functions we prefer to use a different, but equivalent definition which allows us to couple the branching process in a natural way to the AR(1) process introduced above, see (16) and (17) below.
Let d ≥ 1 and let Φ be the set of all measurable functions ψ : [0, 1] → N d 0 . An environment for a multitype Galton-Watson branching process is a sequence Here ψ n determines the reproduction behavior of the individuals in the (n − 1)-st generation, namely, if U is distributed uniformly is interpreted as the probability that an individual of type j in the (n−1)-st generation gets x i children of type i, i = 1, . . . , d.
and independent of Y 0 . The vector ⌊Y n ⌋ of integer parts of the components of Y n gives the numbers of immigrants of the d possible types who join the population at time n. Moreover, let (U j m,n,k ) 0≤m<n;1≤k;1≤j≤d be an i.i.d. family of random variables which are distributed uniformly on [0, 1]. Assume that this family is independent of Ψ and Y . Set We interpret ξ i,j m,n,k as the (random) number of children of type i of the k-th individual of type j in generation n − 1 whose ancestors immigrated at time m, provided that there are at least k individuals of this kind. Define for all m ≥ 0 the process (B m,n ) n≥m by setting B m,m := ⌊Y m ⌋ and Here [B m,n ] j stands for the number of individuals of type j at time n which descended from the individuals who immigrated at time m. Then the process (Z n ) n≥0 defined by Z 0 := ⌊Y 0 ⌋ and (or any other process with the same distribution) is called a branching process with immigration ⌊Y ⌋ in the random environment Ψ. Here [Z n ] j is the number of individuals of type j present at time n. The random matrix contains at position (i, j) the expected value, given the environment, of the number of children of type i of a member of type j of the (n − 1)-st generation. As above in the definition of the AR(1) process X, the sequence (A n , Y n ) n≥1 is i.i.d. and independent of Y 0 . It is well-known that for all 0 ≤ m ≤ n, see for example [Har63, Chapter II, (4.1)]. It follows from (3), (15), and ⌊Y ⌋ ≤ Y that for all n ≥ 0 a.s.
Observe that all four processes X, Z, M, and R defined above are order-preserving.
Let π be the transition kernel of such an order-preserving Markov chain. Then π (and any Markov chain with transition kernel π) is called irreducible for the state Then the processes X, Z, and M are irreducible for the state space [0, ∞) d .
Proof. Without loss of generality we assume that a.s. Y 1 ∈ N d 0 . Let µ be the minimum of the entries of the matrix A K+1 A K . . . A 2 and choose ε > 0 such that (17) and (11) that If π is irreducible then π (and any Markov chain with transition kernel π) is called recurrent iff there exists b ∈ (0, ∞) such that where V is a Markov chain with kernel π starting at 0. In fact, the initial state is not important here. Condition (18)  In order to deduce from the recurrence of one process the recurrence of another process we will need to infer from the divergence of a series of the form n≥0 a n the divergence of another series n≥0 b n . Sometimes we will do this by showing either sup n a n /b n < ∞ or inf n b n /a n > 0. Sometimes we shall use the following lemma instead.
Proof. By Markov's inequality, which is not summable in n by assumption.

Constant environment
Theorem 3. (Subcritical case, constant environment) Assume that there is a primitive matrix A with spectral radius ̺ < 1 such that a.s. A n = A for all n ≥ 1. Let y ∈ (0, ∞) be such that P [ Y 1 ≤ y] > 0. Then the following three assertions are equivalent.
The autoregressive processes X is recurrent.
The max-autoregressive process M is recurrent.
The branching process with immigration Z is recurrent. (ZR) Proof. By Proposition 1, X, Z, and M are irreducible since A is primitive. When checking (XR), (MR), and (ZR) we assume without loss of generality that Y 0 has the same distribution as Y n , n ≥ 1, since the initial state does not matter, see Section 2.3. Recall from Perron-Frobenius theory, see e.g. [KT75, Appendix, Theorem 2.3], that there is a matrix H ∈ (0, ∞) d×d such that We consider the following auxiliary conditions. (Recall (11) for the definition of Therefore, by (3) and since Y is i.i.d. we have for all n ≥ 0 that Applying Lemma 2 to (U, V ) = (N, X) we obtain the claim (XR).
This implies the claim.
(ZR)⇒(RC): Denote by q j,ℓ the probability that a given individual of type j does not have any descendants ℓ generations later, i.e. with a slight abuse of notation, q j,ℓ := P [B 0,ℓ = 0 | ⌊Y 1 ⌋ = e j ], where e j ∈ Z d is the j-th standard unit vector. Due to the moment assumption on ξ i,j 0,1,1 and [JS67, Theorems 2 (3.6) and 4], ̺ −k (1 − q j,k ) tends as k → ∞ for all j = 1, . . . , d, to a strictly positive and finite limit. In particular, there are c, ℓ ∈ N such that Set Z n := n−ℓ m=0 B m,n for n ≥ ℓ. By (ZR) and subadditivity there is some z ∈ N d 0 such that n P [Z n = z] = ∞. Then on the one hand by the Markov property and independence, where G is the cumulative distribution function of ⌊Y 1 ⌋ . Note that G(y) > 0 and setḠ := 1 − G. Then by the above for all n ≥ ℓ, We set f (t) := (ln(c/y) + ln(− ln t)) /(− ln ̺) and use the telescopic form of the sum in (25) for estimating this sum for all t ∈ (0, e −y/c ) from above by since all the differences in (26) are nonpositive. Since exp(−y/c) 0 f (t) dt < ∞, the right-hand side of (25) is bounded from above uniformly in n. Therefore, (24) implies that n m=ℓ G(y̺ −m ) is not summable. Since G(x) ≤ P [ Y 1 ≤ 2x] for all x ≥ 1, (RC) follows.
