Convex hulls of Lévy processes

Let X(t), t ≥ 0, be a Lévy process in R starting at the origin. We study the closed convex hull Zs of {X(t) : 0 ≤ t ≤ s}. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set Zs and find explicit expressions for their means in the case of symmetric α-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of Zs for all s > 0. Limit theorems for the convex hull of Lévy processes with normal and stable limits are also obtained.

The mixed volume is a function of d arguments, and K[j] stands for its j arguments, all being K.The intrinsic volumes of a convex body K are normalised mixed volumes where B d denotes the centred d-dimensional unit ball and κ d−j is the (d − j)-dimensional volume of B d−j .In particular, V d (K) is the volume (or the Lebesgue measure), V d−1 (K) is half the surface area, V d−2 (K) is proportional to the integrated mean curvature, V 1 (K) is proportional to the mean width of K, and V 0 (K) = 1, see [15,Sec. 4.2,5.3].The The j-dimensional volume of the parallelepiped spanned by u 1 , . . ., u j ∈ R d is denoted by D j (u 1 , . . ., u j ).
Let X(t), t ≥ 0, be a Lévy process in R d starting at the origin.We are interested in where cl(•) denotes the closure and conv(•) the convex hull.It is easy to see that Z s is a random closed convex set, see [10].We shortly denote Z = Z 1 .In the case of a Brownian motion, the expected intrinsic volumes of Z are known, see [2].The integrability of intrinsic volumes of Z for X being a symmetric α-stable Lévy process with independent coordinates and the expected mean width of Z were obtained in [6].
We start by establishing the integrability of intrinsic volumes in relation to the properties of the Lévy measure of the general Lévy process and then find explicit expressions for their first moments of all intrinsic volumes for the case of symmetric α-stable Lévy processes with α ∈ (1,2].For this purpose, we generalise results on expected random determinants from [5,16]. It is also shown that the origin a.s.belongs to the interior of the convex hull of symmetric Lévy processes if the scalar product X(t), u is non-atomic for each u = 0.As a direct consequence, we prove that X(s) a.s.belongs to the interior of the convex hull of {X(t) : 0 ≤ t ≤ s}.We also consider expectations of the L p -generalisations of mixed volumes and prove limit theorems for the scaled Z t as t → ∞.

Integrability of the intrinsic volumes
The integrability of V j (Z s ) p for some p > 0, s ≥ 0, and all j = 0, . . ., d is equivalent to Thus, the existence of the pth moment in the left-hand side is equivalent to the existence of the pth moments of all (non-negative) summands in the right-hand side.
Proof.The result is obvious if X(t), t ≥ 0, is a deterministic process, so we exclude this case in the following.The main idea is to split the path of the Lévy process into several parts with integrable volumes of their convex hulls.The random variables T 0 = 0 and form an increasing sequence of stopping times with respect to the filtration generated by the process, see [1,Cor. 8].Since X(t) has unbounded support for each non-trivial Lévy process and any t > 0 [14, Th. 24.3], these stopping times are a.s.finite.The random variables Ti = T i − T i−1 , i ≥ 1, are independent identically distributed.We set T0 = 0 and consider the renewal process It is easy to see that EN j s < ∞ for all j ≥ 1.Let I k be the segment in R d with end-points at the origin and X(min(T k , s)) − X(min(T k−1 , s)), k ≥ 1.
The right-hand side equals the linear combination of the mixed volumes Since the lengths of the segments I 1 , . . ., I j are independent, E( where r > 1 satisfies pr < β ν .Hence, Hölder's inequality yields that where 1/r + 1/q = 1.Since all moments of N s are finite, the series converges. Corollary 1.2.If X is the Brownian motion, then EV j (Z s ) p < ∞ for all p ≥ 0, all j = 0, . . ., d and all s ≥ 0.
An analogue of Theorem 1.1 holds for random walks.Let {ξ n , n ≥ 1} be a sequence of i.i.d.random vectors in R d and let Denote by C n the convex hull of the origin and S 1 , . . ., S n .The following result can be proved similarly to Theorem 1.1.

