Excursions and local limit theorems for Bessel-like random walks

We consider reflecting random walks on the nonnegative integers with drift of order 1/x at height x. We establish explicit asymptotics for various probabilities associated to such walks, including the distribution of the hitting time of 0 and first return time to 0, and the probability of being at a given height k at time n (uniformly in a large range of k.) In particular, for drift of form -\delta/2x + o(1/x) with \delta>-1, we show that the probability of a first return to 0 at time n is asymptotically n^{-c}\phi(n), where c = (3+\delta)/2 and \phi is a slowly varying function given explicitly in terms of the o(1/x) terms.


Introduction
We consider random walks on Z + = {0, 1, 2, . . . }, reflecting at 0, with steps ±1 and transition probabilities of the form (1.1) for x ≥ 1. We call such processes Bessel-like walks, as their drift is asymptotically the same as that of a Bessel process of (possibly negative) dimension 1 − δ. We call δ the drift parameter. Bessel-like walks are a special case of what is called the Lamperti problem-random walks with asymptotically zero drift. A Bessel-like walk is recurrent if δ > −1, positive recurrent if δ > 1, and transient if δ < −1; for δ = −1 recurrence or transience depends on the o(1/x) terms. Here we consider the recurrent case, with primary focus on δ > −1, as the case δ = −1 has additional complexities which weaken our results. Bessel-like walks arise for example when (reflecting) symmetric simple random walk (SSRW) is modified by a potential proportional to log x. Bessel-like walks have been extensively studied since the 1950's. Hodges and Rosenblatt [25] gave conditions for finiteness of moments of certain passage times, and Lamperti [32] established a functional central limit theorem (with non-normal limit marginals) for δ < 1; for −1 < δ < 1 our Theorem 2.4 below is a local version of his CLT. In [33] Lamperti related the first and second moments of the step distribution to finiteness of integer moments of first-return-time distributions. He worked with a wider class of Markov chains with drift of order 1/x, showing in particular that for return times of Bessel-like walks, moments of order less than κ = (1 + δ)/2 are finite while those of order greater than κ are infinite. Lamperti's results were generalized and extended to noninteger moments in [3], [5], and to expected values of more general functions of return times in [4]. "Upper and lower" local limit theorems were established in [34] for certain positive recurrent processes which include our δ > 1. Bounds for the growth rate of processes with drift of order 1/x were given in [35], and the domain of attraction of the excursion length distribution was examined in [18]. Karlin and McGregor ([28], [29], [30]) showed that, for general birth-death processes, many quantities of interest could be expressed in terms of a family of polynomials orthogonal with respect to a measure on [−1, 1]. This measure can in principle be calculated (see Section 8 of [29]) but not concretely enough, apparently, for some computations we will do here. An exception is the case of p x = 1 2 (1 − δ 2x+δ ) considered in [13] (for δ = 1) and [11]; we will call this the rational-form case. Birth-death processes dual to the rational form case were considered in [37]. Further results for birth-death processes via the Karlin-McGregor representation are in [8], [17].
Our interest in Bessel-like walks originates in statistical physics. These walks were used in [12] in a model of wetting. Additionally, in polymer pinning models of the type studied in [20] and the references therein, there is an underlying Markov chain which interacts with a potential at times of returns to 0. The location of the ith monomer is given by the state of the chain at time i. There may be quenched disorder, in the form of random variation in the potential as a function of the time of the return. Let τ 0 denote the return time to 0 for the Markov chain started at 0. For many models of interest, e.g. SSRW on Z d , the distribution of τ 0 for the underlying Markov chain has a power-law tail: (1.2) P (τ 0 = n) = n −c ϕ(n) for some c ≥ 1 and slowly varying ϕ. Considering even n, for d = 1 one has c = 3/2 and ϕ(n) converging to 2/π; for d = 2 one has c = 1 and ϕ(n) proportional to (log n) −2 [27]; for d ≥ 3 one has c = d/2 and ϕ(n) asymptotically constant. In general the value of c is central to the critical behavior of the polymer with the presence of the disorder altering the critical behavior for c > 3/2 but not for c < 3/2 ( [1], [2], [22].) In the "marginal" case c = 3/2, the slowly varying function ϕ determines whether the disorder has such an effect [21]. As we will see, for Bessel-like walks, (1.2) holds in the approximate sense that (1.3) P (τ 0 = n) ∼ n −c ϕ(n) as n → ∞, with c = (3 + δ)/2 and ϕ(n) determined explicitly by the o(1/x) terms. Here ∼ means the ratio converges to 1. Thus Bessel-like walks provide a single family of Markov chains in (1 + 1)-dimensional space-time in which (1.2) can be realized (at least asymptotically) for arbitrary c and ϕ. A related model is the directed polymer in a random medium (DPRM), in which the underlying Markov chain is generally taken to be SSRW on Z d and the polymer encounters a random potential at every site, not just the special site 0. The DPRM has been studied in both the physics literature (see the survey [24]) and the mathematics literature (see e.g. [7], [9], [31].) In place of SSRW, one could use a Markov chain on Z d in which each coordinate is an independent Bessel-like walk. In this manner one could study the effect on the DPRM of the behavior (1.3), or more broadly, study the effect of the drift present in the Bessel-like walk. As with the pinning model, via Bessel-like walks, all drifts and all tail exponents c (not just the half-integer values occurring for SSRW) can be studied using the same space of trajectories. This will be pursued in future work.
