Excursions of excited random walks on integers

Several phase transitions for excited random walks on the integers are known to be characterized by a certain drift parameter delta. For recurrence/transience the critical threshold is |delta|=1, for ballisticity it is |delta|=2 and for diffusivity |delta|=4. In this paper we establish a phase transition at |delta|=3. We show that the expected return time of the walker to the starting point, conditioned on return, is finite iff |delta|>3. This result follows from an explicit description of the tail behaviour of the return time as a function of delta, which is achieved by diffusion approximation of related branching processes by squared Bessel processes.


Introduction
.4]. "Under fairly general conditions, biased walks are strongly transient" [Hug95,p. 127]. In the present paper we study the tail behavior of the depth and the duration of excursions of excited random walks (ERWs). In particular, we show that ERWs can be biased, in the sense of satisfying a strong law of large numbers with non-zero speed, and at the same time be not strongly transient. Precise statements are given later in this section after we describe our model of ERW. (For a recent survey on ERW we refer the reader to [KZ13]. ) An ERW evolves in a so-called cookie environment. These are elements ω = (ω(z, i)) z∈Z,i≥1 of Ω := [0, 1] Z×N . Given ω ∈ Ω, z ∈ Z and i ∈ N we call ω(z, i) the i-th cookie at site z and ω(z, ·) the stack of cookies at z. The cookie ω(z, i) serves as transition probability from z to z + 1 of the ERW upon its i-th visit to z. More precisely, given ω ∈ Ω and x ∈ Z an ERW starting at x in the environment ω is a process (X n ) n≥0 on a suitable probability space (Ω ′ , F ′ , P x,ω ) which satisfies for all n ≥ 0: P x,ω [X 0 = x] = 1, P x,ω [X n+1 = X n + 1 |(X i ) 0≤i≤n ] = ω(X n , #{i ≤ n |X i = X n }), (1) P x,ω [X n+1 = X n − 1 |(X i ) 0≤i≤n ] = 1 − ω(X n , #{i ≤ n |X i = X n }).
The environment ω is chosen at random according to some probability measure P on (Ω, F ), where F is the canonical product Borel σ-field. Throughout the paper we assume that P satisfies the following hypotheses (IID), (WEL), and (BD M ) for some M ∈ N 0 := N ∪ {0}.
If we assumed only (IID) and (WEL) the model would include RWs in random i.i.d. environments (RWRE), since for them P-a.s. ω(0, i) = ω(0, 1) for all i ≥ 1. However, for the ERW model considered in this paper we assume that there is a non-random M ≥ 0 such that after M visits to any site the ERW behaves on any subsequent visit to that site like a simple symmetric RW: (BD M ) P-a.s. ω(z, i) = 1/2 for all z ∈ Z and i > M.
If we average the so-called quenched measure P x,ω defined above over the environment ω we obtain the averaged (often also called annealed) measure P x [·] := E[P x,ω [·]] on Ω × Ω ′ . The expectation operators with respect to P x,ω , P, and P x are denoted by E x,ω , E, and E x , respectively. Several features of the ERW can be characterized by the parameter which represents the expected total average displacement of the ERW after consumption of all the cookies at any given site. Most prominently, the ERW (X n ) n≥0 • is transient, i.e. tends P 0 -a.s. to ±∞, iff |δ| > 1 (see [KZ13,Th. 3.10] and the references therein), • is ballistic, i.e. has P 0 -a.s. a deterministic non-zero speed lim n→∞ X n /n, iff |δ| > 2 (see [KZ13,Th. 5.2] and the references therein), • converges after diffusive scaling under P 0 to a Brownian motion iff |δ| > 4 or δ = 0 (see [KZ13,Theorems 6.1, 6.3, 6.5, 6.6, 6.7] and the references therein). In this paper we are concerned with the finite excursions of ERWs. Let R := inf{n ≥ 1 : X n = X 0 } be the time at which the RW returns to its starting point. Denote for k ∈ Z the first passage time of k by T k := inf{n ≥ 0 : X n = k}.
