How long is the convex minorant of a one-dimensional random walk?

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized.


Introduction and main results
Given a sequence (ξ k ) k∈N of independent and identically distributed (i.i.d.) real-valued random variables with a generic copy ξ, consider the associated random walk (S n ) n∈N 0 , N 0 := N ∪ {0}, defined by S 0 := 0 and S n := ξ 1 + ξ 2 + · · · + ξ n for n ∈ N, and the random piecewise linear function t → S(t), t ≥ 0, obtained by linear interpolation between the values S(n) := S n for n ∈ N 0 . For any fixed T > 0, let t → S ⌣ T (t) and t → S ⌢ T (t) be, respectively, the convex minorant and the concave majorant of the function t → S(t) on the interval [0, T ]. Let us recall that the convex minorant (concave majorant) of a function f on an interval [a, b] is the largest convex (least concave) function f for all x ∈ [a, b]. Clearly, both t → S ⌣ T (t) and t → S ⌢ T (t) are piecewise linear continuous functions and have therefore well-defined finite lengths, here denoted by L ⌣ T and L ⌢ T , respectively. In this paper, we provide distributional limit theorems for L ⌣ n and L ⌢ n as n → ∞ in the following three regimes: (A) Eξ 2 < ∞ and Eξ = 0; (B) the law of ξ lies in the domain of attraction of an α-stable law with α ∈ (1, 2) and Eξ = 0; (C) the law of ξ lies in the domain of attraction of an α-stable law with α ∈ (0, 1). The case Eξ = 0 in parts (B) and (C) turns out to be less intriguing and will be discussed in Section 4.
Let us point out at the outset that it suffices to consider the length of the convex minorant L ⌣ n because it has the same law as L ⌢ n , so Although nonintuitive, this follows fairly easily from the observation that the concave majorant of (S 0 , . . . , S n ) for any n coincides with the negative of the convex minorant of the reflected vector (−S 0 , . . . , −S n ) in combination with a distributional representation of L ⌣ n , stated as (11) and owing to Abramson et al. [1,2], which only involves the squares of the S k .
Before putting our work into some context by pointing out connections with earlier work on convex minorants and the convex hulls of random walks, we present our main results, stated as Theorems 1.1, 1.2 and 1.3.
It is well-known that in each of the cases (A), (B) and (C), there exists a sequence (a n ) n∈N of positive constants such that, with S α = (S α (t)) t∈[0,1] denoting an α-stable Lévy process, (2) S(nt) a n t∈[0,1] n→∞ = == ⇒ (S α (t)) t∈ [0,1] in the Skorokhod space D[0, 1] endowed with the standard J 1 -topology. Note that S 2 is just a centered Brownian motion. Throughout the paper, we always use a n and S α for the normalization and the α-stable Lévy process such that (2) holds. Also, we let N (0, s 2 ) denote the normal distribution with mean zero and variance s 2 .
In order to state our result for case (A), put which may be sharpened to under the additional assumption Eξ 2 log + |ξ| < ∞.
In view of the previous result, one could expect that in case (B) a suitable normalization of L ⌣ n converges in law to some stable law. It may therefore be surprising that the true answer, stated in the next theorem, looks more complicated. Theorem 1.2. Suppose that the following assumptions hold: (B1) The function t → P{|ξ| > t} is regularly varying at infinity with index α ∈ (1,2) and Then where (Z 1 , Z 2 , . . .) has a Poisson-Dirichlet distribution with parameter θ = 1 and the (S α (t)) t∈[0,1] , k = 1, 2, . . . , are independent copies of the α-stable process (S α (t)) t∈[0,1] appearing in (2).
The proofs of both theorems rely on a known representation of the convex minorant in terms of uniform random permutations that will be described in Subsection 2.1, followed by some explanations of the main arguments in Subsection 2.2. The main difference between the cases (A) and (B) is that, roughly speaking, in case (A) the main contributions to the fluctuations of L ⌣ n are due to a large number of "small" segments of the convex minorant, whereas in case (B) they are rather due to few "large" segments.
Our third result deals with the case when ξ lies in the domain of attraction of a stable law with index α ∈ (0, 1). This is the simplest case because rather than making use of the connection with random permutations, a simple comparison argument applies; see Subsection 3.4 below. Theorem 1.3. Suppose that the following assumptions hold: (C1) The function t → P{|ξ| > t} is regularly varying at infinity with index α ∈ (0, 1).
