Convex hulls of random walks and their scaling limits

For the perimeter length and the area of the convex hull of the first $n$ steps of a planar random walk, we study $n \to \infty$ mean and variance asymptotics and establish non-Gaussian distributional limits. Our results apply to random walks with drift (for the area) and walks with no drift (for both area and perimeter length) under mild moments assumptions on the increments. These results complement and contrast with previous work which showed that the perimeter length in the case with drift satisfies a central limit theorem. We deduce these results from weak convergence statements for the convex hulls of random walks to scaling limits defined in terms of convex hulls of certain Brownian motions. We give bounds that confirm that the limiting variances in our results are non-zero.


Introduction
Random walks are classical objects in probability theory. Recent attention has focused on various geometrical aspects of random walk trajectories. Many of the questions of stochastic geometry, traditionally concerned with functionals of independent random points, are also of interest for point sets generated by random walks. Here we examine the asymptotic behaviour of the convex hull of the first n steps of a random walk in R 2 , a natural geometrical characteristic of the process. Study of the convex hull of planar random walk goes back to Spitzer and Widom [21] and the continuum analogue, convex hull of planar Brownian motion, to Lévy [15, §52.6, pp. 254-256]; both have received renewed interest recently, in part motivated by applications arising for example in modelling the 'home range' of animals. See [16] for a recent survey of motivation and previous work. The method of the present paper in part relies on an analysis of scaling limits, and thus links the discrete and continuum settings.
Let Z be a random vector in R 2 , and let Z 1 , Z 2 , . . . be independent copies of Z . Set S 0 := 0 and S n :=  n k=1 Z k ; S n is the planar random walk, started at the origin, with increments distributed as Z . We will impose a moments condition of the following form: Throughout the paper we assume (usually tacitly) that the p = 2 case of (M p ) holds. For several of our results we impose a stronger condition and assume that (M p ) holds for some p > 2, in which case we say so explicitly.
For a subset S of R d , its convex hull, which we denote hull S, is the smallest convex set that contains S. We are interested in hull {S 0 , S 1 , . . . , S n }, which is a (random) convex polygon, and in particular in its perimeter length L n and area A n . (See Fig. 1.) The perimeter length L n has received some attention in the literature, initiated by the remarkable formula of Spitzer and Widom [21], which states that EL n = 2 n  k=1 k −1 E∥S k ∥, for all n ∈ N := {1, 2, . . .}.
(1) Table 1 Each of the simulation estimates is based on 10 5 instances of a walk of length n = 10 5  Much later, Snyder and Steele [20] obtained the law of large numbers lim n→∞ n −1 L n = 2∥µ∥, a.s.; this is stated for the case µ ̸ = 0 in [20] but the proof works equally well in the case µ = 0.
To prove their law of large numbers, Snyder and Steele used the Spitzer-Widom formula (1) and the variance bound [20,Theorem 2.3] n −1 Var L n ≤ π 2 σ 2 2 , for all n ∈ N.
The natural question of the second-order behaviour of L n was left largely open; similar questions may be posed about A n .
In [23], a central limit theorem to accompany (3) was also obtained: provided σ 2 µ > 0, n −1/2 (L n −EL n ) converges in distribution to a normal random variable with mean 0 and variance 4σ 2 µ . If Σ is positive definite, then both σ 2 µ and σ 2 µ ⊥ are strictly positive, but our results are still of interest when one or other of them is zero (the case where both are zero being entirely trivial).
The aims of the present paper are to provide second-order information for L n in the case µ = 0, and to study the area A n for both the cases µ = 0 and µ ̸ = 0. For example, we will show that The quantities v 0 and v + in (4) are finite and positive, as is u 0 ( • ) provided σ 2 ∈ (0, ∞), and these quantities are in fact variances associated with convex hulls of Brownian scaling limits for the walk. These scaling limits provide the basis of the analysis in this paper; the methods are necessarily quite different from those in [23]. The result lim n→∞ n −1 Var L n > 0 in the case µ = 0 answers a question raised by Snyder and Steele [20, §5]. For the constants u 0 (I ) (I being the identity matrix), v 0 , and v + , Table 1 gives numerical evaluations of rigorous bounds that we prove in Proposition 3.7, plus estimates from simulations. See also Section 4 for an explicit integral expression for u 0 (I ). Table 2 Results originate from: a [21]; b [20]; c [23]; d [1] (in part); the rest are new. The limit laws exclude degenerate cases when associated variances vanish.
Limit exists for E Limit exists for Var Limit law Furthermore, we show below that distributional limits accompanying the three variance asymptotics in (4) are non-Gaussian, excluding trivial cases, by contrast to the central limit theorem accompanying (3) from [23]. Also notable is the comparison between the variance asymptotics for µ ̸ = 0 in (3) and (4): each of the components σ 2 µ and σ 2 µ ⊥ of σ 2 contributes to exactly one of the asymptotics for Var L n and Var A n . Other results that we present below include asymptotics for expectations.

