The potential function and ladder heights of a recurrent random walk on Z with inﬁnite variance

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function, a ( x ), of the random walk. Applying it, we derive an asymptotic estimate of a ( x ) and thereby a criterion for a ( x ) to be bounded on a half-line. The application is also made to estimate some hitting probabilities as well as to derive asymptotic behaviour for large times of the walk conditioned never to visit the origin. 1


Introduction
Let S n = S 0 + X 1 + • • • + X n be a random walk on Z where the starting position S 0 is an unspecified integer and the increments X 1 , X 2 , . . .are independent and identically distributed random variables defined on some probability space (Ω, F , P ) and taking values in Z.Let X be a random variable having the same law as X 1 .We suppose throughout the paper that the walk S n is recurrent and irreducible (as a Markov chain on Z).
For a subset B of the whole real line R such that B ∩ Z = ∅, put σ B = inf{n ≥ 1 : S n ∈ B}, the first entrance time of the walk into B. Let Z be the first strictly ascending ladder height that is defined by Z = S σ [S 0 +1,∞) − S 0 .
We also define Ẑ = S σ (−∞,S 0 −1] − S 0 , the first strictly descending ladder height.Because of recurrence of the walk Z is a proper random variable whose distribution is concentrated on positive integers x = 1, 2, . . .and similarly for − Ẑ.Let E indicate the integration by P as usual.If Section 17], [3,Theorem 8.4.7]).Denote by P x the probability of the random walk with S 0 = x and E x the expectation by P x .Put p n (x) = P 0 [S n = x], p(x) = p1 (x) and define a(x) = ∞ n=0 p n (0) − p n (−x) ; the series on the RHS is convergent (cf.Spitzer [19, P28.8]).The function a(x), called potential function, plays a central role in the potential theory of recurrent random walks.(This is true for two dimensional walks but here we restrict our discussion below to the one dimensional walks).Spitzer [18] established fundamental facts concerning a(x)-its existence, positivity, asymptotic behaviour etc.-and based on them Kesten and Spitzer [12] obtained certain ratio limit theorems for the distributions of the hitting times and sojourn times of a finite set and the transition probabilities of the walk stopped as it hits the set, which were refined by Kesten [10] under mild additional assumptions.An excellent exposition for the principal contents of [18], [12], [10] is given in Chapter 7 of Spitzer's book [19]; extensions to non-lattice random walks are obtained by Ornstein [14], Port and Stone [15], Stone [20].Kesten [11] conjectured that the series that defines a(x) converges absolutely and provided certain mild sufficient conditions for the absolute convergence.
According to Theorems 6a and 7 of [10] lim n→∞ P x [S n = y, S k = 0 for 1 ≤ k < n] P 0 [S n = 0, S k = 0 for 1 ≤ k < n] = a † (x)a † (−y) + xy σ 4  (1.1) (x, y ∈ Z).Here (and in the sequel) a † (0) = 1 and = a(x) for x = 0 and 1/∞ is understood to be zero.The asymptotic estimates valid uniformly in x and y of the ratio under the limit above are studied by the present author in [21], [24] in case σ 2 < ∞ and in [25] for the stable walks with exponent 1 < α < 2. The denominator of the ratio in (1.1), which equals the probability that the walk starting at zero returns to zero at n for the first time, are estimated with some exact asymptotics in these articles.