There is an interesting connection of our results, notably Theorem 1.1, with the work by Wade and coauthors [16,21,22] on the convex hulls of planar random walks. Assuming Eξ = 0, one can regard the bivariate sequence {(n, S n )} n∈N 0 as a degenerate walk in the plane whose increments are supported on the line orthogonal to the mean vector (1, 0) and thus to the x-axis. Except for this degenerate case, it was shown by Wade and Xu [22, Thms. 1.1 and 1.2] that the perimeter of the convex hull of the first n steps of any square-integrable planar random walk satisfies a central limit theorem and has linearly growing variance as n → ∞. But in the degenerate case, their approach only provides that this growth is sublinear. Although Theorem 1.1 does not hold any information on the asymptotic behavior of the moments of L ⌣ n , as it does not claim any type of uniform integrability, some moment asymptotics are specified in Proposition 1.5 below, most notably logarithmic growth of the aforementioned variance if ξ has a finite fourth moment, see (8).
Note further that the perimeter L n of the convex hull of {(j, S j ) : j = 0, . . . , n} equals L ⌣ n +L ⌢ n . Under the assumptions of our Theorem 1.3 (Case (C)), we therefore immediately infer a distributional limit result for L n , viz.
On the other hand and despite relation (1), we do not know in the other cases whether joint convergence of (L ⌣ n , L ⌢ n ) holds which would give a limit theorem for the perimeter L n in all cases. The connection with random permutations seems to be insufficient for this purpose and we must leave this as an open problem. Proposition 1.5. Suppose that Eξ = 0 and σ 2 = Eξ 2 ∈ (0, ∞). Then as n → ∞. Additionally assuming E|ξ| p < ∞ for some p ∈ (2, 4), even  The asymptotics of EL ⌣ n in (6) have been known before and follow, for example, from Theorem 1.8 in [16] and the fact that L ⌣ n d = L ⌢ n . The main results here are about the asymptotic behavior of Var L ⌣ n and the corollary on the behavior of Var L n . A recent weaker result, Theorem 6.2.6 in [15], asserts that Var L n grows sub-polynomially if ξ is centered and bounded.
In view of our discussion about the joint law of (L ⌣ n , L ⌢ n ) in Remark 1.4, it is not clear if, under the assumption Eξ 2 < ∞, (8) could lead to the stronger result Var L n ≃ c log n for a constant c > 0 as suggested by Conjecture 1.13 in [16]. Although the bound (7) for the variance may not be tight, it appears that Var L ⌣ n might have non-logarithmic behavior when Eξ 4 = ∞, and the same is very plausible for Var L n . Finding the asymptotics of Var L ⌣ n in the case p ∈ (2, 4) remains another open problem. Remark 1.7. In view of the functional convergence (2), it is natural to ask whether the above theorems can be obtained by applying the continuous mapping theorem to the functional F ⌣ : D[0, 1] → (0, ∞) which assigns to each function f in D[0, 1] the length of its convex minorant. An approach of this kind has been applied in [14], [16], [21] to various functionals of convex hulls of multidimensional random walks. As we are interested in the graph of a one-dimensional random walk, this functional limit approach cannot be used here because time and space are scaled by different sequences in (2). In fact, in the cases (A) and (B), the scaling in time (which is n) is stronger than the scaling in space (which is a n and thus regularly varying with index 1/α, with α = 2 in case (A)), whereas in case (C) the scaling in space is the stronger one.
There is just one "critical" case where the two scalings coincide and the functional limit approach does work. Assume that the law of ξ is such that (9) S(nt) a n 1] in the Skorokhod space D[0, 1] with the standard J 1 -topology, where a n /n → c ∈ (0, ∞) and (S 1 (t)) t∈[0,1] is the standard symmetric Cauchy process. Since the functional F ⌣ is continuous on a set of measure 1 with respect to the law of the Cauchy process, the continuous mapping theorem implies that where L ⌣ ∞ (c) is the length of the convex minorant of the Cauchy process (c S 1 (t)) t∈[0,1] . Note that the above argument does not completely cover the domain of attraction of the symmetric Cauchy distribution. Even if we assume that there is no centering, the sequence a n in (9) is in general of the form a n = nℓ(n) with some slowly varying function ℓ.

Proofs explained
2.1. Connection with uniform random permutations. Our approach relies crucially on the following representation of the convex minorant of a random walk, observed already in 1950th by Sparre Andersen [18] as well as Spitzer [19]. The version presented below is borrowed from [2, Thms. 1 and 2], see also [1, Thm. 1.1], and valid under the assumption that the law of the increment ξ is continuous.