Examples.
Here are some examples to illustrate a range of asymptotic behaviours exhibited by some simple models. We summarize what now is known in general in Table 2.
The outline of the rest of the paper is as follows. In Section 2 we describe our scaling limit approach, and carry it through after presenting the necessary preliminaries; the main results of this section, Theorems 2.5 and 2.7, give weak convergence statements for convex hulls of random walks in the case of zero and non-zero drift, respectively. Armed with these weak convergence results, we present asymptotics for expectations and variances of the quantities L n and A n in Section 3; the arguments in this section rely in part on the scaling limit apparatus, and in part on direct random walk computations. This section concludes with upper and lower bounds for the limiting variances. Section 4 collects some final remarks and open questions. Finally, the Appendix collects some auxiliary results on random walks that we use.

Overview
We describe the general idea of our approach. Recall that S n =  n k=1 Z k is the location of our random walk in R 2 after n steps. Write S n := {S 0 , S 1 , . . . , S n }. Our strategy to study properties of the random convex set hull S n (such as L n or A n ) is to seek a weak limit for a suitable scaling of hull S n , which we must hope to be the convex hull of some scaling limit representing the walk S n .
In the case of zero drift (µ = 0) a candidate scaling limit for the walk is readily identified in terms of planar Brownian motion. For the case µ ̸ = 0, the 'usual' approach of centering and then scaling the walk (to again obtain planar Brownian motion) is not useful in our context, as this transformation does not act on the convex hull in any sensible way. A better idea is to scale space differently in the direction of µ and in the orthogonal direction.
In other words, in either case we consider φ n (S n ) for some affine continuous scaling function φ n : R 2 → R 2 . The convex hull is preserved under affine transformations, so the convex hull of a random set which will have a weak limit. We will then be able to deduce scaling limits for quantities L n and A n provided, first, that we work in suitable spaces on which our functionals of interest enjoy continuity, so that we can appeal to the continuous mapping theorem for weak limits, and, second, that φ n acts on length and area by simple scaling. The usual n −1/2 scaling when µ = 0 is fine; for µ ̸ = 0 we scale space in one coordinate by n −1 and in the other by n −1/2 , which acts nicely on area, but not length. Thus these methods work in exactly the three cases corresponding to (4).
In view of the scaling limits that we expect, it is natural to work not with point sets like S n , but with continuous paths; instead of S n we consider the interpolating path constructed as follows. For each n ∈ N and all t ∈ [0, 1], define Note that X n (0) = S 0 = 0 and X n (1) = S n . Given n, we are interested in the convex hull of the image in R 2 of the interval [0, 1] under the continuous function X n . Our scaling limits will be of the same form.

