Deterministic time intervals on which a class of persistent processes are away from their origins

There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur. The ﬁrst result gives a sufﬁcient condition for a persistent random walk with a ﬁnite fourth moment. In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin. The second result gives a necessary and a sufﬁcient condition for similar excursions in the case of Brownian motion. The third result gives a necessary and a sufﬁcient condition for time intervals to be free of the zeros of a class of persistent natural scale linear diffusions on the line and is equivalent to the determination of recurrent sets at inﬁnity of the inverse local time.


The results
A random walk S n = S 0 + ξ 1 + · · · + ξ n , n = 0, 1, . . . , on the line with steps a sequence {ξ n } of i.i.d. random variables with mean 0 returns infinitely often to arbitrarily small neighborhoods of its origin, but given non-random pairwise disjoint time intervals J k moving out to infinity, steadily increasing in length, the possibility arises of finding the walk far away from its origin during these particular time intervals. Theorem 1.1. Let J k = [a k , a k + b k ] and define events A k = A(a k , b k , c k ) = |S n | < c k for some n ∈ J k . The method selected for proving this theorem gives a very short, concise demonstration but it does not lend itself to proving a converse. However, for Brownian motion a converse presents itself at the cost of a little tediousness in the proof. * University of Washington, United States of America. E-mail: kbe@u.washington.edu Theorem 1.2. Let {B(t), t ≥ 0} be standard Brownian motion on the line with probabilities P x [ · · · ] = P [ · · · B(0) = x] . Define A k = |B(t)| ≤ c k for some t ∈ [a k , a k + b k ] . for all x in the state interval. If [a k , a k + b k ], k = 1, 2 . . . are disjoint intervals marching out to infinity, and and if the series 1 gives an answer to the implied question at the beginning and allows the conclusion that a finite random variable κ exists such that w.p.1 for all k ≥ κ, |S n | ≥ c k for every n ∈ J k . This result relates to a result of K. L. Chung and P. Erdös, [3]. They show that under mild growth conditions on a deterministic sequence of times, {2n i }, a simple random walk (unit step with probability 1/2 in either direction) returns to its starting position at infinitely many of these times, w.p.1, if and only if n −1/2 i diverges.
Theorem 3 is equivalent to the assertion that the set B = k [a k , a k + b k ] is recurrent/transient at infinity (terminology of [7]), for the inverse local time at 0 of the diffusion. Inverse local time at 0 for a diffusion is a special type of subordinator and recurrence of sets at infinity is a complicated problem for general subordinators. Though Wiener type tests in terms of capacities are known, except in special cases, it seems difficult to get concrete expressions in terms of the sequences {a k }, {b k } and the fundamental object associated with subordinators, namely the Lévy measure. Some preliminary work is included in §4, Theorem 4.1, which is used in the proof of Theorem 1.3 in the last section. The problem of recurrence at infinity of sets for Lévy processes is closely related, but not does not seem equivalent, to the gap problems discussed in [5] or [11].

Proof of Theorem 1.1
Without loss of generality we may suppose that the a k and b k are actulally positive integers. Next, note that the event " A k occurs infinitely often k ↑ ∞ " belongs to the σfield of permutable events based on the i.i.d. sequence {ξ n } and therefore has probability 0 or 1.
According to the Skorokhod Embeding Theorem, [2], on some probability space Times away from the origin stopping times {T n } with mutually independent identically distributed increments {T n − T n−1 } exist such that, T 0 = 0 and {S n ; n = 0, 1, . . .} d = {B(T n ); n = 0, 1, . . .} . (2.1) Moreover, E T n = n E ξ 2 1 = n and because the ξ i have a finite moment of order 4, the increments of T have a finite moment of order 2. Clearly, However, the Hartman-Wintner Law of the Iterated Logarithm entails existence of a finite random variable ρ and a finite positive number q such that |T n − n| ≤ q n log log n ∀ n ≥ ρ .
It follows that if infinitely many of the events A k occur, then infinitely many of the events D k must occur where The Borel-Cantelli Lemma takes care of the rest.
Proof of (2.2) For future use here and in the next section, we note the inequalities: Times away from the origin (These follow quickly from [9], page 17, Problem 1, or do it yourself.) Let α > 1, β > 1; then by symmetry and the Markov property one gets

Proof of Theorem 1.2
The main tools are strong Markov, (2.3), some simple inequalities, a standard extension of the Borel-Cantelli Lemma, [10]. I omit some details.
and for c > 0, c > 0 , denote by A, A the events: |B s | < c, (< c ) for some s ∈ I, (s ∈ I ) , respectively. Assume that the parameters satisfy the conditions of the theorem: for some θ, 0 < θ < 1. The key to Theorem 1.2 is the following: for a constant M , which may depend on θ, but not on a, a , b, b , c, c .