Set [n] := {1, 2, . . . , n} and let Π n be a permutation of [n] picked uniformly at random from the symmetric group S n , that is Denote by Z n,1 , Z n,2 , . . . , Z n,Kn the nonincreasingly ranked cycle lengths of Π n , with K n being the total number of cycles. The convex minorant t → S ⌣ n (t) being a piecewise linear function, let F n denote the number of intervals where it is linear. Denote by C n,1 , . . . , C n,Fn the nonincreasingly ordered lengths of these intervals (on the horizontal axis). Then the basic result we shall rely on states that (F n , C n,1 , . . . , C n,Fn , 0, 0, . . .) d = (K n , Z n,1 , Z n,2 , . . . , Z n,Kn , 0, 0, . . .).
Furthermore, given (F n , C n,1 , . . . , C n,Fn ), the increments of the convex minorant over the linearity intervals are conditionally independent and the conditional law of any such increment over an interval of length ℓ equals the law of S ℓ . In what follows, we formally put Z n,k := 0 for k > K n . For j ∈ [n], let K n,j be the number of cycles of length j in Π n , that is K n,j := #{k : Z n,k = j}, j = 1, . . . , n.
Note that n k=1 Z n,k = n K n = n j=1 K n,j , and n j=1 jK n,j = n.
From the above observations, we immediately derive two equivalent distributional representations for the length of the convex minorant, namely where the S i,j for i ∈ N and j ∈ N 0 are independent random variables that are also independent of (Z n,k ) n,k∈N and satisfy S i,j d = S j for all i and j. Note that the summand −n on the left-hand side (matched by the summands −Z n,k and −j, respectively, on the right-hand sides) corresponds to the length of the horizontal interval [0, n] and should be viewed as a very rough first order approximation to the total length L ⌣ n in the cases (A) and (B).
The following smoothing argument shows that (10) and (11) do not require that the random walk has continuous increment law. In other words: The representations (10) and (11) remain valid without the continuity assumption. Fixing any n ∈ N, consider the random walk S k;ε := ξ ε,1 +· · ·+ξ ε,1 for 1 ≤ k ≤ n and any ε > 0, where ξ ε,k := ξ k + εN k and (N k ) k∈N are i.i.d. standard normal random variables independent of (ξ k ) k∈N . Let (S ε (t)) t∈[0,n] be its linear interpolation, defined the same way as S(t) above. The distribution of ξ ε,1 is continuous, hence the representations (10) and (11) hold for L ⌣ ε,n , the length of the convex minorant of (S ε (t)) t∈[0,n] . As ε ↓ 0, the process (S ε (t)) t∈[0,n] converges to (S(t)) t∈[0,n] weakly in the space C[0, n], and since the functional assigning to each continuous function the length of its convex minorant is continuous on C[0, n] (by the Cauchy-Crofton formula), the continuous mapping theorem implies that L ⌣ n;ε converges in distribution to L ⌣ n , as ε ↓ 0. Finally, the claim follows because the right-hand sides of (10) and (11) for (S ε (t)) t∈[0,n] converge, as ε ↓ 0, to the corresponding expressions for (S(t)) t∈[0,n] .

Explanation of the proofs in the cases (A) and (B).
The structure of uniform random permutations is well understood. In particular, see Theorem 1.3. in [3], it is known that (12) (K n,1 , K n,2 , . . . , K n,n , 0, 0, . . .) where the P j are mutually independent and the law of P j is Poisson with mean 1/j. Moreover, the convergence is fast: there exists a coupling (called the Feller coupling) such that and thus . This coupling will be crucial for the proof in case (A). Put (14) in conjunction with representation (11) strongly suggests that the asymptotic behavior of L ⌣ n − n should be well approximated by that of the sum which is simply a partial sum of independent (but not identically distributed) random variables. The corresponding limit laws are usually called distributions of class L, but in our case (A) the limit turns out to be normal. Below we will prove Theorem 1.1 by showing that the distributions of normalized random variables L ⌣ n − n and V n are asymptotically close, and then checking the classical conditions for convergence in distribution of the V n after suitable normalization, which are the row sums of triangular arrays whose rows consist of independent random variables.