Paths, hulls, and hulls of paths
We introduce the setting in which we will describe our scaling limit results. At this point, it is no extra difficulty to work in R d for general d ≥ 2. Let ρ(x, y) = ∥x − y∥ denote the Euclidean distance between x and y in R d . For T > 0, let C([0, T ]; R d ) denote the class of continuous Usually, we work with T = 1, in which case we write simply is the section of the path up to time t ∈ [0, T ].
We need some notation and concepts from convex and integral geometry: we found [9,19,21] to be very useful. For a set A ⊆ R d , write ∂ A for its boundary and int(A) := A\∂ A for its interior.
Let K d denote the collection of convex compact sets in R d , and K 0 d := {A ∈ K d : 0 ∈ A} those that contain the origin. Given A ∈ K d , for r ≥ 0 set Two equivalent descriptions of ρ H (see e.g. Proposition 6.3 of [9]) are For the rest of this section we study some basic properties of the map from a continuous path to its convex hull.
is compact, and hence Carathéodory's theorem for convex hulls (see Corollary 3.1 of [9, p. 44]) shows that hull f [0, t] is also compact. So hull f [0, t] ∈ K d is convex, bounded, and closed; in particular, it is a Borel set.
It mostly suffices to work with paths parametrized over The next result shows that the function H : Proof. Let f, g ∈ C 0 d . Then H ( f ) and H (g) are non-empty, as they contain Since the convex hull of a set is the set of all convex combinations of points of the set (see Lemma 3.1 of [9, p. 42]), there exist n ∈ N, weights λ 1 , . . . , , we have that y ∈ H (g) and, by the triangle inequality, Thus, writing r = ρ ∞ ( f, g), every x ∈ H ( f ) has x ∈ π r (H (g)), i.e., H ( f ) ⊆ π r (H (g)). The symmetric argument starting with x ∈ H (g) shows that H (g) ⊆ π r (H ( f )) as well. Hence, by (5), we obtain (7).

We end this section by showing that the map
, so that the continuous trajectory t  → f (t) is accompanied by a continuous 'trajectory' of convex hulls. This observation was made by El Bachir [4, pp. 16-17]; we take a different route based on the path-space result Lemma 2.1. First we need a lemma.
Proof. First we fix t ∈ [0, T ] and show that s  → g t (s) is continuous, so that g t ∈ C d as claimed.
Since f is continuous on the compact interval [0, T ], it is uniformly continuous, and admits a monotone modulus of continuity µ f : , and µ f (r ) ↓ 0 as r ↓ 0 (see e.g. [12, p. 57]). Hence which tends to 0 as ρ(t 1 , t 2 ) → 0, again using the uniform continuity of f .
Here is the path continuity result for convex hulls of continuous paths; cf [4, pp. 16-17].
is the composition of two continuous functions, hence itself a continuous function.

Functionals of planar convex hulls
Now, and for the rest of the paper, we return to d = 2 to address our main questions of interest; parts of what follows carry over to general d ≥ 2, but we do not pursue that generality here. We consider functionals A : K 2 → R + and L : K 2 → R + given by the area and the perimeter length of convex compact sets in the plane. Formally, we define A as Lebesgue measure on R 2 , and then The limit in (8) exists by the Steiner formula of integral geometry (see e.g. [21]), which expresses A(π r (A)) as a quadratic polynomial in r whose coefficients are given in terms of the intrinsic volumes of A: In particular, with H d denoting d-dimensional Hausdorff measure, It follows from Cauchy's formula that L is increasing in the sense that if A, B ∈ K 0 2 satisfy A ⊆ B, then L(A) ≤ L(B); clearly the functional A is also increasing. The next result shows that the functions L and A are both continuous from (K 0 2 , ρ H ) to (R + , ρ).
Proof. First consider L. By Cauchy's formula and the triangle inequality, which with (6) gives (10).