Proof of (3.2)
For a path to be in A and A we have B(a) = x, (for some x) then a (first after time a) hit in (−c, c) at some time σ = s ∈ I. It must happen also that B(a ) = y for some y followed by a first hit after time a in (−c , c ) at some time σ in I . But the strong Markov property and time homogeneity implies that B starts from scratch at time s and the subsequent development is identical in law to the process time shifted by s. Given B(a) = x, the event σ = s ∈ [a, a + b] is the same as σ = r = s − a ∈ [0, b] for a Brownian B starting at place x at time 0. Similarly, given the position y of the path at time a ( = time a − s = a − a − r for B), the probability of a hit of (−c , c ) during [a , a + b ] is the same as an unconditional hit during [0, b ] for a brand new Brownian starting at place y.
Hitting an interval, [−c, c] say, from outside the interval is, by path continuity, the same as hitting one or the other boundary points ±c. Hence for x > c: . Now we can begin the estimations. The letter K, with or without subscripts or superscripts, denotes a positive numerical constant independent of a, a , b, b , c, c , r, θ but these Ks are not necessarily the same at each appearance.
Times away from the origin To reduce some of the notation, we will frequently omit the differentials dx, etc. The indicated integral limits suffice to specify the integration variables. The necessity to keep down the length of this paper (12 pages) required brevity in proofs so the reader may need to take pencil in hand to check details he or she does not quite believe.
From symmetry: We will write 1 for the first term on the RHS of (3.5) and 2 for the second term.
The triple integral 1 in (3.5) The y part does not involve x and x > c.
We make the change of variables x → x + c with a new x ≥ 0. (One ought never to introduce a new symbol with a trivial change of variables.) In light of (3.4), the y integral splits into into three integrals corresponding to y ≥ c , y ≤ −c , and |y| ≤ c . The first two of these combine into one with y ≥ c but with an additional integrand and in that integral we replace y by y + c with a new y ≥ 0: Applying (2.4) we have in the integral J 1 of (3.6) For J 2 in (3.6) note that for 0 ≤ r ≤ b ≤ a − a and any y, It follows from these inequalities and (3.1) that for a number K 1 independent of θ, a , b , etc.
First integral on RHS of (3.7) Second integral on RHS of (3.7) for some number K 2 and putting the estimates for 1 , 2 together in (3.5), we get: Next, a lower bound for P {A}: Times away from the origin

The rest of the proof of Theorem 1.2 A well known extension of the Borel-Cantelli
Lemma, [10], states that for events {A k } in a probability space, if

Recurrence at infinity of sets for a class of subordinators
In this section S = {S t = S(t)} t≥0 denotes a pure jump (no continuous "drift") subordinator with infinite lifetime. References for this section are [1], [7], and [8] though our notation is different. Also in this section P x denotes probabilites for S with S(0) = x, but P 0 = P . Under these assumptions the Laplace transform of S is given by a k + b k < a k+1 ∀ k and a k , b k → ∞, as k → ∞ . In addition to ν we will also need the renewal function: meaning that the LHS is bounded above and below by a positive constant multiple of the RHS.
The event Z a ≤ b occurs if and only if the range of the subordinator has non-empty intersection with [a, a + b]. Hence by [1], Lemma 5.5, page 45, we have: Times away from the origin We now assume that U has a positive monotone (decreasing) density, i.e.

U (x) =
x 0 u(y) dy, ∀ x > 0, and 0 < u(x 2 ) ≤ u(x 1 ) whenever x 1 ≤ x 2 .  For the converse it seems most straighforward to make use of Lamperti's extension of the B.-C. Lemma again and for this it suffices to establish an inequality for some constant C independent of j and k.
This task requires an explicit formula for the distribution of Z a , not simply the inequalities (4.4). Here is the formula from [8], Lemma 2.3: In addition to (4.2) we now also impose lim sup Fix k 0 , θ so that 0 < θ < 1 and (a k + b k )/a k+1 ≤ 1 − θ, ∀ k ≥ k 0 . Let a, b; a , b stand for a k , b k ; a j , b j , with j > k ≥ k 0 , respectively. Note that a + b < a , and θa ≤ a − a − b .
The stationary independent increments property implies that the conditional law of the process {S(t + t 0 ), t ≥ 0 } given {S(s), s ≤ t 0 , & S(t 0 ) = x}, coincides with the unconditional law of {x + S(t), t ≥ 0 } given S(0) = 0, t 0 < ∞ a stopping time. Since σ a ≤ σ a < ∞, it follows that where F σa is the usual "stopped" σ-field generated by {S(t), t ≤ σ a }. Assuming (4.5), it follows from (4.8) that F (z; a) = z 0 ν (x)u(a + z − x) dx is non-increasing in a, and, because a − a − x ≥ θa for 0 ≤ x ≤ b, we conclude To make more progress, it becomes necessary to somehow replace F (b ; θa ) by the smaller F (b ; a ) in the last inequality. From (4.6) we find that Times away from the origin We now make one final assumption: lim sup x→∞ u(x/2) u(x) < ∞ (4.12) and this, along with b k = O(a k ) and monotonicity of u, implies that the RHS of (4.11) is bounded independent of a , b , i.e., F (b ; θa ) ≤ CF (b ; a ) with C independent of a , b (but not independent of θ). (See [6], page 289, the definition and the exercises on that page.) Therefore (4.7) is indeed correct, the RHS being the same as C F (b k ; a k )F (b j ; a j ), and our work to this point almost completes the proof of the: ] is a tail event for the subordinator and therefore has probability 0 or 1. Consequently, divergence implies the probability equals 1, and now the proof is complete.