The following basic properties of a(x) are found in [19]: and a(x) − x σ 2 = 0 for all x > 0 if P [X ≤ −2] = 0, > 0 for all x > 0 otherwise (1.3) (the strict positivity in the second case of (1.3) is implicit in [19] if σ 2 < ∞; see e.g., [21, Eq(2.9)]).When σ 2 < ∞ (1.2) entails the exact asymptotics a(x) ∼ |x|/σ 2 , whereas in case σ 2 = ∞ it gives only a(x) = o(|x|) and sharper asymptotic estimates are desired.For the stable walks exact results are given in [1] for the case 1 < α < 2 apart from an extreme case and in [26,Section 8.1.1]for all the cases 1 ≤ α ≤ 2 under some natural side conditions.In the general case of σ 2 = ∞ there seems to have been no results of asymptotic estimates of a(x) other than those mentioned above.Very recently the present author gave some relevant results.Let σ 2 = ∞ and E|X| < ∞ and put with a universal constant C * > 0; the upper bound is also given so that under a reasonable side condition, which is satisfied if e.g., lim sup x→∞ xm ′ (x)/m(x) < 1 or m + (x)/m(x) → 0. (Here b x ≍ c x means that b x /c x is bounded away from zero and infinity.) In this paper we shall show, supposing where V ds denotes the renewal function for the weakly descending ladder process, and that there exists lim x→ ∞ a(−x) ≤ ∞, and provided P [X ≥ 2] > 0. In (1.6) the lim inf may be expected to be replaced by lim (see Remark 2.1(e)).Note that if E|X| = ∞, then EX + = EX − = ∞ because of the assumed recurrence of the walk.(Here X + = max{X, 0} and X − = X + − X.) Applying (1.6), we derive asymptotic estimates of some hitting probabilities as well as asymptotic behaviour for large times of the random walk conditioned never to visit the origin.As an intelligible manifestation of the significance of the condition EZ < ∞ in the sample path behavior of the walk, we shall observe that EZ < ∞ if and only if the walk conditioned never to visit the origin approaches the positive infinity with probability one (Section 7).
The main results (Theorems 1 and 2) of the present paper are derived from those given in Spitzer's book (that are stated in Section 3 of the present paper) independently of those of [26] in which the proof is solely based on the Fourier integral representation of a(x).In the proof of our main results, we could apply those from [26] whose usage, however, we avoid in order not to cause any suspicion of circular arguments, some of our results (1.6) being used in [26]. .For the sake of comparison, we include the case of finite variance when all the results are known or easily derived from known ones.

Statements of results
Let S n be the random walk specified in Introduction and Z, Ẑ, σ B , m ± (x) and a(x) be as given there.In order to state the results of the paper we further bring in the following notation.Put (where (α, β] denotes the interval α < x ≤ β as usual) and define the hitting distribution of (−∞, 0] for the walk starting at x ∈ Z. Likewise let H x B be the hitting distribution of a non-empty set ] There exists lim x→∞ H x (−∞,0] (y), which we denote by H +∞ (−∞,0] (y) and similarly for ) is a probability distribution if EZ < ∞ and vanishes identically otherwise [19, P24.7].Let V ds (x), x = 0, 1, 2, . . ., be the renewal function of the weak descending ladder-height process (see (3.5) or Appendix).For our present purpose it is convenient to bring in the function f r , the shift of V ds to the right by 1, namely According to [19, P19.5, E27.3], [8, Section XII.3] f r is a positive harmonic function on [1, ∞), i.e., a positive solution of the equation f r (x) = E x [f r (S 1 ); S 1 ≥ 1], which may be written as and the solution is unique apart from a constant factor; it turns out that the distribution of Z is expressed as (see Theorem A and (3.10) in Section 3 for more details).Define for any non-negative function ϕ(y), y ≤ 0, For a set B ⊂ R such that B ∩ Z = ∅ let g B (x, y) denote the Green function of the walk killed as it hits B: where δ(x, y) = 1 if x = y and = 0 otherwise.This definition is different from that in [19], where the corresponding one agrees with our g B (x, y) if x / ∈ B, but vanishes if x ∈ B whereas according to our definition This relation shows that g B (x, y) equals the hitting distribution of B by the dual (or time-reversed) walk started at y which fact is expressed as Here −B = {−z : z ∈ B} and σB = σ B if S 0 / ∈ B and σB = 0 otherwise.In case B = (−∞, 0], g B (x, y), x, y ∈ B is expressed explicitly by means of the renewal functions of ascending and descending ladder height processes (cf.Theorem A in Section 3), by which it follows immediately that there exists lim y→∞ g (−∞,0] (x, y) which is denoted by g (−∞,0] (x, ∞) and given by if EZ = ∞, the RHS vanishes so that g (−∞,0] (x, ∞) = 0 for all x (cf.(3.7)).