Remark 4. (Once-excited RWs) In the case of once-excited RWs with identical cookies (i.e. M = 1, P-a.s. ω(z, 1) = ω(0, 1) ∈ (0, 1) for all z ∈ Z), results (3) and (4) have been obtained in [AR05, Section 3.3]. Note that the case M = 1 is very special, since at time T k all the cookies ω(z, i) = 1/2 between the starting point 0 and the current location k of the ERW have been "eaten". This allows to use simple symmetric RW calculations between 0 and k. For M ≥ 2 such simplification is not available.
Problem 5. Find necessary and sufficient criteria under which RWRE in one dimension is strongly transient.
Our approach is based on the connection between ERWs and a class of critical branching processes (BPs) with random migration (see Section 2 for details). It is close in spirit to the (second) Ray-Knight theorem (see, for example, [Tóth96], where similar ideas were used for other types of self-interacting RWs). This approach was proposed for ERWs in [BS08] and, since then, seemed to dominate the study of one-dimensional ERWs under the (IID) assumption. The main benefits gained from this connection are: (i) BPs associated to ERWs are markovian, while the original processes do not enjoy this property; (ii) after rescaling, these BPs are well approximated by squared Bessel processes of generalized dimension. From these diffusion approximations one can immediately conjecture such important properties of BPs as survival versus extinction, the tail asymptotics of the extinction time and of the total progeny (conditioned on extinction where appropriate). Rigorous proofs of these conjectures are somewhat technical, but, in a nutshell, they are based on standard martingale techniques applied to gamblerruin-like problems.
Diffusion approximations for BPs associated to ERWs and the mentioned above martingale techniques were used in [KM11] to study the tail behavior of regeneration times of transient ERWs, which led to theorems about limit laws for these processes. In the current work we extend some of the results and techniques of [KM11] and, in addition, apply the Doob transform to treat BPs conditioned on extinction. The results for conditioned BPs are then readily translated into the proof of Theorem 1.
While the majority of results in the BPs literature rely on generating functions approach, diffusion approximations of BPs is also a well-developed subject. Its history goes back to [Fel51] (see [EK86,Chapter 9] for precise statements and additional references). But it seems that diffusion approximations for our kind of BPs are not available in the literature. Moreover, among a wealth of results (obtained by any approach) about conditioned BPs we could not find those which would cover our needs (but see the related work [Mel83] and the references therein).
We would like to point out one more aspect of the relationship between ERWs and BPs. At first (see, for example, [BS08], [KZ08]) there was a tendency to use known results for BPs to infer results about ERWs. Gradually, as we mentioned above, the study of ERWs required additional results about BPs, not covered by the literature. In [KM11] all BP results needed for ERWs were obtained directly. In this work we continue the trend. Theorem 21 gives asymptotics of the tails of extinction time and the total progeny of a class of critical BPs with random migration and geometric offspring distribution conditioned on extinction. We believe that this result might be of independent interest and that our methods are sufficiently robust to be applicable to more general critical BPs with random migration.
Let us now describe how the present article is organized. We close the introduction with some notation. In the next section we recall how excursions of ERWs are related to certain BPs. Section 3 deals with diffusion approximations of these BPs. In Section 4 we prove that BPs conditioned on extinction can be approximated by the diffusions from Section 3 conditioned on hitting zero. In Section 5 we use these results to obtain tail asymptotics of the extinction time and of the total progeny of BPs conditioned on extinction. Short Section 6 translates the obtained asymptotics into the proof of Theorem 1. In the Appendix we collect and extend as necessary several auxiliary results from the literature, which we quote throughout the paper and which do not depend on the results from Sections 3-6.
Notation. For any I ⊆ [0, ∞) and f : I → R we let σ f y := inf {t ∈ I : f (t) ≤ y} and τ f y := inf {t ∈ I : f (t) ≥ y} be the entrance time of f into (−∞, y] and [y, ∞), respectively. (Here inf ∅ := ∞.) If Z is a process with P [σ Z 0 < ∞] > 0 then we denote by Z any process which has the same distribution as Z under P [ · | σ Z 0 < ∞]. Whenever X is a Markov process starting at time 0 we indicate the starting point X(0) = x in expressions like P x [X ∈ A] by the subscript x to P . The space of real-valued càdlàg functions on [0, ∞) is denoted by D[0, ∞) and convergence in distribution by ⇒.