For the DPRM, an essential feature is the overlap, that is, the value where {X i }, {X ′ i } are two independent copies of the Markov chain; see ( [7], [9], [31].) To determine the typical behavior of the overlap one should know the probabilities P (X i = y), y ∈ Z d , as precisely as possible, with as much uniformity in y as possible..
For this paper we thus have two goals: given the transition probabilities p x , q x of a Bessel-like walk, determine (i) the value c and slowly varying function ϕ for which (1.3) holds, and (ii) the probabilities P (X i = y), y ∈ Z, asymptotically as i → ∞, as uniformly in y as possible. We will not make use of the methods of Karlin and McGregor ([28], [29], [30]) due to the difficulty of calculating the measure explicitly enough, and obtaining the desired uniformity in y. Instead we take a more probabilistic approach, comparing the Bessellike walk to a Bessel process with the same drift, while the walk is at high enough heights. This leads to estimates of probabilities of form P (τ 0 ∈ [a, b]) when a/b is bounded away from 1. Then to obtain (1.3) we use special coupling properties of birth-death processes which force regularity on the sequence {P (τ 0 = n), n ≥ 1}. These properties, given in Lemma 6.1 and Corollary 6.2, may be of some independent interest.

Main Results
Consider a Bessel-like random walk {X n } on the nonnegative integers with drift parameter δ ≥ −1, with transition probabilities p x = p(x, x + 1), q x = p(x, x − 1) = 1 − p x . The walk is reflecting, i.e. p 0 = 1. We assume uniform ellipticity: there exists ǫ > 0 for which Define R x by where R x = o(1/x). Note that in the rational-form case we have The drift at x is Let λ 0 = 1, M 0 = 0 and for x ≥ 1, M x is the scale function. Note M 1 = 1, and M Xn∧τ 0 is a martingale. It is easily checked that the assumption R x = o(1/x) ensures L is slowly varying. By linearly interpolating between integers, we can extend L to a function on [1, ∞) which is still slowly varying. Let τ j be the hitting time of j ∈ Z + , let P j denote probability for the walk started from height j and let be the height of an excursion from 0. From the martingale property we have In place of δ, a more convenient parameter is often Our assumption of recurrence is equivalent to M x → ∞. Define the slowly varying function ν(n) = l≤n, l even 1 lL( √ l) .
Throughout the paper, K 0 , K 1 , . . . are constants which depend only on {p x , x ≥ 1}, except as noted; for example, K i (θ, χ) means that K i depends on some previouslyspecified θ and χ. Further, to avoid the notational clutter of pervasive integer-part symbols, we tacitly assume that all indices which appear are integers, as may be arranged by slightly modifying various arbitrarily-chosen constants, or more simply by mentally inserting the integer-part symbol as needed.