Proof of Theorem 1.3
The proof makes use of the observation mentioned before and proved in [9], § 6, page 225, that the set of zeros of the diffusion X corresponds to the closure of the range of a subordinator, to wit, the inverse local time, L −1 , of X at 0. The assertion of the theorem is equivalent to the statement that B = k [a k , a k + b k ] is either a transient or recurrent set (at infinity) for L −1 depending on the behavior of the series (1.5).
For the Browninan motion B, denote by (t, x) or x (t) the standard Brownian local time, jointly continuous in (t, x), normalized as in [9]. We may suppose, without loss of generality, that the diffusion X has the representation: the integral being over the entire state interval. (What is being set-up here is quite general -the assumption (1.4) will not come into play for awhile.) Let L denote local time for X at 0. Then L(t) = 0 • µ −1 (t) and The Lévy-Khintchine formula for L −1 has the form (4.1) (no drift or killing terms as occur in [1], Theorem 1.2) because m has no atom at 0 and the diffusion is persistent. The measure ν also has infinite mass. (For a proof see [9], Chapter 6, and in particular §6.2 and Problem 2 on page 218. See also Chapter 14 in [12].) As before ν(x) = ν{(x, ∞)} which is finite for x > 0 From (5.1) we get: the integral being over the state interval of X. Let X * be another diffusion on natural scale with the same state interval and with the associated objects decked out with * . We may suppose X * is also represented as a time change of the same Brownian motion as X. If for some constant K > 0, dm(x) ≤ K dm * (x) for all x, then for all s. Thus, for any x > 0, the number of jumps in L −1 (t) of magnitude greater than x during some interval 0 ≤ t ≤ s, does not exceed the number of jumps of magnitude greater than x/K of L * −1 during the same interval. The expected number of such jumps per unit time equals ν (x) for X and ν * (x/K) for X * 1 and this argument proves: If ν q denotes the Lévy measure of the inverse local time at 0 of the diffusion on natural scale associated with the speed measure |x| q dx with q as in (1.4), then for some constant κ depending only on q. See [9], §6. Applying Lemma 5.1 with dm * equal to Cx q dx for the uppper bound and equal to cx q dx for the lower bound, gives: There exist numbers C 1 > 0, C 2 > 0 depending only on c, C, and q of (1.4), such that The inverse local time at a point of a diffusion (and more generally of a gap diffusion) is a special kind of subordinator and its Lévy measure and renewal measure have very nice properties. In particular the renewal function U has a density u(x) = dU (x)/ dx which is non-increasing on (0, ∞). See [12], in particular, Theorem 10.3, (11.3 in 2nd ed.) See also [1] Corollary 9.7, page 78. Lemma 5.4. There exists a number C 5 > 0 such that C 5 x β−1 ≤ u(x) ≤ C 4 x β−1 , x > 0 (C 4 as above) . Remark. If X is a mean 0 stable process of index α ∈ (1, 2], then the conclusion of Theorem 1.3 remains valid for the zeros of X but with the exponent (q + 1)/(q + 2) replaced by 1/α. The inverse local time at 0 of X is a stable subordinator of index 1 − 1/α ([1], §8.1), so its Lévy measure ν(x) = Const. × x −1+1/α and the renewal density u(x) = Const. × x −1/α . Our proof easily adapts to get the conclusion or one might also make a proof using the Wiener style test of [7], but the calculations of capacities of {∪ k [a k , a k + b k ]} ∩ [2 n , 2 n+1 ) required (see Lemma 2, p. 58 of [7]) will be left to the interested reader. Similar results involving regular variation of Lévy measures are perhaps possible but will also be left to the interested reader.