3.1. An application to mortal frog processes. For a survey on frog processes we refer to [Pop03]. The following application is related to [Pop03, Theorem 4.3]. Let (Y n ) n≥0 be an i.i.d. sequence of N 0 -valued random variables. Put on each n ≥ 0 a number Y n of sleeping frogs. Fix p, r ∈ (0, 1). Wake up the frogs at 0 (if there are any). Once woken up, every frog performs a nearest-neighbor random walk, jumping independently of everything else with probability r to the right and with probability 1−r to the left, until it dies after an independent number of steps which is geometrically distributed with parameter 1 − p and may be 0. Whenever a frog visits a site with sleeping frogs those frogs are woken up as well and start their own independent lives.
Almost surely only finitely many different frogs visit 0.
Almost surely only finitely many frogs are woken up.
Proof. Let a ± ∈ (0, 1) be the probability that a frog which starts at 0 ever hits ±1 before it dies.
(28)⇔(29): By conditioning on the first step we see that a + satisfies a + = pr + p(1−r)a 2 + and get a + = ̺. Assign to each frog an a.s. finite trajectory which the frog will follow once it has been woken up. For any 0 ≤ m ≤ n let B m,n be the number of frogs originally sleeping at m whose trajectories reach the site n. Then for all m ≥ 0, B m,m = Y m and (B m,m+k ) k≥0 is a Galton-Watson branching processes with offspring distribution Bernoulli(a + ). Moreover, the processes (B m,m+k ) k≥0 , m ≥ 0, are independent. Hence, if we denote by Z n , n ≥ 0, the total number of frogs originating in {0, 1, . . . , n} whose trajectories visit n then (Z n ) n≥0 is a subcritical branching process with immigration. By Theorem 3, (Z n ) n≥0 is recurrent iff (29) holds. On the other hand, (Z n ) n≥0 is recurrent iff there is a.s. an n ≥ 1 such that Z n − Y n = 0, i.e. iff there is a site n which is never visited, which is equivalent to (28).
We need the following rather mild regularity condition on the distribution of Y 1 .
The autoregressive processes X is recurrent.

(XR)
The max-autoregressive process M is recurrent.
If we assume in addition (BD2) then (XR), (MR), and (RR) are equivalent to the following statement.
The branching process with immigration Z is recurrent.
Proof of Theorem 5. By Proposition 1 and (PR), X, Z, and M are irreducible. As in the proof of Theorem 3 we assume that Y 0 has the same distribution as Y n , n ≥ 1.
Denote by A ⊆ [0, ∞) d×d the support of A 1 . Due to (BD1) and (PR) the assumptions of Lemma 9 from the appendix are satisfied.
(ZR)⇒(R+): Set P Ψ [·] := P [ · | Ψ]. Denote by q Ψ,j,m,n the probability that in the environment Ψ a given individual of type j who immigrated at time m does not have any descendants at time n, i.e. with a slight abuse of notation, q Ψ,j,m,n = P Ψ [B m,n = 0 | ⌊Y m ⌋ = e j ]. Proposition 10 and (BD2) yield that for all j = 1, . . . , d, a.s.
In summary, it follows from (37) and (40)-(43) that if the expression in (43) is finite then n≥0 n i=0 L(a + λi + g(i)) = ∞. We shall use this fact for two functions g of the form g(i) = ζi η with ζ > 0 and 0 < η ≤ 1. For any such g we have due to Lemma 12 for all i ≥ 0, First, fix ε > 0 and let g(i) := εi for i ∈ N 0 . In this case the term in (43) can be bounded from above by c 8 E exp 2L(a)T g , which is finite for large enough a since T g has some finite exponential moment due to (44). Therefore, the assumptions of Lemma 6 are satisfied with ⌊Y 1 ⌋ instead of Y 1 . Consequently, a can be chosen large enough such that (32) holds withL instead ofF . Consider now g(i) := λi α . Then the expression in (43) is finite due to the same computation as in (33) and (34), where we use (44) instead of (30). This proves (R+) with L instead of F . Since L(x) ≤ F (x + 1) for all x ≥ 0, (R+) follows.
Therefore, it is enough to show that F (g + (i)) − F (g − (i)) is summable in i. Let m ∈ N be large enough such that t ≥ 2t α for all t ≥ m and set η := ln Y 1 . Then However, on the event {g −1 − (η) ≥ m}, by definition of g ± and our choice of m, Therefore, the expression in (45) is finite due to E[η α ] < ∞.