An interesting observation is that the above argument, based on replacing K n,j by P j in representation (11), fails to work in the cases (B) and (C). In order to heuristically explain our approach in case (B), we recall another classical fact from the theory of random permutations, namely (see Vershik and Schmidt [20] or Kingman [13]) where the random vector (Z 1 , Z 2 , . . .) has a Poisson-Dirichlet distribution with parameter θ = 1. Furthermore, there exists a coupling such that see [4,Theorem 8.10]. Put b n := a 2 n /n and note that regular variation of (a n ) n∈N with index 1 α implies regular variation of the sequence (b n ) n∈N with index 2 α − 1 > 0. In Recalling relation (10), we can argue heuristically as follows: where the first approximation stems from the one-term Taylor expansion, the second is a consequence of (16) and the asserted convergence follows from (2), the S (k) α being independent copies of S α which are also independent of (Z 1 , Z 2 , . . .). Proposition 3.2 below will show that the series in the last line is a.s. finite. Moreover, the above heuristic turns out to be correct and this will provide the proof of Theorem 1.2.

Proofs
3.1. Proof of Theorem 1.1. LetL ⌣ n denote the random variable on the right-hand side of (11) increased by n so thatL ⌣ n d = L ⌣ n . As discussed in Section 2.2, the first step of the proof is to show that We have Using the definition of V n , this entails where the penultimate inequality follows from (14) and (19). Now (17) follows from the last line and the Markov inequality. The second step is to simplify V n . Put Observe that by the Borel-Cantelli lemma. Then which implies that (V n − W n ) n∈N is bounded in probability. Hence, by (17), it suffices to prove the theorem for W n instead of L ⌣ n − n. The third step is to simplify W n . Let us rewrite it as where θ : [0, ∞) → [0, 1] is a decreasing continuous function resulting from the remainder in the Taylor expansion of x → √ 1 + x at x = 0. We claim that for which it obviously suffices to verify that Put p j := P{P j = 1} and note that As Eξ = 0 and σ 2 = Eξ 2 < ∞, it follows by [ √ j log log j, P j = 1}, j ∈ N, occur with probability 1. Now the proof of (23), which in turn implies (22), can be completed as follows: Note the inequality The first sum on the right-hand side is finite a.s. since it contains a.s. only finitely many non-zero terms, and the second sum is finite a.s. because it has finite expectation.
In view of (21) and (22), it suffices to prove Theorem 1.1 with L ⌣ n − n replaced by Therefore, as the fourth step of the proof, we will show that Even though the W ′ n are sums of independent random variables of a rather simple structure, the derivation of a central limit theorem for these variables without any additional assumptions is not elementary. For example, when trying to verify the Lindeberg condition, the fourth moment of S 1,j appears which is not assumed to be finite here. Instead, we use general conditions ensuring convergence of row sums in triangular arrays of random variables to an infinitely divisible law, see [9,Thm. 1 in §25].
First of all, we check that the infinitely divisible limiting distribution has vanishing Lévy measure and must therefore be Gaussian. According to the aforementioned result in [9], this amounts to showing that for any ε > 0. Since P j is independent of S j and p j ≤ j −1 , we infer and then (27) immediately because the right-hand limit is zero by the dominated convergence theorem. The latter applies upon noting that for all n ∈ N and j ≤ n, and that the right-hand side is summable over j ∈ N by (25) and another appeal to [ Var The second sum on the right-hand side is increasing and bounded by n j=1 σ 4 4j 2 (use (24)). Therefore it converges for any fixed ε > 0 as n → ∞. As a consequence, denoting provided that the two iterated limits exist and coincide. We have and then, upon integration by parts, We will need below that (see [7, p. 1480] and [17, p. 130 which was proved in [8] but stated there without uniformity in x. In slight abuse of notation, let N (0, σ 2 j ) denote a centred normal variable with variance σ 2 j , where the latter quantity is given by (3). By using (29) to replace S j √ j by N (0, σ 2 j ) in the above formulae for A(n, ε), we see that (28) is equivalent to The family (N 4 (0, σ 2 j )) j∈N is uniformly integrable because σ 2 j → σ 2 . Therefore, the above iterated limits are both equal to We have thus proved ς 2 = 3σ 4 4 and a central limit theorem of the form where the centering term is as per Theorem 1 in §25 in [9].