Expected intrinsic volumes
Let X(t), t ≥ 0, be a symmetric α-stable Lévy process in R d .In the Gaussian case all moments of V j (Z), j = 0, . . ., d, exist.If α < 2, then its Lévy measure is ν(dx) = c x −d−α for a constant c > 0, so that EV j (Z) p < ∞ for each p ∈ [0, α).In this section we calculate the expected intrinsic volumes of Z assuming that α > 1.
Recall that the characteristic function of X(t) can be represented as where X(t), u is the scalar product and is the support function of a convex body K called the associated zonoid of X( 1), see [6,11].Zonoids are convex bodies that are obtained as limits (with respect to the Hausdorff metric) of zonotopes, i.e.Minkowski sums of segments.
Recall that a random compact set Y in R d is said to be integrably bounded if Y = sup{ u : u ∈ Y } is integrable.Its selection expectation EY is defined as the convex body with support function Eh(Y, u), u ∈ R d , see [10,Sec. 2.1].Each zonoid K can be obtained as the selection expectation of the segment [0, ξ] with a suitably chosen ξ.
We start by proving an auxiliary result on the expected j-dimensional volume of a parallelepiped spanned by random vectors ξ 1 , . . ., ξ j ∈ R d .Theorem 2.1.Let j ∈ {1, . . ., d}.If ξ 1 , . . ., ξ j ∈ R d are independent integrable random vectors, then Proof.For all k = 1, . . ., j, and We define the measure ρ k on Borel sets A in the unit sphere S d−1 by letting meaning that ρ k is the generating measure of the zonoid E[−ξ k , ξ k ], see [15,Th. 3.5.3].
Let M j be a d × j matrix composed of j columns being i.i.d.copies of a random vector ξ ∈ R d .Corollary 2.2.If ξ ∈ R d is an integrable random vector, then ECP 21 (2016), paper 69.
Proof.The main idea is to approximate the Lévy process with the random walk S i = X(i/n), i = 0, . . ., n. Denote by C n the convex hull of S 0 , S 1 , . . ., S n .It is shown in [17] that where S n , . . ., S n are i.i.d.random walks that arise from i.i.d.copies X (1) , . . ., X (d) of the Lévy process.The determinant is known as the Gram determinant and is always non-negative, so that the absolute value can be omitted.
Since X(t) coincides in distribution with t 1/α X(1) for any t > 0, where the last equality follows from Corollary 2.2.It is shown in [11] that where K is the associated zonoid of X(1).Thus, It follows from the Stolz-Cesàro theorem [12,Th. 1.22] that ECP 21 (2016), paper 69. Since for all x > 0, it remains to find the limit of As n → ∞, the limit can be written using the multinomial Beta function related to the Dirichlet distribution, see [4, p. 11], as . Finally, and the result follows from the homogeneity of the intrinsic volumes.
Remark 2.4.By the self-similarity property, Z s coincides in distribution with s 1/α Z for s > 0, whence EV j (Z s ) = s j/α EV j (Z).