Excursions of RWs and branching processes
We recall a relationship between nearest neighbor paths from 1 to 0, representing RW excursions to the right, and BPs. Among the first descriptions of this relation is [Har52, Section 6]. We refer to [KZ08, Sections 3, 4] and [Pet, Section 2.1] for detailed explanations in the context of ERW.
Assume that the nearest neighbor random walk (X n ) n≥0 starts at X 0 = 1, set U 0 := 1 and let for k ≥ 1, be the number of upcrossings from k to k + 1 by the walk before time T 0 . If we set ∆ (k) is, if finite, the time of the completion of the m-th downcrossing from k to k − 1 prior to T 0 . We define to be the number of upcrossing from k to k + 1 between the (m − 1)-th and the m-th downcrossing from k to k − 1 before T 0 . Then U k+1 can be represented in BP form as can be interpreted as the number of children of the m-th individual in the k-generation. The joint distribution of these numbers depends on the RW model under consideration. In the case of ERW it may be quite complicated, especially in the case where T 0 = ∞ with positive probability. Therefore, we study instead of U a slightly different BP V , the so-called forward BP described in the following statement.
Proposition 6. (Coupling of ERW and forward BP) Assume we are given M ∈ N and an ERW X = (X n ) n≥0 which satisfies (IID), (WEL) and (BD M ). Then one may assume without loss of generality that there are on the same probability space N 0 -valued random variables ξ (k) m , m, k ≥ 1, which define a Markov chain V = (V k ) k≥0 by V 0 := 1 and such that under P 1 the following holds: The random quantities (ξ The random vectors (ξ where U is defined by (7).
Proposition 6 follows from the so-called coin-toss construction of ERW described in [KZ08, Section 4], see also [Pet,Section 2]. Note that the above conditions (8)-(12) do not completely characterize the distribution of V . For this statement (10) would have to be made stronger. However, we refrain from doing so, since the conditions (8)-(12) are the only ones we need for our proofs to work. (The moment condition in (10) is inherited from the proof of [KM11, Lemma 5.2] and could be relaxed.) Indeed, we only make the following assumptions on V .
Assumptions on the offspring ξ and the BP V . For the remainder of the paper we assume that the Markov chain V is defined by (8), where the offspring variables ξ (k) m , m, k ≥ 1, satisfy (9)-(12).
Remark 7. (Cookies of strength 1 and BPs with immigration) In [KZ08, p. 1960] we describe how the above process V can be viewed as a BP with migration, i.e. emigration and immigration. If (IID) and (WEL) hold, but not necessarily (BD M ), and if there is P-a.s. some random K ∈ N ∪ {∞} such that ω(0, i) = 1 for all 1 ≤ i < K and ω(0, i) = 1/2 for all i ≥ K then one can couple the ERW in a way similar to the one described in Proposition 6 to a BP with immigration without emigration, see e.g. [ [SW69] about the extinction of Galton-Watson processes in random environment, see also [AN72, Ch. VI.5]. In [Afa99,p. 268] this relationship is shown to imply that for recurrent RWRE P 1 [T 0 > n] ∼ c/ log n as n → ∞ for some constant 0 < c < ∞. In [KZ08, Th. 1] we used this correspondence and results from [FYK90] for a proof of the recurrence/transience result about ERW mentioned above, see also Corollary 34 below. In [Bau13] and [Bau] this connection is used to determine how many cookies (of maximal value ω(x, i) = 1) are needed to change the recurrence/transience behavior of RWRE. And in [Pet, Th. 1.7] strict monotonicity with respect to the environment of the return probability of a transient ERW is shown to be inherited from monotonicity properties of this BP.

.2], and [Pet, Section 2.2].