Theorem 2.1. Assume (2.2) and (2.1). For δ > −1, and for n even, For the case of SSRW, in contrast to (2.8), the excursion length distribution is easily given exactly [19]: for n even, By (2.7) we have for fixed η ∈ (0, 1) that Heuristically, one expects that conditionally on the event on the left side of (2.10), τ 0 should be approximately uniform over even numbers in the interval [(1−η)n, (1+η)n], leading to (2.8). The precise statement we use is Lemma 5.1. It follows from (2.4), (2.6) and Theorem 2.1 that τ 0 and H 2 have asymptotically the same tail, to within a constant: This says roughly that the typical height of an excursion becomes a large multiple of the square root of its length (i.e. duration), as κ grows, meaning the downward drift becomes stronger. In this sense the random walk climbs higher to avoid the strong drift. By reversing paths we see that (2.13) P k (X n = 0) = p k λ k P 0 (X n = k).
Hence to obtain an approximation for P 0 (X n = k), we need an approximation for P k (X n = 0), and for that we first need an approximation for P k (τ 0 = m). In this context, keeping in mind the similarity between τ 0 and H 2 , for a given constant χ < 1 we say that a starting (or ending) height k is low if k < √ χm, midrange if √ mχ ≤ k ≤ m/χ and high if k > m/χ. Theorem 2.2. Suppose δ > −1. Given θ > 0, for χ > 0 sufficiently small, there exists m 0 (θ, χ) as follows. For all m ≥ m 0 and 1 ≤ k < √ χm (low starting heights) with m − k even, For all √ mχ ≤ k ≤ m/χ (midrange starting heights) with m − k even, For all k > m/χ (high starting heights) with m − k even, In general, for high starting heights, as in (2.16) we accept upper bounds, rather than sharp approximations as in (2.14) and (2.15).
Note that by (2.6), when k is large (2.14) and (2.15) differ only in the factor e −k 2 /2m , which is near 1 for low starting heights. (Here "large" does not depend on m.) Further, by (2.8), one can replace (2.14) with We will see below that the left and right sides of (2.15) represent approximately the probabilities for a Bessel process, with the same drift parameter δ and starting height k, to hit 0 in [m − 1, m + 1]. But the Bessel approximation is not necessarily valid for low starting heights, where (2.14) holds, because the analog of M k for the Bessel process may be quite different from its value for the Bessel-like RW, and because L( √ m)/L(k) need not be near 1, whereas the analog of L(·) for the Bessel process is a constant. Even if a RW has asymptotically constant L(·), the constant K 0 may be different from the related Bessel case. From (2.15), for midrange starting heights the distribution of τ 0 is nearly the same as for the approximating Bessel process. For low starting heights, this is not true in general-the Bessel-like RW in this case will typically climb to a height of order √ m for paths with τ 0 = m, and this climb is what is affected by the dissimilarity between the two processes, as reflected in the errors R x .
(i) For k < √ χn (low starting heights) with n − k even, (midrange starting heights) with n − k even, and for k > n/χ (high starting heights) with n − k even, and there exists K 1 (κ) such that for k > n/χ (high starting heights) with n − k even, and for k > n/χ (high starting heights) with n − k even, From [23], the integral that appears in (2.23) and (2.27) is the probability that the approximating Bessel process started at k hits 0 by time n.
We now combine (2.13) with Theorem 2.3 to approximate the left side of (2.13).
For large k we can use the approximation (2.5) in (2.29). For example, in the case −1 < δ < 1, there exists k 1 (θ) such that for n ≥ n 0 and k 1 ≤ k < √ χn we have We can use Theorem 2.4 to approximately describe the distribution of X n only because its statement gives uniformity in k. This requires uniformity in k in Theorems 2.2 and 2.3, which points us toward our probabilistic approach.
We are unable to extend our results to random walks with drift which is asymptotically 0 but not of order 1/x, because we rely on known properties of the Bessel process.