Theorem 1. (i) For all x, y ∈ Z, where (ii) If EZ < ∞, then as x → ∞, a(x)/f r (x) → A/EZ and a(−x)/a(x) → 0, and It is natural to extend f r (x) to a function on Z by means of (2.2) (so as to make (2.2) valid for all x ∈ Z), or what amounts to the same thing (in view of (2.3)), for x ≤ 0. (2.7) Since f r (0) = 1 < V ds (0) = f r (1), f r is increasing.According to this extension of f r together with the identity [8,(XI.4.10)] or the remark following (3.10))relation (2.5) is expressed simply as (2.9) For any integer k and for any non-negative ϕ, The following corollary will be often useful in application of Theorem 1.For brevity of expression we write The statement a(−x)/a(x) → 0 in Theorem 1 follows under the weaker condition m + (x)/m(x) → 0 (x → ∞) (cf.[26,Theorem 4]).By Corollary 1(i) a(x) ∼ Af r (x)/EZ, which is shown in [21] in case σ 2 < ∞ and generalized in [26] as We include proofs of these parts of Theorem 1 which are much simpler than the proofs in [26]-although the latter do not depend on our Theorem 1.
(b) By (2.3) it follows that This together with Theorem which may be effectively used to derive Chow's criterion for EZ < ∞ in a quite different way from [4] (see Remark 3.1 for more details).
(c) The process M n := a(S n∧T ) is a non-negative martingale under (d) As another application of Theorem 1 we shall consider the random walk conditioned never to visit the origin and observe that the conditional walk distinguishes +∞ and −∞ if and only if either EZ or E Ẑ is finite, although its Martin compactification does not whenever σ 2 = ∞ (see Section 7).
The first part of the following theorem provides asymptotic estimates of a(x) as |x| → ∞, and its third part an answer to the open question stated at the very end of Spitzer's book [19] (see Remark 2.2(d) below). (2.15)

17)
[with all the members vanishing if P [X ≥ 2] = 0]; and there exists lim x→∞ a(−x) ≤ ∞ where the limit is finite if and only if and if this is the case, 18) is possibly true even under the condition that for some δ > 0, P [X > x] > x −1 (log x) −1−δ for all sufficiently large x.
(c) Condition (2.18) implies EZ < ∞, the latter being equivalent to the integrability condition because of the recurrence assumption), Theorem 2(iii) gives an exact criterion for the trichotomy of (This trichotomy itself is stated at the end of [19].)Corollary 2. Suppose σ 2 = ∞.There exists M ± := lim x→±∞ a(x) ≤ ∞ where M − = 0 if and only if P [X ≥ 2] = 0 and in order that M − < ∞ each of the following conditions are necessary and sufficient.
Proof.The existence of the limit and the condition for M − = 0 follows immediately from Theorem 2. Each of conditions (i) and (ii) implies EZ < ∞ (see Remark 3.1(b) for (i) and note M + > 0 under (ii)).The assertion of the corollary then follows from Theorems 1 and 2 and the identity For y ∈ Z write σ y for σ {y} .The results (i) and (ii) given below are taken from Sections 7.3 and 7.5 of [26].
(ii) If m + (x)/m(x) converges to 0 or to 1 as x → ∞, then uniformly for 0 (The second relation of (2.20) is the dual of the first.)These results are supplemented by the following propositions.
where in case EZ < ∞ the convergence in (2.21) is uniform for 0 ≤ x ≤ R.
and Af r (x)/EZ = a † (x) for all x ∈ Z, where all the terms vanish for x ≤ −1, and the formula (2.21) (necessarily the first case) is still reasonable.The dual statement of (2.21) for the case EZ < ∞ may be written as The corresponding one for (2.22) will be stated as Lemma 6.3 in Section 6.