We notice that all results of [KM11] about backward BPs have the corresponding analogs for forward BPs, which are obtained by replacing δ (which is assumed to be positive in [KM11]) with 1−δ < 1 throughout. The proofs carry over essentially word for word without any additional changes. In what follows we simply quote such results. All additional results about forward BPs, in particular, for δ > 1, are supplied with detailed proofs or comments as appropriate.

Diffusion approximation of unconditioned branching processes
The main result of this section is Theorem 13 about diffusion approximations of the process V . It extends [KM11, Lemma 3.1], which only considered the process V stopped at σ V εn with ε > 0. The limiting processes are defined in terms of solutions of the stochastic differential equation (SDE) where (B(t)) t≥0 is a one-dimensional Brownian motion.
In order to obtain these diffusion approximations we first introduce a modification V of the original process V and state in Proposition 11 a functional limit theorem for this process. The advantage of this process V is that it admits some nice martingales. Note that (8) can be rewritten as This recursion is modified below in (17).
Proof. By (10)-(12), This implies the first statement. To find the Doob decomposition of the sub- Recalling (19) we obtain the second claim.
Proposition 11. Let (x n ) n≥1 be a sequence of positive numbers which converges to x > 0, let δ ∈ R and assume that ξ satisfies (9)-(12). For each n ∈ N define V n = ( V n,k ) k≥0 and Y n = ( Y n (t)) t≥0 by setting V n,0 := ⌊nx n ⌋ and Proof. We are going to apply [EK86, Th. 4.1, p. 354]. To check the assumptions of this theorem we first let µ be a distribution on R and consider the C where Gf := (a/2)f ′′ + δf ′ and a(x) := 2x + . This martingale problem is well posed due to [EK86,Cor. 3.4, p. 295] and our discussion above after (15) concerning existence and distributional uniqueness of solutions of (15). Now define for each n ∈ N, (M n,k ) k≥0 and (A n,k ) k≥0 in terms of V n,k as in (18) and (19) We are now going to check conditions (4.1)-(4.7) of [EK86, Th. 4.1, p. 354]. By Lemma 10, M n − B n and M 2 n − A n are martingales for all n ∈ N, i.e. conditions (4.1) and (4.2) are satisfied. To verify the remaining conditions (4.3)-(4.7) we fix r, T ∈ (0, ∞) and set τ n,r : This is a consequence of (12) and the fact that the geometric distribution has exponential tails. More precisely, The first term in the last line goes to 0 as n → ∞ and the second term is equal to The first term in the last line vanishes as n → ∞. Applying the union bound and Lemma 28 to the last probability we find that the second term does not exceed 4rT y>(rn) 3/2 e −y/(6(rn∨ √ y)) ≤ 4rT This finishes the proof of (21 For (4.7) we consider for all t ≤ T ∧ τ n,r , which does not depend on t any more and converges to 0 as n → ∞. Thus, (4.7) holds as well. The theorem follows now from [EK86, Th. 4.1, p. 354].
To be able to apply the continuous mapping theorem to Proposition 11 we need the following statement. Define for every f ∈ D[0, ∞) and y ∈ R by Lemma 12. Let δ ∈ R, 0 < ε < x < ∞ and let ψ be any of the following three mappings defined on D[0, ∞):  For , which converges to 0 as n → ∞, see e.g. [JS87, Ch. VI, Prop. 1.17b]. If σ f ε = ∞, then for any T < ∞, σ fn ε ≥ T for n large and thus Theorem 13. (Convergence of unconditioned processes) Let (x n ) n≥1 be a sequence of positive numbers which converges to x > 0, let δ ∈ R\{1}, and assume (9)-(12). For each n ∈ N define V n = (V n,k ) k≥0 and Y n = (Y n (t)) t≥0 by setting V n,0 := ⌊nx n ⌋ and Let Y be a solution of (15) with Proof. Let V n and Y n be defined as in Proposition 11, where V n,0 = V n,0 for all n.