Coupling
Let us consider the random walk with steps ±1 imbedded in a Bessel process Y t ≥ 0 with drift −δ/2Y t : where B t is Brownian motion. (We need only consider this process until the time, if any, that it hits 0, which avoids certain technical complications.) The imbedded walk is defined in the standard way: we start both the RW and the Bessel process at the same integer height k. The first step of the RW is to k ± 1, whichever the Bessel process hits first, at some time S 1 . The second step is to Y S 1 ± 1, whichever the Bessel process hits first starting from time S 1 , and so on. Let g(x) = x 1+δ ; then g(Y t ) is a martingale, in fact a time change of Brownian motion (see [36].) Write P Be for probability for the Bessel process, P BI for the imbedded RW and P sym for symmetric simple random walk (not reflecting at 0.) For the imbedded RW, for x ≥ 1, the downward transition probability is We write {X n }, {X BI n } and {X sym n } for the Bessel-like RW, imbedded RW, and symmetric simple RW, respectively, and τ j , τ BI j , τ sym j for the corresponding hitting times. Here is a special construction of n } ignores the alarm and always takes its step according to ξ i . A second special construction, coupling {X n } to {X BI n }, is as follows; a related coupling appears in [10]. If X i = x, the alarm sounds independently with probability a(x) given by Whenever the alarm sounds, {X i } takes a step up in the case p x ≥ p BI x , and down in the case p x < p BI x . If there is no alarm, {X n } goes up if ξ i > q BI x and down if ξ i ≤ q BI x . By contrast, {X BI n } ignores the alarm and always takes its step according to ξ i . Under this construction, if p x ≥ p BI x , the probability of an up step for which shows that this second construction does indeed couple {X n } to {X BI n }. Note that in the second construction, unlike the first, the frequency of alarms is o(1/x). The coupling to {X BI n } is more complicated because the transition probabilities for {X BI n } depend on location. Even when no alarm sounds, the two walks may take opposite steps if X i = x, X BI i = y and ξ i falls between q BI x and q BI y . When (i) there is no alarm, (ii) X i = x, X BI i = y for some x, y, and (iii) ξ i falls between q BI x and q BI y , we say a discrepancy occurs at time i. A misstep means either an alarm or a discrepancy. For h sufficiently large, for x ≥ h, y ≥ h, conditioned on X i = x, X BI i = y and no alarm, the probability of a discrepancy is We let N(k) denote the number of missteps which occur up to time k. Note that if δ = 0, the imbedded RW is symmetric and there are no discrepancies. When we couple {X n } and {X BI n } in the above manner, with both processes starting at k, we denote the corresponding measure by P * k . Where confusion seems possible, for hitting times we then use a superscript to designate the process that the hitting time refers to, e.g. τ Be 0 and τ BI 0 for the Bessel process and its imbedded RW, respectively.

Proof of the tail approximation (2.7)
Recall that for (2.7) we have δ > −1. Let θ > 0, 0 < ρ < 1/8, 0 < ǫ 1 < ǫ 2 < √ ρ To prove (2.7) we will show that provided ρ, θ are sufficiently small, one can choose the other parameters so that the following sequence of six inequalities holds, for large m: These may be viewed as three "sandwich" bounds on P 0 (τ 0 ≥ m), with the outermost sandwich readily yielding the desired result, as we will show. The innermost sandwich (the 3rd and 4th inequalities) may be interpreted as follows. For convenience we assume the h i are even integers. Recall H from (2.3); when H ≥ h 2 , we let T denote the first hitting time of h 1 after τ h 2 . We can decompose an excursion of height at least h 2 and length at least m into 3 parts: 0 to τ h 2 , τ h 2 to T , and T to the end. The idea is that for a typical excursion of length at least m, most of the length τ 0 of the full excursion will be in the middle interval [τ h 2 , T ]; the first and last intervals will have length at most ρm. The middle sandwich (2nd and 5th inequalities) comes from approximating the original RW by the imbedded RW from a Bessel process, during the interval [τ h 2 , T ]. Then the outermost sandwich (1st and 6th inequalities) comes from approximating the imbedded RW by the actual Bessel process, and from showing that the third interval, from T to excursion end, is typically relatively short. A useful inequality is as follows: for h > k ≥ 0 and m ≥ 1, As a special case we have which establishes the 3rd inequality in (4.1).
By (2.6) there exists l 1 ≥ 1 such that for all x ≥ l 1 , If δ = 0, enlarging l 1 if necessary, we also have We turn to the 4th inequality in (4.1). We have The main contribution should come from the first probability on the right. To show this, we first need two lemmas. We begin with the following bound on stripconfinement probabilities.