By exponentiating R we obtain from Theorem 5 the following generalization of Proposition A.
Corollary 7. (General random exchange process) Assume that T 1 is bounded and E[T 1 ] > 0. Moreover, suppose that there exists β ∈ (2/3, 1) such that lim 4.1. An application to random walks in random environments perturbed by cookies of maximal strength. We consider the same version of excited random walks in random environment as Bauernschubert in [Bau13]. Let ω = (ω x ) x∈Z be an i.i.d. family of (0, 1)-valued random variables and Y = (Y x ) x∈Z be an i.i.d. family of N 0 -valued random variables such that P [Y 0 = 0] > 0. We call ω x the environment at x and Y x the number of cookies at x. The random walk ξ = (ξ n ) n≥0 in the random environment ω perturbed by the cookies Y is defined as follows. The walk starts at ξ 0 = 0. Upon any of the first Y x many visits to a site x the walker reduces the number of cookies at that site by one and then moves in the next step deterministically to x + 1. Upon the (Y x + 1)-st or any later visit to x, i.e. when there are no cookies left at x, the walker jumps independently of everything else with probability ω x to x + 1 and with probability 1 − ω x to x − 1. More formally, for all n ≥ 0 and z = ±1 a.s.
We consider the case E[ln ρ 0 ] > 0 in which the underlying RWRE is transient to the left and ask how many cookies are needed in order to make this walk recurrent or even transient to the right. Using (8), (9), and a well-known relationship between excursions of random walks and branching processes, Bauernschubert obtained in [Bau13] the following result.
Replacing in the proof of this theorem (8) and (9) by Theorem 5 we obtain the following complete characterization of recurrence/transience of ξ in the so-called uniformly elliptic case where the transition probabilities ω x are bounded away from 0 and 1.
Theorem 8. Assume that (ω x , Y x ) x∈Z is independent and that there is an ε > 0 such that a.s.
is infinite then ξ is a.s. recurrent. (c) If the series in (46) is finite then ξ is a.s. transient to the right.
Appendix. Bounds for the case of random environment Lemma 9. Let γ, K ∈ N, 0 < κ ≤ 1 and A ⊆ [0, ∞) d×d . For n ∈ N 0 set G n := {A 1 . . . A n : A 1 , . . . , A n ∈ A} and G := n≥0 G n . Assume that A ≤ γ for all A ∈ A and G K ⊆ [κ, ∞) d×d . Then there is a constant c = c(γ, K, κ, d) such that for all n ≥ K, A ∈ G n , and (49) Proof. For any matrix A let µ(A) := min j max i [A] i,j . The following two quantities are used to measure the variation among the entries of A.
for A ∈ [0, ∞) d×d \{0} and We first show the following relations.
To prove (54) let k be such that max i [A] i,k = µ(A). Then This concludes to proof of (51)-(54). Next we show that sup{∆ A : A ∈ G n , n ≥ K} < ∞ and (56) First note that c 9 := sup{δ A : A ∈ A} < ∞. Indeed, let A ∈ A and B ∈ G K−1 .
no element of A has a column of zeros. Hence, G n ⊆ (0, ∞) d×d for all n ≥ K. Therefore, if we let K ≤ n = mK + r with m ≥ 1 and 0 ≤ r < K then for all A 1 , . . . , A n ∈ A, where we used in the last step that ∆ B ≤ B /κ ≤ γ K /κ for any B ∈ G K . This implies (56). Moreover, (53) implies δ A ≤ c K 9 for all A ∈ G n , n ≤ K, and (54) and (58) imply δ A ≤ dc 10 for all A ∈ G n , n ≥ K. Together this yields (57).
For the proof of the first claim of the Lemma, (47), let k be such that x = x k . Then for all A ∈ G, Along with (57) this implies (47). The second claim, (48), follows from (47) and the definition of the matrix norm · . For the proof of (49) let n ≥ K and A ∈ G n . Then This along with (56) implies (49). The last claim, (50), follows from κ ≤ A K ≤ A K .
The following result provides bounds on the extinction time of multitype branching process in varying environment. The easy bound is standard and based on a first moment method, i.e. Markov's inequality. To the best of our knowledge the opposite bound appeared first in a similar form in [Agr75, Theorem 1]. We prove it by the second moment method. For precise asymptotics under different assumptions see e.g. [JS67], [Dya08]. Define the matrices A n := E ξ i,j n,1 i,j=1,...,d , n ≥ 1, and suppose that γ, K ∈ N, κ ∈ (0, 1] and A := {A n | n ≥ 1} satisfy the assumptions of Lemma 9. Denote for n ≥ 1 and j = 1, . . . , d by V j n the covariance matrix of the vector (ξ i,j n,1 ) i=1,...,d and suppose that c 11 := sup n≥1,j=1,...,d V j n < ∞. Then there is a constant c 5 = c 5 (γ, K, κ, d, c 11 ) such that for all n ≥ 1, Substituting this into (60) yields the claim.
Since η ≤ 1 this yields the claim.