Now we consider the case δ < 1. Our first goal is to show that We aim to use [Bil99, Th. 3.2], quoted as Lemma 25 in the Appendix, for this purpose. First observe that for all m ∈ N, ϕ 1/m ( Y n ) as n → ∞ due to Proposition 11, Lemma 12 and the continuous mapping theorem. Moreover, ϕ 1/m (Y ) =⇒ ϕ 0 (Y ) as m → ∞ since Y has a.s. continuous paths. For the proof of (22) it therefore suffices to show, due to Lemma 25, that For the proof of (23) we use [Bil99,(12.16)] and see that for all ε > 0, n ∈ N, and y ∈ (0, x ∧ ε), The first term in (24) is 0 for large enough n since Lemma 32 now yields (22) if we choose y = 1/m. If we choose y = M/n then the above estimate and Lemma 32 give that Consequently, by (22) and [Bil99, However, recall that V n,k = V n,k for all 0 ≤ k ≤ σ Vn M and therefore ϕ M/n Y n = ϕ M/n (Y n ). Hence, ϕ M/n (Y n ) J 1 =⇒ ϕ 0 (Y ) as n → ∞. As in (25), d • ∞ (ϕ M/n (Y n ), ϕ 0 (Y n )) tends to 0 in distribution as n → ∞. The claim for δ < 1 now follows from another application of [Bil99, Th. 3.1]. (Note that ϕ 0 (Y n ) = Y n since 0 is absorbing for V .)

Diffusion approximation of conditioned branching processes
The main result of this section is the following. Recall that V is obtained from V by conditioning on {σ V 0 < ∞}. In particular, by Corollary 33, V = V if δ < 1. Theorem 14. (Convergence of conditioned processes) Assume the conditions of Theorem 13 and let Y = (Y (t)) t≥0 be a solution to Then as n → ∞, In the proof the (harmonic) function h defined by will play an important role. Note that it follows from (8) that h(x) is nonincreasing in x.
Remark 15. (Doob transform) Recall that V is Doob's h-transform of V with h as defined in (30), see e.g. [LPW09, Chapter 7.6.1]. By this we mean that V is a Markov chain with transition probabilities P x [V n = y] = P x [V n = y] h(y) h(x) . More generally, it follows from the strong Markov property, that for any stopping time σ ≤ σ V 0 and all x, y ∈ N 0 , In many cases, a Doob transform of a process belongs to the same class of processes as the process itself. For example, the asymmetric simple RW on Z with probability p ∈ (1/2, 1) of stepping to the right, start at 1 and absorption at 0 is, conditioned on hitting 0, an asymmetric simple RW on Z with probability p of stepping to the left. If similarly X, i.e. X conditioned on hitting 0, were under P 1 an ERW satisfying (IID), (WEL) and (BD M ) for some M , or, equivalently, if V were of the form described in Proposition 6 then Theorem 14 would follow from Theorem 13. However, we do not expect that the conditioned processes X and V are of this form on the microscopic level. Theorem 14 shows that nevertheless on a macroscopic scale V does behave as V with drift parameter δ = 1 − |δ − 1|.
Lemma 18. Let δ ∈ R\{1}. Then there is c 5 ∈ (0, 1) such that for all k, x ∈ N and n ≥ 0, By the strong Markov property and monotonicity with respect to the starting point we have for all 2M ≤ x ≤ z < 2x, The last expression is strictly positive for all x > 0 and converges due to Proposition 11 and Lemma 12 as x → ∞ to P 2 [σ Y 1/2 < 1] > 0, where Y solves the SDE (15). This proves (35).
Next we show that there is c 7 > 0 such that for all x ∈ N, For the proof of (36) note that by the strong Markov property and monotonicity of h for all 0 ≤ z ≤ x/2, z ∈ N 0 , which converges due to Proposition 16 to 2 1−δ < 1 as x → ∞. Since the left hand side of (36) is strictly positive for all x this implies (36). Now define ρ i := inf{n > ρ i−1 + 2x : V n ∈ [x, 2x)} for all i ∈ N. Then the left hand side of (34) is less than or equal to By the strong Markov property for V , by (35) and (36). Substituting this into (37) and iterating gives the claim with c 5 := 1 − c 6 c 7 .