Proof. Consider first δ = 0, h > l 1 . We claim that is bounded away from 1 uniformly in q, h with l 1 ≤ q < h. In fact, from the definition of l 1 , the drift p x − q x has constant sign for x ≥ l 1 . Suppose the drift is positive; then {X n } and {X sym n } can be coupled so that X n ≥ X sym n for all n up to the first exit time of {X n } from (l 1 , h). Therefore Since X sym n is a non-reflecting symmetric RW, for Z a standard normal r.v. we have as h → ∞, so P sym 0 (τ h ≤ h 2 − l 1 ) is bounded away from 0 uniformly in h > l 1 , and the claim follows. Similarly if the drift is negative, we can couple so that X n ≤ X sym n until the time that {X n } hits l 1 , and therefore and the claim again follows straightforwardly. Then since q x ≥ ǫ for all x ≤ l 1 , we have which together with the claim shows that there exists γ = γ(l 1 , ǫ) such that for all Therefore by straightforward induction, since m ≥ 2h 2 , completing the proof for δ = 0, h > l 1 .
For δ = 0, it seems simplest to proceed by comparison. Instead, in place of (4.5) we have We can change the value of the (downward) drift parameter from δ = 0 toδ ∈ (−1, 0) by subtractingδ/4x from p x for each x ≥ 1. By an obvious coupling, this reduces the probability on the right side of (4.8). But by Proposition 6.3 below, this reduced probability is bounded away from 0 in q ≥ 1. Thus (4.6) and then (4.7) hold in this case as well.
It should be pointed out that the proof of Proposition 6.3 makes use of Theorem 2.1 which in turn makes use of Lemma 4.1. Since the application of Proposition 6.3 in the proof of Lemma 4.1 is only forδ = 0, and since this application is only used to prove the lemma in the case δ = 0, this is not circular-all proofs can be done for nonzero drift parameter first, and then this can be applied to obtain the result for 0 drift parameter.
If we start the RW at 0, we can strengthen the bound in Lemma 4.1, as follows.
Lemma 4.2. Assume δ > −1. There exist K 3 (ǫ, l 1 ), K 4 (ǫ, l 1 ) as follows. For all h > l 1 and m ≥ 4h 2 , Proof. Let k 1 = min{k : 2 k−2 > l 1 } and k 2 = max{k : 2 k−1 < h}. Then for some constants K i (ǫ, l 1 ), and the lemma follows from this and (2.6). Here in the 2nd inequality we used the ellipticity condition (2.1), in the 4th inequality we used Lemma 4.1 and reversal of the path from time 0 to time τ 2 k−1 , in the 5th inequality we used (2.3), in the 6th inequality we used Lemma 4.1, in the 8th inequality we used (2.5), and in the last three inequalities we used the fact that L is slowly varying.
We return to the proof of the 4th inequality in (4.1). We have for m sufficiently large that The 4th inequality in (4.10) uses (2.5) and Lemmas 4.1 and 4.2. We want to show that (II), (III), (IV ) are much smaller than (I). We will show that if ǫ 1 ≪ ǫ 2 the probability in (I) is of the same order as This means that (III) ≪ (I) provided ǫ 1 ≪ ǫ 2 2 . To complement (4.10) we have the following bound from Lemma 4.2: We will later prove the following lower bound for (I).
Our next task is to use the coupling of {X n } to {X BI n }, from Section 3, to prove the 2nd and 5th inequaltites in (4.1). Here h 1± should be viewed as substitutes for h 1 which allow an error of η √ m in the coupling construction. Fix m/2 ≤ l ≤ m. We begin with the 5th inequality. From the coupling construction we have We need to bound the last probability. Consider first δ = 0. Let A(x) = sup y≥x a(y), so A(x) = o(1/x), and let d 0 = h 2 1− A(h 1− )/|δ|. Suppose that for some time i and some even integers Provided h 1− is large, by (3.1) the misstep probability for the next step is then at most The gap |X i −X BI i | can change (always by 2) only at times of missteps. Therefore if we start from the time (if any) before τ BI h 1− when the gap first reaches d 0 , the time until the next misstep (if any) before τ BI h 1− is stochastically larger than G d 0 , and then the time until the misstep after that (if any) before τ BI h 1− is stochastically larger than G d 0 +2 , and so on. It follows that . Note that for h 1− large (depending on η/ǫ 1 ), which grows to infinity as m → ∞; thus E(S) ≫ m. In fact by standard computations using exponential moments, we obtain that for some K 11 (η, δ, ǫ 1 ) we have √ m for all sufficiently large m, and hence by Claim 1, In the case δ = 0, {X BI n } is a symmetric simple RW so there are no discrepancies, only alarms, which have probability at most A(h 1 ) when the original RW is above height h 1 . Hence in place of (4.16) we have the left side of (4.16) bounded above by the probability that a Binomial(l, A(h 1 )) exceeds η √ m, and this probability is also bounded by e −K 11 √ m , and then the same argument applies. Now (4.13), (4.15), (4.16) and (4.18) show that provided m is large, the 5th inequality in (4.1) holds.