The formula (2.21) says that if EZ < ∞, then being equal to the ratio on the LHS, approaches unity as x becomes large independently of how R is large, while if EZ = ∞ this is not the case: this conditional probability tends to zero as R → ∞, in other words, for R large enough the walk-even if it is conditioned on σ R < σ 0 -reaches R only after entering the negative half line with overwhelming probability as far as its starting position x is fixed.If m + /m → 0 and ℓ + (x) = [26,Lemma 46], so that by (2.19) it follows that as x → ∞ under x < R (2.23) This shows that for each ε > 0, the ratio above approaches 1 as R → ∞ uniformly for x > εR.The same holds true if E| Ẑ| < ∞ at least under some regularity condition on the tails of F but can fail in general (see Remark 6.2 of Section 6).The rest of the paper is organized as follows.In Section 3 we collect fundamental facts used in this paper about f r , a(x), g (−∞,0] etc. given in Spitzer [19] and advance several lemmas that are directly derived from them.The proofs of Theorems 1 and 2 are given in Sections 4 and 5, respectively.The proofs of Propositions 1 and 2 are given in Section 6.In Section 7 we briefly study large time behaviour of the walk conditioned never to visit the origin.In Section 8 (Appendix) we present a few facts about strictly and weakly ascending ladder height variables.

Preliminary lemmas
In this section we collect fundamental results of the recurrent random walks on Z given in Spitzer's book [19] and then derive some consequences of them that are used the later sections.
For B ⊂ Z we have defined the first hitting time by σ B = inf{n ≥ 1 : S n ∈ B}.For a point x ∈ Z write σ x for σ {x} .For typographical reason we sometimes write σB for σ B .
Let u as (x), x = 0, 1, 2, . . .be the renewal sequence of the strictly ascending ladder variables, namely u as (0) = 1 and and similarly v ds (x), x = 0, 1, 2, . . .denotes the renewal sequence of the weak descending ladder variables, which may be given by v ds (0) = 1/c and where (See Appendix for (3.2) as well as for the probabilistic meaning of the constant c.) Owing to the renewal theorem [8], there exist limits The Green function g B (x, y) ( x, y ∈ Z ) defined in (2.4) may be written as: The following theorem follows from the propositions P18.8, P19.3, P19.5 of [19].For two real numbers s and t write s ∧ t = min{s, t} and s ∨ t = max{s, t}.
The formulae in Theorem A will often be used in combination with the following representation of the hitting distribution H x (−∞,0] (y) of (−∞, 0]: and analogous one for H x [0,∞) (see (5.2) for another representation).The function f r may be written as and its dual as . By Theorem A(ii) and u as (y) ≤ 1 it follows that In particular the three conditions (a and, by summation by parts, Theorem B. The series ∞ n=0 [p n (0) − p n (−x)] converges for each x ∈ Z and if a(x) denotes the sum, then the following relations hold.
a(x + y) ≤ a(x) + a(y) and a † (x) + a(−x) ≥ 1 (x, y ∈ Z), (3.12)If the walk is left-continuous (i.e.P [X ≤ −2] = 0), then a(x) = x/σ 2 for x > 0; analogously a(x) = −x/σ 2 for x < 0 for right-continuous walks; except for left-or right-continuous walks with infinite variance a(x) > 0 for all x = 0. [(3.11) with x = 0 and the second inequality of (3.12), not given in [19], follows from (3.13) and g {0} (x, x) ≥ 1, respectively.]We put By (3.11) it follows that g {0} (y, y) = 2ā(y) + δ(0, y) > 0 and that The equation (3.13) states that a(x) is harmonic on x = 0, which together with a(0) = 0 entails that the process M n := a(S σ ξ ∧n − ξ) is a martingale, provided that S 0 = ξ ∈ Z a.s.Using the optional sampling theorem and Fatou's lemma we obtain first the inequality ] valid whenever x = ξ, and then by using In the rest of this section we prove several lemmas that are derived more or less directly from the results presented above.gives lim inf a(x) ≥ lim sup a(x) so that lim a(x) exists.If this limit is zero, then the RHS of (3.20) must be zero for all x > 0, which is possible only if the walk is left-continuous.Now suppose lim sup x→∞ a(x) = ∞ and put M = lim inf x→∞ a(x)(≤ ∞).Contrary to what is to be shown let M < ∞.Then one can choose R such that a(x) + a(−x) > 4M + 6 for x > R. In view of (3.14) there must exist x 1 > R such that 2M + 2 ≤ a(x 1 ) < 2M + 3, which entails a(−x 1 ) > 2M + 3. Combined with (3.20) these lead to the absurdity Hence M must be infinite.[in (3.19) take −y and x in place of x and y respectively], which, after simple rearrangements, becomes the left-hand inequality of (3.21); the case x = 0 is obvious.The right-hand one is the same as g {0} (x, −y) ≥ 0.