To finish the proof of (26) and recalling (24), it remains to show that Integration by parts provides us with The subtracted sum in the previous line converges to zero by (27). As for the sum in the line before, the substitution x = √ 2t(log n) 1/4 leads to By (29) and the dominated convergence theorem, the first summand converges to zero as n → ∞. By combining the previous estimates, we arrive at as n → ∞ and thus see that (31) is equivalent to which in turn is the same as But the latter follows easily with the help of the Cauchy-Schwarz inequality and the well-known tail estimate The proof of (26) is herewith complete. If Eξ 2 log + |ξ| < ∞, then σ 2 j in (29) can be replaced by σ 2 , see Lemma 1 and the Theorems on p. 1480 in [7]. Repeating the above calculations with σ 2 instead of σ 2 j , we obtain (5). This finishes the proof of Theorem 1.1.

3.2.
Proof of Proposition 1.5. We first give explicit formulae for EL ⌣ n and Var L ⌣ n and put for simplicity Proof. It will be used that for any integers 1 ≤ j = k ≤ n, and E(K n,j K n,k ) = ½ {j+k≤n} jk ; (32) see [3, Lemma 1.1]. Thus, K n,j and K n,k are uncorrelated whenever j + k ≤ n.
The formula for EL ⌣ n follows immediately from representation (11) and (32). For the variance of L ⌣ n , we obtain with the help of the formula for the variance of a random sum of i.i.d. random variables: Similarly, by conditioning on K n,j and K n,k and setting γ (n) jk := Cov(K n,j , K n,k ), Combining these formulas with (11) and (32), we obtain The last term on the right-hand side can be estimated by using inequality (19), viz.
where the last passage follows from formula (21) in [12] with b = k = 1. This proves the formula for Var L ⌣ n because the last integral is finite. We are ready to prove our claims on the asymptotics of the moments of L ⌣ n .
Proof of Proposition 1.5. By the equality in (18), the law of large numbers and the central limit theorem, Under the assumptions Eξ 2 < ∞ and Eξ = 0, the family (S 2 n /n) n∈N is uniformly integrable [10, Thm. 1.6.3] whence, by the inequality in (18), the same holds for (η n ) n∈N . In conjunction with (33), this yields Combined with Lemma 3.1, this gives the asympotics of EL ⌣ n − n stated in (6), and it also implies Var L ⌣ n → ∞ by the inequality Eη 2 j ≥ (Eη j ) 2 and divergence of the harmonic series.
To prove the remaining claims will use the estimates for truncated moments of S j under the assumption E|ξ| p < ∞ for p ∈ [2, 4). First, by uniform integrability of (|S n | p /n p/2 ) n∈N , see again [10, Thm. 1.6.3], for all j ∈ N. Second, by Hölder's inequality applied for p > 2 with parameter p/2 and its conjugate, Note that the term in square brackets converges to zero as j → ∞ by uniform integrability of (|S n | p /n p/2 ) n∈N and the fact P{|S j | > εj} → 0 as j → ∞. Now we are able to derive the asymptotic expansion of EL ⌣ n up to O(1). By (18) and the Taylor formula already used in (21) and applied on the event {|S j | ≤ εj}, we get Dividing both sides by j 1−p/2 and sending first j → ∞ and then ε → 0, we obtain from (35) and (36) that Note that, by (34), this relation also holds for p = 2. Hence for any p > 2, This proves the first equality in (7). It remains to prove the claims on Var L ⌣ n . Note that Thus, by Lemma 3.1 and (37), This completes the proofs of (6) and (7). We are left with a proof of (8). If Eξ 4 < ∞, then the family (S 4 n /n 2 ) n∈N is uniformly integrable and the same holds for the family (η 2 n ) n∈N by the inequality in (18). Hence, α (t)) t∈[0,1] , k = 1, 2, . . . be a sequence of independent copies of the strictly stable process S α with index α ∈ (1, 2). Then, Proof. Fix an arbitrary δ ∈ (0, α/2). Using that x → x δ is subadditive and then the self-similarity of the process S α , we obtain is finite. To this end, formula (2.1) from [4] with φ(x) = x (2/α−1)δ can be used to see that This completes the proof and we note that the same argument applies to any strictly stable process of index α ∈ (0, 2).
Before passing to the proof of Theorem 1.2, we give an auxiliary lemma.
for any i, j ∈ N.