Turning to the 2nd inequality in (4.1), the analog of (4.16) is still valid, so from the coupling construction, (4.17) and (4.14) (trivially modified to allow h 1+ in place of h 1 ), we have The next step is to prove the first and last inequalities in (4.1), by relating the probabilities for {X BI n } to probabilities for the continuous-time Bessel process Y t . We need to establish the following.
Proof of Claim 2. Let T 0 = 0 and let T 1 , T 2 , . . . be the stopping times when the Bessel process reaches an integer different from the last integer it has visited, so that X BI n = Y Tn . Denote the hitting times of h 1− in the two processes by τ BI h 1− and τ Be h 1− and let σ i = min{t : Y t ∈ {i − 1, i + 1}}. Given k and x 1 , x 2 , . . . , x k , with .
We need estimates for the quantities be the scale function for the Bessel process and let Lf given by be its infinitesmal generator. For fixed x and z ∈ [x − 1, x + 1] the functions f = f x , g = g x , h ± = h ± x given by These can be solved explicitly, yielding that for δ > −1, Note the formulas here for δ = 1 are determined by the formulas for δ = 1, by continuity in δ.
Here B x is given by B ′ x is given by and B ′′ x and B ′′ x are given by Finally, A x , A ′ x and D x , D ′ x and D ′′ x , D ′′′ x are determined by g(x − 1) = 0, h + (x − 1) = 0 and h − (x + 1) = 0, respectively, but we do not need these values because we can use for example g(x) = g(x) − g(x − 1), and A x or A ′ x cancels in the latter expression. From these computations we readily obtain (4.27) f and then also It is easily seen by comparison to "Brownian motion plus small constant" that P Be z (σ x > 1) is bounded away from 1 uniformly in (large) x and in z ∈ [x − 1, x + 1]. Hence by the Markov property P Be x σ x > t) decays exponentially in t, uniformly in large x. By (4.27), this means there exist K 13 , K 14 such that , for all t ≥ 0 and all (large) x. Therefore for m sufficiently large, for all A and t, By standard methods, it follows from (4.28) and (4.29) that for some K 15 (ρ), K 16 (ρ) not depending on A, 16 k . Therefore the same bound holds unconditionally, so where the last inequality follows from (4.23) and (4.24), for large m. Thus (4.20) is proved. We have similarly from (4.30) (with τ h 1− trivially replaced by τ h 1+ ) that so (4.21), and thus Claim 2, are also proved. Proof of Claim 1. From (4.21), (4.25) and then(4.24), we have 2 , and it is straightforward to replace τ h 1+ here by τ h 1 , proving the second inequality in (4.14). The first inequality there is trivial.
The first inequality in (4.13) is also trivial, so we prove the second one. Using (4.14) and slight variants of (4.16) and (4.17) we get that for large m, √ m ≥ 1 2 K 10 ǫ 2κ 2 , completing the proof of Claim 1.
This also completes the proof of (2.7), as noted after Claim 2.

5.
Proof of (2.8) and (2.9) For even numbers 0 < m < n, let A m,n is the average of the even-index f j 's with j ∈ [m, n]. We use (2.10) and the following convexity property of {f m }.
It follows that Equation (5.2) is an immediate consequence of (5.1).
Let θ > 0. Provided η is sufficiently small (depending on θ), we have from (2.10), (2.11) and Lemma 5.1 that for n large and even and k = 2⌊ηn/2⌋, In the reverse direction, suppose f n < (1 − θ)A n−k,n−2 for some 0 < k < n/2, with k, n even. By Lemma 5.1 we have and therefore which in the case k = 2⌊ηn/2⌋ gives For small η and large n, this contradicts (2.7), showing that we cannot have f n < (1 − θ)A n−k,n−2 . Therefore for large n, using (2.7) we have This and (5.3) prove (2.8).