Interior of the convex hull
It is well known that the convex hull of the Brownian motion in R d contains the origin as interior point with probability 1 for each s > 0, see [3].We extend this result for symmetric Lévy processes.Theorem 3.1.Let X(t), t ≥ 0, be a symmetric Lévy process in R d , such that X(1), u has a non-atomic distribution for each u = 0. Then P{0 ∈ Int Z s } = 1 and P{X(s) ∈ Int Z s } = 1 for each s > 0.
Proof.Note that where ∂Z s is the topological boundary of Z s .We approximate Z s with the convex hull C n = conv{X(is/n), i = 0, . . ., n} of the random walk embedded in the process.Observe that the sequence of events {0 ∈ ∂C n } is decreasing to {0 ∈ ∂Z s }.Denote by Y n the number of faces of C n that contain the origin as a vertex.Since X(t) is symmetric and, in view of the imposed condition, P{X(t) ∈ H} = 0 for any affine hyperplane H in R d and t > 0, [17, Eq. ( 14), (15)] yield that Thus, EY n → 0 as n → ∞, so that Now we prove the second statement and make the convention that X(0−) = 0.The time reversal of X(t), t ∈ [0, s], is defined as It is well known that X(t) coincides in distribution with −X(t), see [1,Sec. II.1].By symmetry of the process and the fact that the origin almost surely belongs to the interior of the convex hull, This relation is equivalent to The distribution of X(1), u is non-atomic for each u = 0 if projections of the Lévy measure ν on any one-dimensional linear subspace of R d is infinite outside the origin, see [14,Th. 27.4], or if X(1) has a non-trivial full-dimensional Gaussian component.

L p -geometry of the convex hull
For convex bodies L and M that both contain the origin and p ∈ [1, ∞), the L p -sum L + p M is defined via its support function as The L p -generalisation of this mixed volume is defined by where it is required that L and M are convex bodies with the origin as interior point, see [15,Eq. (9.11)].
We generalise the expression for EV (L[d − 1], Z [1]) from [6] for the L p -case and symmetric α-stable Lévy processes in R d with α ∈ (1, 2].The case α = 2 is considered separately, since the integrability condition is different and in this case we obtain an explicit expression. ECP 21 (2016), paper 69.
(ii) Assume that (5.1) does not hold, ν({x : x ≥ r}), r > 0, is a regularly varying function at infinity with exponent −α for α ∈ (0, 2), and ν({x as r → ∞ for all Borel subsets A of the unit sphere S d−1 such that Λ(∂A) = 0, where Λ is a finite Borel measure on the unit sphere which is not supported by any proper linear subspace.Then X(T 1 ) belongs to the domain of attraction of an α-stable law.
Proof.It is well known that a Lévy process X can be expressed as the sum of three independent Lévy processes X 1 , X 2 , X 3 , where X 1 is a linear transform of a Brownian motion with drift, X 2 is a compound Poisson process having only jumps of norm strictly larger than 2 and X 3 is a pure jump process having jumps of size at most 2.This decomposition is known as the Lévy-Itô decomposition.It is obvious that X(T 1 ) = X(T 1 −) + X 2 (T 1 ) − X 2 (T 1 −) + X 3 (T 1 ) − X 3 (T 1 −).
Since the norm of X(T 1 −)+X 3 (T 1 )−X 3 (T 1 −) is at most 3, it is in the domain of attraction of the normal distribution.The compound Poisson process X 2 (t) = i: τi≤t ξ i is given by the sum of i.i.d.random vectors ξ i , i ≥ 1, with the distribution given by ν restricted onto the complement of the ball of radius 2 and normalised to become a probability measure.Here {τ i , i ≥ 1} is a homogeneous Poisson process on R + .Thus, T 1 ≤ τ 1 a.s., and X 2 (T 1 ) − X 2 (T 1 −) is zero if T 1 < τ 1 , while otherwise it equals ξ 1 .If (5.1) holds, then E ξ 1 2 < ∞, so that X 2 (T 1 ) − X 2 (T 1 −) belongs to the domain of attraction of a normal law.Otherwise, the conditions of the theorem guarantee that X 2 (T 1 ) − X 2 (T 1 −) belongs to the domain of attraction of an α-stable law, see [9,Th. 8.2.18].Then X(T 1 ) also belongs to the domain of attraction of the same law.
the scaled translate of I k that is symmetric with respect to the origin and such that ζ k = 1.It is well known that the mixed volumes are translation invariant.In view of (0.2), the McMullen-Matheron-Weil formula [7, Eq. (1.4)] yields