Brownian convex hulls as scaling limits
The two different scalings outlined in Section 2.1, for the cases µ = 0 and µ ̸ = 0, lead to different scaling limits for the random walk. Both are associated with Brownian motion.
In the case µ = 0, the scaling limit is the usual planar Brownian motion, at least when Σ = I , the identity matrix. Let b := (b(s)) s∈[0,1] denote standard Brownian motion in R 2 , started at b(0) = 0. For convenience we may assume b ∈ C 0 2 (we can work on a probability space for which continuity holds for all sample points, rather than merely almost all). For t ∈ [0, 1], let h t := hull b[0, t] ∈ K 0 2 denote the convex hull of the Brownian path up to time t. By Proposition 2.3, t  → h t is continuous. Much is known about the properties of h t : see e.g. [2,4,6,13]. We also set ℓ t := L(h t ), and a t := A(h t ), the perimeter length and area of the standard Brownian convex hull. By Lemma 2.4, the processes t  → ℓ t and t  → a t have continuous and non-decreasing sample paths.
We also need to work with the case of general covariances Σ ; to do so we introduce more notation and recall some facts about multivariate Gaussian random vectors. For definiteness, we view vectors as Cartesian column vectors when required. Since Σ is positive semidefinite and symmetric, there is a (unique) positive semidefinite symmetric matrix square-root Σ 1/2 for which The map x  → Σ 1/2 x associated with Σ 1/2 is a linear transformation on R 2 with Jacobian det N (0, Σ ), a bivariate normal distribution with mean 0 and covariance Σ ; the notation permits Σ = 0, in which case N (0, 0) stands for the degenerate normal distribution with point mass at 0. Similarly, given b a standard Brownian motion on R 2 , the diffusion Σ 1/2 b is correlated planar Brownian motion with covariance matrix Σ . We write '⇒' to indicate weak convergence. Proof. Donsker's theorem implies that n −1/2 X n ⇒ Σ 1/2 b on (C 0 2 , ρ ∞ ). Now, the point set Remark. If a real-valued random variable X is Gaussian and non-degenerate (i.e., Var X > 0), then ess inf X = −∞ and ess sup X = +∞. The distributional limits for n −1/2 L n and n −1 A n in Corollary 2.6 are supported on R + and, as we will show in Proposition 3.7, are non-degenerate if Σ is positive definite; hence they are non-Gaussian excluding trivial cases.
(ii) The limit in Corollary 2.8 is non-negative and non-degenerate (see Proposition 3.7) and hence non-Gaussian.