Proof.Let Λ B (y) be the number of visits to y in the time interval {1, 2, . . ., σ B − 1}: Then g {0} (x, y) = δ(x, y) + E x [Λ {0} (y)] and similarly for g B (x, y), and (3.22) can be written as provided that 0 ∈ B which entails σ B ≤ σ 0 a.s.Recall that g(0, y) = 0 and for z = 0, g(z, y) = g {0} (x, y), the expected number of visits to y before entering.If y / ∈ B, then by the strong Markov property the above equality follows immediately, It therefore suffices to show (3.23) for y ∈ B.
Let y ∈ B, when one always has Λ B (y) = 0 a.s.For x / ∈ B, (3.23) then follows immediately.For x ∈ B, one observes that of which the RHS equals g(z, y) for z = 0. Thus If the walk is not right-continuous and Proof.Take B = (−∞, 0] in (3.22) and use the inequality a(z)−a(z−y) ≤ a(z)/a(−z) a(−y) (z ≤ −1) that follows from (3.21) to see that the difference on the middle member of (3.24) is not larger than E[g(S x σ(−∞,0] , y)] ≤ (1 + k + )a(−y), hence the right-hand inequality of (3.24).The left-hand one is trivial.
The right-hand inequality of (3.25) is also derived from (3.22) but this time we use the inequality g(z, y) ≤ 2ā(z) to have By definition 2ā(z . In [26], Lemma 3.4 plays a significant role for the proof of (2.19).In this article we apply it only to obtain the next result.Lemma 3.5.If either a(−x)/a(x) → 0 or a(x)/a(−x) → 0 as x → ∞ (with the understanding that a(x) > 0 (a(−x) > 0) for x > 0 in the former (latter) case), then and The identities in (3.27) and (3.28) are valid whenever σ 2 < ∞.
Proof.Under the assumption of the lemma it follows from Lemma 3.
Proof.As a dual relation of (3.10) we have for t ≥ 0 tends to zero as x → ∞ for each y ≥ 0. Replacing u as (y) by u as (∞) + o(1) in (3.29) and recalling v ds (0)u as (∞) = 1/cEZ we then infer that 1 m − (x) Thus (i) is verified.Noting that cf r (x + 1) is the renewal function for the variable − Ẑ we use the first inequality of Lemma 1 of Erickson [7] which may read combining this with (i) we can readily deduce (ii).The last assertion is obvious, for m the necessity part of the Chow's criterion for EZ < ∞ (this half of it is also proved by Doney [6]).Combined with (2.13) this shows that if EZ < ∞, then both of the following summability conditions hold The converse as well as the implication (♭) ⇒ (♯) is proved in [22,  (b) For a positive integer R let τ R = σ Z\(−R,R) .Then and the function u B defined in (3.30) is given by in particular the limit appearing in (3.35) is independent of the choice of ξ.These formulae are verified as follows.For x = ξ, M n := a(S n∧σ B − ξ) being a non-negative martingale under P x that is uniformly bounded on n < τ R , one obtains the identity a(x − ξ) = E x M τ R .As for the case x = ξ suppose that ξ = 0 ∈ B for simplicity so that M τ R = a(S τ R ∧σ B ). Then Thus one has (3.34).For the proof of (3.35) write it as (3.36) On passing to the limit as R → ∞ the equality (3.35) then comes out in view of (3.32), the last expectation converging to . When X is of finite range, the identity (3.35) (restricted to x / ∈ B) is shown in the proof of Proposition 4.6.3 of [13] (in a different way from ours).

Let Ĥx
B stand for the hitting distribution of B for the dual (time-reversed) walk, in other words Ĥx and for ξ ∈ B, x / ∈ B \ {ξ}. (3.39) Define the operators P and P − by ) and let G n also denote the corresponding operator.We may suppose inf x>0 a(x) > 0, otherwise a(x) vanishing for all x > 0 so that (4.1) is plainly evident.Owing to (4.2) relation (4.1) then follows if we can show lim n→∞ P n a † (0) for if (4.1) does not hold, a † = a + δ(•, 0) must dominate a positive multiple of f r so that (4.3) is impossible.