Proof. It is obviously enough to show that Proof of Theorem 1.2. Fixing a coupling between (Z n,k ) and (Z k ) such that (16) holds, we show first that Recall that b n = a 2 n /n with a n given by (2). By Lemma 3.3 and use of the triangle inequality, the expectation of the difference in (39) is bounded by a constant times where {x} denotes the fractional part of x ∈ R. Then use (16) to assess that the first summand on the right is of the same order as a −2 n n log n/4 and thus tending to zero as n → ∞ because (a 2 n ) n∈N is regularly varying with index 2/α > 1. The second summand tends to zero by the same reasoning and the fact that In view of (39) and representation (10), it remains to prove To this end, we use Theorem 3.2 in [5] for which the following two assertions must be verified: First, for any fixed m ∈ N, and, second, for any ε > 0, But this is true by the continuous mapping theorem and the joint convergence  which in turn holds because a n = o(n) as n → ∞ and (S i,j ) i,j∈N and (Z k ) k∈N are independent. For (42), we argue as follows. Using formula (18), it is enough to check that Fix δ ∈ (0, α/2). Then, using the subadditivity of x → x δ and Markov's inequality, we see that (43) is a consequence of (44) lim m→∞ lim sup By Lemma 5.2.2 in [11], there exists a constant C = C δ,α > 0 such that Therefore, and (44) will follow once having shown that Fix an arbitrary δ ′ ∈ (0, ( 2 α − 1)δ). Since (b δ n ) n∈N is regularly varying with index ( 2 α − 1)δ, we can apply Potter's bound to the slowly varying sequence (b δ n /n (2/α−1)δ ) n∈N , see [6, Thm. 1.5.6(ii)]. In combination with ⌊nZ k ⌋ ≤ n, this gives Moreover, by formula (2.1) in [4], Hence (46) follows and the proof of (42) is complete. Our argument is geometric and based on a simple comparison argument. First of all note that the concave majorant consists of two piecewise linear subparts: one connecting the origin and the point (τ n , M n ), and the other connecting (τ n , M n ) and (n, S n ). Denote their lengths by L ⌢ 1,n and L ⌢ 2,n , respectively, thus L ⌢ n = L ⌢ 1,n +L ⌢ 2,n . The triangle inequality applied to every segment of the concave majorant provides M n ≤ L ⌢ 1,n ≤ M n + τ n ≤ M n + n, and also M n − S n ≤ L ⌢ 2,n ≤ M n − S n + n − τ n ≤ M n − S n + n. Consequently, 2M n − S n ≤ L ⌢ n ≤ 2M n − S n + 2n. Similarly, S n − 2m n ≤ L ⌣ n ≤ S n − 2m n + 2n. To complete he proof of Theorem 1.3, it remains to note that n/a n → 0 because (a n ) n∈N is regularly varying with index 1/α > 1 and S n a n , M n a n , m n a n d −→ n→∞ (S α (1), sup as an immediate consequence of (2) and the continuous mapping theorem.

The case of nonzero mean
Throughout this section, we assume that µ := Eξ exists and is nonzero which rules out case (C). We are left with two possibilities: (A ′ ) Eξ 2 < ∞ and µ = 0; (B ′ ) Eξ 2 = ∞, µ = 0, and the law of ξ lies in the domain of attraction of an α-stable law with α ∈ (1, 2]. We stress that case (B ′ ) includes α = 2, as opposed to case (B). The essence of the next theorem is that in both cases (A ′ ) and (B ′ ), the distributional behavior of L ⌣ n as n → ∞ coincides with that of S n up to a linear centering and a scaling as in Theorem 1.7 in [16]. We will see that the same holds for L n , the perimeter of the convex hull of {(j, S j ) : j = 0, . . . , n}. where a n := √ n, σ 2 := Eξ 2 − µ 2 , and S 2 (1) := N (0, σ 2 ) in case (A ′ ), and (a n ) n∈N is a sequence and S α (1) an α-stable random variable such that (S n − µn)/a n d → S α (1). The convergence (48) does also hold with L ⌢ n and L n /2 in the place of L ⌣ n . The result for L n in case (A ′ ) is a particular case of Theorem 1.2 in [22] (combined with Theorem 1.8 in [16]), which applies because (n, S n ) n∈N 0 can be regarded as a random walk in the plane. The approach of [16,22] is different from the one employed here: we combine the formula L n = L ⌢ n + L ⌣ n with the limit result (48) (or, to be more precise, (49)) for L ⌣ n and its version for L ⌢ n , both obtained from representation (11). Recall that this argument does not work when µ = 0; cf. Remark 1.4.
Since the representation (50) is also valid for L ⌢ n instead of L ⌣ n , the convergence (48) and the coupling (49) (between L ⌣ n and a normalization of S n ) remain true for L ⌢ n in the place of L ⌣ n . By combining (49) for both L ⌣ n , L ⌢ n finally provides (48) for L n /2 = (L ⌣ n + L ⌢ n )/2.