Expectation asymptotics
We start with asymptotics for EL n and EA n in the case µ = 0. These results, Propositions 3.1 and 3.3, are in part already contained in [21] and [1] respectively; we give concise proofs here since several of the computations involved will be useful later. The first result, essentially given in [21, p. 508], is for L n . Cauchy's formula applied to the line segment from 0 to Y with Fubini's theorem implies Here Y · e = e ⊤ Y is univariate normal with mean 0 and variance )e ⊤ Σ e = ∥Σ 1/2 e∥ 2 , so that E[(Y · e) + ] is ∥Σ 1/2 e∥ times one half of the mean of the squareroot of a χ 2 1 random variable. Hence E∥Y ∥ = (8π ) −1/2  S 1 ∥Σ 1/2 e∥de, which in general may be expressed via a complete elliptic integral of the second kind in terms of the ratio of the eigenvalues of Σ . In the particular case Σ = I , E∥Y ∥ = √ π/2 so then Proposition 3.1 implies that matching the formula Eℓ 1 = √ 8π of Letac and Takács [14,22]. We also note the bounds the upper bound here is from Jensen's inequality and the fact that E[∥Y ∥ 2 ] = tr Σ . The lower bound in (12) follows from the inequality where ∥ • ∥ op is the matrix operator norm and λ Σ is the largest eigenvalue of Σ ; in statistical terminology, λ Σ is the variance of the first principal component associated with Y .
Then by Lemma A.1(ii) we have sup n E[(n −1/2 L n ) 2 ] < ∞. Hence n −1/2 L n is uniformly integrable, so that Corollary 2.6 yields lim n→∞ n −1/2 EL n = EL(Σ 1/2 h 1 ). It remains to show that lim n→∞ n −1/2 EL n = 4E∥Y ∥. One can use Cauchy's formula to compute EL(Σ 1/2 h 1 ); instead we give a direct random walk argument, following [21]. The central limit theorem for S n implies that n −1/2 ∥S n ∥ → ∥Y ∥ in distribution. Under the given conditions, It follows that n −1/2 ∥S n ∥ is uniformly integrable, and hence lim n→∞ n −1/2 E∥S n ∥ = E∥Y ∥. The result now follows from some standard analysis based on (1) and the fact that lim n→∞ n −1/2  n k=1 k −1/2 = 2. Now we move on to the area A n . First we state some useful moments bounds.  Proof. First we prove (ii). Since hull{S 0 , . . . , S n } is contained in the disk of radius max 0≤m≤n ∥S m ∥ and centre 0, we have A p n ≤ π p max 0≤m≤n ∥S m ∥ 2 p . Lemma A.1(ii) then yields part (ii). For part (i), it suffices to suppose µ ̸ = 0. Then, bounding the convex hull by a rectangle, Hence, by the Cauchy-Schwarz inequality, we have Now an application of Lemma A.1(i) and (iii) gives part (i).
The asymptotics for EA n in the case µ = 0 are given in the following result, which is in part contained in [1, p. 325].
Given Corollary 2.6, one may also deduce the limit result in Proposition 3.3 from the formula Ea 1 = π 2 of El Bachir [4, p. 66] with a uniform integrability argument; however, the naïve approach seems to require a slightly stronger moments assumption, such as (M p ) for some p > 2 (cf Lemma 3.2). The proof of Proposition 3.3 is based on an analogue for EA n of the Spitzer-Widom formula, due to Barndorff-Nielsen and Baxter [1]. To state the formula, let T (u, v) (u, v ∈ R 2 ) be the area of a triangle with sides of u, v and u + v. Note that for α, β > 0, T (αu, βv) = αβT (u, v). The formula of [1] states Proof of Proposition 3.3. First we show that, under the given conditions, where Y 1 and Y 2 are independent N (0, Σ ) random vectors. Indeed, it follows from the central limit theorem in R 2 and the continuity of T that which is uniformly bounded for k ≥ m + 1 ≥ 0, by Lemma A.1. It follows that m −1/2 (k − m) −1/2 T (S m , S k − S m ) is uniformly integrable over (m, k) with m ≥ 1, k ≥ m + 1, and the claim (15) follows.
Next we move on to the case µ ̸ = 0. The following result on the asymptotics of EA n in this case is, as far as we are aware, new.
Proposition 3.4. Suppose that (M p ) holds for some p > 2, µ ̸ = 0, and σ 2 µ ⊥ > 0. Then In particular, Proof In light of (18), it remains to identify Eã 1 = 1 3 √ 2π . It does not seem straightforward to work directly with the Brownian limit; it turns out again to be simpler to work with a suitable random walk. We choose a walk that is particularly convenient for computations.
Let ξ ∼ N (0, 1) be a standard normal random variable, and take Z to be distributed as Z = (1, ξ ) in Cartesian coordinates. Then S n = (n,  n k=1 ξ k ) is the space-time diagram of the symmetric random walk on R generated by i.i.d. copies ξ 1 , ξ 2 , . . . of ξ .

Variance asymptotics
We are now able to give formally the results quoted in (4), and to explain the constants that appear in the limits. Indeed, these are defined to be If, in addition, (M p ) holds for some p > 4, then Proof. From (13)