We now prove (2.9). LetP denote the distribution of the Bessel-like RW dual to P , that is, the walk with transition probabilitiesp x = q x ,q x = p x for x ≥ 1. In [14] it is proved that for n even, For δ = −1, the dual walk has drift parameterδ = 1, so (2.9) follows by applying (2.8) and (2.20) to the dual walk.

Proof of Theorem 2.2
We want to use (2.13) so we need to approximate f (k) n = P k (τ 0 = n) and P k (X n = 0). We sometimes omit the superscript (k) when it is equal to 0. We start with the following relative of Lemma 5.1. Lemma 6.1. Let 0 ≤ p ≤ q ≤ r ≤ s ≤ ∞ and 0 ≤ k < l. Then for l − k even, and for l − k odd, Proof. Suppose first that l − k is even. Consider a lattice path x starting at (0, k) in space-time which first hits the horizontal axis at a time in [r, s], and a lattice path x ′ starting at (0, l) which first hits the axis at a time in [p, q]. Since l − k is even, there must be a t ∈ (0, q] with x t = x ′ t . Switching the two trajectories after the first such t and proceeding as in Lemma 5.1 we obtain (6.1).
For l − k odd we repeat this argument but with the path x shifted one unit to the right, that is, started from (1, k).
Here are some special cases of interest for Lemma 6.1, particularly when comparing point versus interval probabilities for τ 0 . Corollary 6.2. (i) For all 0 ≤ k < l ≤ n and j > 0 with n − l and n + j − k even, (ii) For all 0 ≤ l ≤ m, (iii) For all l > 0 and 0 ≤ p < q < m, if l is even, and (6.8) if l is odd.
By Corollary 6.2, to show that P l (τ 0 = m) can be well approximated by M l P 0 (τ 0 = m) (or M l P 0 (τ 0 = m−1), depending on parity), it is sufficient to find, given m, values p < q ≤ m for which the fraction in (6.7) or (6.8) is almost 1. We will see that this can be done for m ≫ l 2 .
Proof of Corollary 6.2. (i) Take p = q = n and r = s = n + j in Lemma 6.1 to get f n in the case of even l − k, and similarly for odd l − k.
(iii) Consider even l. We may assume m is even, for otherwise the right side of (6.7) is 0. Applying Lemma 6.1 with k = 0, [r, s] = {m} we obtain while by (2.3), and together these prove (6.7). For odd l we may again assume m is odd and take [r, s] = {m − 1}, so that in place of (6.12), using (6.13) we get Note that if we take k = 0 and j ≪ n in (6.3), we see from (2.8) that the left side of (6.3) is close to 1, so the right side cannot be much less than 1 for any l > 0.
For the Bessel process we have by (4.24) that for 0 < a < b, recalling κ = (1 + δ)/2, As a step toward approximating P k (X n = 0) we have the following "interval" version of Theorem 2.2, for midrange starting heights (k of order √ m); for these we apparently cannot get sharp results from Corollary 6.2(ii) and (iii). Proposition 6.3. Let θ > 0, χ > 0, 0 < ∆ min < ∆ max and 0 < a < b. Provided χ is sufficiently small (depending on θ), ∆ max is sufficiently small (depending on θ, χ), the starting height k is midrange, that is, and a is sufficiently large (depending on θ, χ, ∆ min , ∆ max ), we have Proof. Let 0 < ρ < ∆ min /8 and ζ > 2β > 0. We always select our constants in the following manner: θ is given; we choose χ then ∆ max , and then ∆ min < ∆ max is arbitrary, then we choose ρ and then ζ and β (which appear in (6.20) below.) Finally we choose [a, b] as specified. Each choice may depend only on the preceding choices, and when we say a parameter is "sufficiently large" (or small), the required size may depend on the previous choices. The general outline is similar to the proof of (2.7). Analogously to (4.1), we will establish the following sequence of ten inequalities: . As with (4.1), this should be viewed as five "sandwich" bounds on P k (τ 0 ∈ [a, b]), with the outermost sandwich yielding the desired result.