The next lemma states that Lemma 32 also holds for V .
Proof. By the strong Markov property and monotonicity of h, for all t > 1, due to Proposition 16. Since t 1−δ → 0 as t → ∞ and P n [σ V 0 < ∞] = 1 for all n ∈ N by definition of V this finishes the proof.
Thus for large enough n the right-hand side of (43) is less than or equal to Here we used in the last step that P [ϕ n,ε < ε/2] decays exponentially fast as n → ∞ due to (68). Letting γ ց 0 yields lim n A n = 0. This proves (27). For the next statement, (28), we shall use Lemma 25. It follows from (27), Lemma 12, the continuous mapping theorem, and the a.s. continuity of Y that To verify the condition corresponding to (63) fix ε > 0 and let c 9 (·) be the function of Lemma 20. Choose g : N → (0, ∞) such that c 9 (g(k)) ≤ εk for all k ∈ N and g(k) → 0 as k → ∞. Then for all k, n ∈ N, due to Lemma 20. Letting first n → ∞ and then k → ∞ implies the condition corresponding to (63). Consequently, Lemma 25 shows (28). The final statement, (29), is obtained similarly. As above, it follows from (27), Lemma 12, the continuous mapping theorem, and the a.s. continuity of Y that Moreover, for all ε ∈ (0, 1), as in (24) and due to (45). Letting first n → ∞ and then k → ∞ this converges to 0 due to Lemma 19 and the choice of g. Having verified the assumptions of Lemma 25 statement (29) follows from this lemma.

Asymptotics of diffusions and branching processes
In this section we derive certain asymptotics of the conditioned BP V from those of the approximating squared Bessel process as stated in Lemma 26. Our goal is the following result: Theorem 21. Let V = (V n ) n≥0 be defined as in (8) such that (9)-(12) holds with V 0 = 1 and let δ ∈ R\{1}. Then there are c 10 , c 11 ∈ (0, ∞) such that Remark 22. In the special case described in Remark 7 formula (46) follows for 0 < δ < 1 from the tail behavior of the extinction time of recurrent critical BPs with immigration described in [Zub72, Th. 2, first part of (21)]. In this case it is for δ > 1 reminiscent of the unproven claims made in [IS85,  Lemma 23. Let δ ∈ R\{1}. Then there is c 12 ∈ (0, ∞) such that lim n→∞ n |δ−1| P 1 τ V n < σ V 0 = c 12 .
Proof. For δ < 1 the statement follows from [KM11, (C), Lem. 8.1]. Now let δ > 1. Then by the strong Markov property and monotonicity of h, due to Proposition 16. On the other hand, for all γ > 1 by the strong Markov property and monotonicity of h, where we used in the last identity (67) and Proposition 16. Lower bound in (46). Fix x > 0. Then by the strong Markov property for V we have for all n ≥ 0, Now we choose suitable y n ∈ [x, 2x] where the infimum in (49) is attained. Then the expression in (49) can be estimated from below by The second term above vanishes as n → ∞ due to (69). Therefore, by Lemma 23, for all x > 0, for some increasing sequence (n k ) k≥0 . Choose a subsequence (m k ) k≥0 of (n k ) k≥0 along which (y m k ) k≥0 converges to some y ∈ [x, 2x]. Then by (28), the limit in (50) is equal to P y [σ Y 0 > 1] where Y solves the SDE (26). By scaling (Lemma 26), as x ց 0 by (64). This proves the lower bound in (46).