Variance bounds
The next result gives bounds on the quantities defined in (19).
Proposition 3.7. We have u 0 (Σ ) = 0 if and only if tr Σ = 0. The following inequalities for the quantities defined at (19) hold.
0 < 4 49 Finally, if Σ = I we have the following sharper form of the lower bound in (20): For the proof of this result, we rely on a few facts about one-dimensional Brownian motion, including the bound (see e.g. equation (2.1) of [11]), valid for all r > 0, We let Φ denote the distribution function of a standard normal random variable; we will also need the standard Gaussian tail bound (see e.g. [3, p. 12]) We also note that for e ∈ S 1 the diffusion e · (Σ 1/2 b) is one-dimensional Brownian motion with variance parameter e ⊤ Σ e. The idea behind the variance lower bounds is elementary. For a random variable X with mean EX , we have, for any θ ≥ 0, If EX ≥ 0, taking θ = αEX for α > 0, we obtain and our lower bounds use whichever of the latter two probabilities is most convenient.
Proof of Proposition 3.7. We start with the upper bounds. Snyder and Steele's bound (2) with the statement for Var L n in Proposition 3.5 gives the upper bound in (20). Boundingã 1 by the area of a rectangle, we havẽ where r 1 := sup 0≤s≤1 w(s) − inf 0≤s≤1 w(s). A result of Feller [7] states that E[r 2 1 ] = 4 log 2. So by the first inequality in (26), we have E[ã 2 1 ] ≤ 4 log 2, and by Proposition 3.4 we have 2π ; the upper bound in (22) follows.
Similarly, for any orthonormal basis {e 1 , e 2 } of R 2 , we bound a 1 by a rectangle and the two (orthogonal) components are independent, so E[a 2 1 ] ≤ (E[r 2 1 ]) 2 = 16(log 2) 2 , which with the fact that Ea 1 = π 2 [4] gives the upper bound in (21). We now move on to the lower bounds. Let e Σ ∈ S 1 denote an eigenvector of Σ corresponding to the principal eigenvalue λ Σ . Then since Σ 1/2 h 1 contains the line segment from 0 to any (other) point in Σ 1/2 h 1 , we have from monotonicity of L that Here e Σ · (Σ 1/2 b) has the same distribution as λ 1/2 Σ w. Hence, for α > 0, using the fact that λ Σ ≥ 1 2 tr Σ and the upper bound in (12). Applying (25) to X = L(Σ 1/2 h 1 ) ≥ 0 gives, for α > 0, using the lower bound in (12) and the fact that P[sup 0≤s≤1 w(s) ≥ r ] = 2P[w(1) ≥ r ] = 2(1 − Φ(r )) for r > 0, which is a consequence of the reflection principle. Numerical curve sketching suggests that α = 1/5 is close to optimal; this choice of α gives, using (24), which is the lower bound in (20). We get a sharper result when Σ = I and L(h 1 ) = ℓ 1 , since we know Eℓ 1 = √ 8π explicitly. Then, similarly to above, we get Var ℓ 1 ≥ 8π α 2 P  sup 0≤s≤1 w(s) ≥ (1 + α) √ 2π  , for α > 0, which at α = 1/4 yields the stated lower bound. For areas, tractable upper bounds for a 1 andã 1 are easier to come by than lower bounds, and thus we obtain a lower bound on the variance by showing the appropriate area has positive probability of being smaller than the corresponding mean.
Remark. The main interest of the lower bounds in Proposition 3.7 is that they are positive; they are certainly not sharp. The bounds can surely be improved, although the authors have been unable to improve any of them sufficiently to warrant reporting the details here. We note just the following idea. A lower bound forã 1 can be obtained by conditioning on θ := sup{s ∈ [0, 1] : w(s) = 0} and using the fact that the maximum of w up to time θ is distributed as the maximum of a scaled Brownian bridge; combining this with the previous argument improves the lower bound on v + to 2.09 × 10 −6 .
We have not been able to deal with this integral analytically, but numerical integration gives E[ℓ 2 1 ] ≈ 26.1677, which with the fact that Eℓ 1 = √ 8π gives u 0 (I ) = Var ℓ 1 ≈ 1.0350, in reasonable agreement with the simulation estimate in Table 1.
Another possible approach to evaluating u 0 is suggested by a remarkable computation of Goldman [8] for the analogue of u 0 (I ) = Var ℓ 1 for the planar Brownian bridge. Specifically, if b ′ t is the standard Brownian bridge in R 2 with b ′ 0 = b ′ 1 = 0, and ℓ ′ 1 = L(hull b ′ [0, 1]) the perimeter length of its convex hull, [8,Théorème 7] states that

Final comments
The framework of Section 2 shows that whenever a discrete-time process in R d converges weakly to a limit on the space of continuous paths, the corresponding convex hulls converge. It would be of interest to extend the framework to admit discontinuous limit processes, such as Lévy processes with jumps [13] that arise as scaling limits of random walks whose increments have infinite variance.
The idea used in the proof of Proposition 3.4, first establishing the existence of a limit for a class of models and then choosing a particular model for which the limit can be conveniently evaluated, goes back at least to Kac; see [5, p. 293].