From the identity P a = a + δ(•, 0) one deduces by induction that On the other hand one obtains that P f r = f r + P − f r and by induction again , which can be rewritten as Since y≤0 f r (y) = y≥0 P [Z > y] = ∞ due to the assumption of the lemma, for any K > 0 one can choose a positive integer M so that −M ≤y≤0 f r (y) ≥ K; and hence if n is large enough, for the recurrence of the walk implies lim n→∞ G n (z, 0)/G n (0, 0) = 1 (cf.[19, P2.6]).Combined with (4.4) the inequality derived above implies that P n f r (0) ≥ 1 2 KP n a † (0) for all sufficiently large n and we can conclude the required relation (4.3).Proof of Theorem 1.By Lemma 3.9 the formula of (i) follows if we verify its special case y = 0. Note that g (−∞,0] (x, 0) = δ(x, 0).It is then obvious that if EZ = ∞, (i) and (iii) follow from Lemma 4.1 and (4.1), respectively.Let EZ < ∞.Then by Lemma 3.9 the difference a † (x) − H x (−∞,0] {a} is non-negative and harmonic on x > 0, so that it is a constant multiple of f r (x).The constant factor is determined by using Lemmas 3.5, 3.7, and 3.8.(Note that ā(x) ∼ a(x) if σ 2 < ∞.)By these the second assertion (ii) also follows.

5
Proof of Theorem 2 Proof of (i).The first half of (i) of Theorem 2 follows from Lemma 3.6 and Corollary 1. Suppose that EZ < ∞ and σ 2 = ∞.The formula (2.16), what is asserted in the second half, may be written as The hitting distribution of the half line [0, ∞) by the random walk started at a negative site −x agrees with that by the ascending ladder height process started at −x.Since G(x ′ , x ′′ ) := u as (x ′′ − x ′ ) (x ′ ≤ x ′′ ) is the Green function of the ladder height process it accordingly follows that Since for each w, ∞ k=0 P [Z = k + x − w]a(k) → 0 as x → ∞ and u as (w) → 1/EZ (w → ∞), we can conclude (5.1) owing to Corollary 1(ii) that gives the identity H {a} is bounded, so that EZ cannot be finite, for otherwise by Corollary 1(ii) a(x) + a(−x) must be bounded which is impossible in view of (3.14).This shows (ii) by virtue of Lemma 3.1.
Proof of (iii).Let σ 2 = ∞ and EZ < ∞.Then E| Ẑ| = ∞ so that a(−x) = H −x [0,∞) {a} according to Corollary 1 and by the first equality of (5.2) we have In view of (2.15) (or Lemma 3.6(ii)) one can replace f r (j) by j/m − (j), showing (2.17), the desired asymptotics of a(−x).The rest of (iii) is readily ascertained to be true by (2.17 Proof.If E| Ẑ| < ∞, then f r (x) ∼ Cx and the assertion of the lemma is obvious.Let E| Ẑ| = EZ = ∞ and we show that if a(x) is almost increasing, then a(−x)/f l (x) → 0, which by duality amounts to the same as the assertion of the lemma.For x ≥ 1, a(−x) = H −x (∞,0] {a} so that we have (5.3)  for some x 1 .On noting ∞ w=0 v ds (w) ∞ k=0 a(k)p(k + w) = b(0) = Ea(Z) < ∞ according to (5.4), the dominated convergence therefore concludes that b(y) → 0, as desired.

Proofs of Propositions 1 and 2
We employ the following identities:
and for the present purpose it suffices to show that b(y) → 0. Rewrite the expression of b(y) in (5.6) as b(y) = ∞ w=0 v ds (w) ∞ k=0 a(k)p(y + k + w).Now suppose that a(x) is almost increasing.Then ∞ k=0 a(k)p(y + k + w) ≤ C ∞ k=y a(k)p(k + w) + x 1 k=0 a(k)p(y + k + w) (5.7)
.22)The second equivalence in(2.22)followsfrom (2.19) if x ranges over a set depending on R in which f r