Upper bound in (46). Fix ε ∈ (0, 1) and estimate for all x > 0 and n ≥ 0, The expression in (53) vanishes as n → ∞ due to (69). The term in (54) vanishes as well due to Lemma 24 if we let first n → ∞ and then x ց 0. For the treatment of (52) let B n := σ V 0 > (1 − ε)n for n ≥ 0. Then by the strong Markov property the quantity in (52) is less than or equal to By choosing suitable y n ∈ [x, (1 + ε)x] where the supremum in (55) is attained the expression in (55) can be written as As above the first factor converges to c 12 as n → ∞ by Lemma 23. Summarizing we get that for all ε ∈ (0, 1), for suitable increasing sequences (n k,x ) k≥0 , x > 0. Choose for all x > 0 a subsequence (m k,x ) k≥0 of (n k,x ) k≥0 along which (y m k,x (x)) k≥0 converges to some y(x) ∈ [x, (1 + ε)x] and let Y solve the SDE (26). Then by (28), the expression in (56) is equal to This completes the proof of (46).
Indeed, for all x > 0 due to (46) Lower bound in (57). Fix x > 0. Then (58) This can be estimated from below by the expression in (48). By the same argument as after (48), using (29) instead of (28), we obtain for a suitable y ∈ [x, 2x] and a solution Y of (26), Upper bound in (57). We estimate the term in (58) from above for all n ≥ 0 and x, ε ∈ (0, 1) by As above, see (53) and the second term on the right-hand side of (51), the sum in (60) is negligible. The expression in (59) can be estimated from above by the strong Markov property by the expression in (55), where B n := By the same argument as after (55), using (29) instead of (28), we obtain for suitable y(x) ∈ [x, (1 + ε)x] and Y a solution of (26), −→ c 11 as ε ց 0.
As for the remaining claims about the tail of R consider the first step of the walk and use the fact that P x,ω [n ≤ T 0 < ∞] does not depend on ω(0, ·) to obtain that due to (IID). Because of (WEL) both E[ω(0, 1)] > 0 and E[1 − ω(0, 1)] > 0.
and ω is distributed under P like 1 − ω under P. The parameter δ for P is equal to −δ.

Lemma 26. (Scaling of squared Bessel processes)
Then there are constants a(δ), b(δ) ∈ (0, ∞) such that Proof. Statements (64)  In fact, with respect to (64) much more is known. In [GY03, (15)] a formula for the density of the first passage time to 0 of a Bessel process with dimension in [0, 2) is given. This formula implies (64) in the case 0 ≤ δ < 1. It has been noticed in the remark after [Ale11, (4.24)] that the same formula also holds for negative dimensions. Then for all x, y ∈ N, Proof. We are going to use a special case of Azuma's inequality, which states that for the simple symmetric RW (S n ) n≥0 on Z starting at 0 and any a, n ≥ 0, P [S n ≥ a] ≤ exp(−a 2 /(2n)), see e.g. [AS00, Theorem A.1.1]. Let (Y i ) i≥1 be an independent sequence of Bernoulli(1/2)-distributed random variables. By interpreting ξ i + 1 as the time of the first appearance of "heads" in a sequence of independent fair coin flips we obtain ≤ e −y 2 /(2(2x+y−1)) ≤ e −y 2 /(6(x∨y)) by Azuma's inequality. Similarly, for x ≥ y, = P [S 2x−y ≥ y] ≤ e −y 2 /(2(2x−y)) ≤ e −y 2 /(6(x∨y)) again by Azuma's inequality. For x < y the quantities in (66) are 0. A union bound now yields the claim.
The following lemmas about the BPs V, V , and V are slight modifications of results from [KM11]. The first one controls how much the BPs V and V "overshoot" x at the times τ x and σ x .
Lemma 29. (Overshoot) There are constants c 13 , c 14 > 0 and N ∈ N such that for every x ≥ N, y ≥ 0, and ε > 0, For the third statement note that by definition of V , x since by the strong Markov property and monotonicity with respect to the starting point, the fraction above does not exceed 1 for all 0 < z < x. An application of (67) with y = εx completes the proof.
Proof. The proof is identical to that of [KM11, Lemma 5.2] if one replaces δ with 1 − δ throughout.
Proof. For the proof of the first claim about V see the proof of [KM11, (5.5)] replacing δ with 1 − δ throughout and using Lemma 31 instead of [KM11, Lem.
As a result we recover, except for the critical case |δ| = 1, the recurrencetransience criterion [KZ08, Th. 1] in a more self-contained way without using [FYK90].