Provided ∆ max /θχ is sufficiently small and the second inequality in (6.17) holds, the gamma density Then by (6.15), and, using also the second inequality in (6.17), Therefore (6.19) follows from (6.18). The inequalities (6.21) and (6.22), with minor modifications made to θ, a and b, also prove the first and last inequalities in (6.20), provided ρ is suficiently small (depending on θ, ∆ min , χ.) Turning to the 2nd and 9th inequalities in (6.20), provided ζ 2 /ρχ is sufficiently small (depending on θ), using (6.15) we have This proves the 9th inequality in (6.20). In the other direction, From (6.16), (6.17) and (6.21), and hence by (4.24), provided ζ 2 /ρ is sufficiently small (depending on θ, ∆ min , χ), . With (6.24) this shows that which proves the 2nd inequality in (6.20). Next we consider the 3rd and 8th inequalities in (6.20), in which Bessel-process probabilities are compared to similar probabilities for the imbedded RW. First, for the 8th inequality, analogously to (4.31) we have for some By (6.17), (6.27) and (6.25), there exist K 20 = K 20 (ρ, ∆ min , χ) and K 21 = K 21 (ρ, ∆ min , χ, θ) such that for a ≥ K 21 , which with (6.28) and (6.23) shows that so the 8th inequality in (6.20) is proved. For the 3rd inequality, similarly to (6.28) and (6.30) we get which together with a slight modification of (6.29) gives yielding the desired result. Now we consider the 4th through 7th inequalities in (6.20), comparing probabilities for the imbedded RW to similar probabilities for the original RW, and comparing the hitting times of (ζ − 2β)k and 0; for this we use the coupling of {X n } and {X BI n }. First, for the 4th inequality, observe that for walks starting at k, if τ BI ζk ∈ [a, (1 − 2ρ)b] and the number of missteps by time τ BI ζk is less than βk, then at time τ BI ζk , the stopping time τ (ζ−2β)k for the RW {X n } has not yet occurred and this RW is located in ((ζ − 2β)k, (ζ + 2β)k). Therefore denote the last event in (6.33). When D occurs, the RW {X BI n } reaches height ζk at some time l, and when it does, the RW {X n } is at some height j close to ζk, so {X n } has a high probability to reach height (ζ − 2β)k within an additional time ρb. More precisely, for j ∈ ((ζ − 2β)k, (ζ + 2β)k) and l ∈ [a, (1 − 2ρ)b], provided ζ 2 /ρχ is sufficiently small (depending on θ), using (2.6), (2.7), (4.2) and our assumption a ≥ χk 2 we have Since l, j are arbitrary, the same bound holds if we just condition on D. From this and (6.33) we get Reasoning similarly to (4.17) using (6.17), and then using (6.21) and (6.32), we get that for some K 22 (ζ, β) and K 23 (ζ, β, θ, ∆ min , χ), for a ≥ K 23 , With (6.35) this shows that (6.37) which yields the 4th inequality in (6.20).
Next, to prove the 7th inequality in (6.20), we can repeat (6.33)-(6.37) with {X n } and {X BI n } interchanged, and with ζk, (ζ −2β)k replaced by (ζ +2β)k, ζk, respectively, to obtain first the following analog of (6.35) and (6.36): and from this the analog of (6.37): so the 7th inequality is proved. Here for the second inequality in (6.39), analogously to (6.36), we require a lower bound for P k τ (ζ+2β)k ∈ [(1 − ρ)a, b] , and this follows from (6.21) and the inequality which is contained in the first four inequalities of (6.20), with trivial modification to replace ζ − 2β with ζ + 2β and [a, For the 6th inequality, we have Let us show that the last probability in (6.41) is much smaller than the first one. The Markov property at τ (ζ+2β)k , together with (2.6), (2.7), (4.2) and (6.17), yields that for some K 24 , provided a is sufficiently large, From (6.17), (6.21) and the first half of (6.20) we have that for some K 25 , From this and (6.42) we obtain that provided ζ 2 /ρ is sufficiently small (depending on ∆ min , θ, χ), the ratio of the last to the first probability in (6.41) is at most θ, which with (6.41) shows that proving the 6th inequality in (6.20), which completes the full proof of (6.20). Statement (6.18) is then immediate, and then, as we have noted, (6.19) follows.
j ∈ [0, m/8] we obtain that for m large and 0 < h < k/3 we have To bound the last probability we couple our Bessel-like RW to a symmetric simple RW. Recall that N(t) denotes the number of alarms by time t, and let ≥ h, and an alarm occurs at time i .

Acknowledgements
The author thanks P. Baxendale, F. Dunlop, T. Huillet, J. Pitman and A. Wade for helpful discussions, and thanks a referee for multiple references and suggestions.