Exchangeable stable random vectors and their simulations

This work concerns the simulation of an exchangeable stable random vector. A characterization for exchangeability of a stable random vector, in terms of its spectral measure, is given. The Modarres and Nolan’s simulating method on stable random vectors is modified to the exchangeable case. FORTRAN subroutines to simulate a desirable exchangeable stable random vector and to create an exchangeable partition are written.

The Authors would like to thank a referee for careful reading of the earlier version and providing valuable comments. Thanks also goes to professor J. P. Nolan for providing the authors the literature on the subject, subroutine VSTAB, and its driver program.

Authors and Affiliations

1. Department of Statistics, College of Sciences, Shiraz University, Shiraz, 71454, Islamic Republic of Iran

   Adel Mohammadpour & A. Reza Soltani

5 Appendix

A1. Constructing an exchangeable partition To prove Theorem 3.3 in Section 3 we need to modify Theorem 1 in Byczkwski, Nolan and Rajput (1993). To do this, we provide an exchangeable partition for S_d−1. We call \({{\cal S}^ \ast }\) a collection of disjoint sets in S_d−1, an exchangeable partition if

1. (i)

   \({S_{d - 1}} = \bigcup\nolimits_{E \in {{\cal S}^ \ast }} {} E\)

2. (ii)

   \(E \in {{\cal S}^ * } \Rightarrow \pi E \in {{\cal S}^ * }\), for any π.

In order to construct an exchangeable partition we proceed as follows: let s = (s₁, …, s_d)′, define

Now define

and \({{\cal S}^ \ast } = \{ \pi E,\;\;E \in {{\cal M}_{d - 1}},\;\;\forall \pi \} \). We call \({{\cal M}_{d - 1}}\) a minimal exchangeable generator for S_d−1 and \({{\cal S}^ \ast }\) a minimal exchangeable partition for S_d−1.

Lemma 5.1 The elements of \({{\cal S}^ \ast }\) form an exchangeable partition of S_d−1.

Proof It is straightforward to see that \({{\cal S}^ \ast }\) satisfies (ii). To prove (i) and that members of \({{\cal S}^ \ast }\) are disjoint, note that \(S_{jk}^ \ast \cap S_{jl}^ \ast = \emptyset ,\;\;k \ne l,\;\;S_{jk}^ \ast \cap S_{lk}^ \ast = \emptyset ,\;\;j \ne l\), and

Also \(\pi S_{jk}^ \ast \bigcap \eta S_{jl}^ \ast = \emptyset \), k ≠ l for any permutations π and η, to see this, assume that there are π, η, k < l such that \(\pi S_{jk}^ \ast \bigcap \eta S_{jl}^ \ast \ne \emptyset \). If \({\bf{z}} \in \pi S_{jk}^ \ast \bigcap \eta S_{jl}^ \ast \), then there are \({\bf{z}} \in S_{jk}^ \ast \), and \({\bf{y}} \in S_{jk}^ \ast \) for which \({\bf{z}} = \pi {\bf{x}} = ({x_{{\pi ^{ - 1}}1}},{x_{{\pi ^{ - 1}}2}}, \cdots ,{x_{{\pi ^{ - 1}}d}})\prime \in S_{jk}^ \ast \), and \({\bf{z}} = \eta {\bf{y}} = ({y_{{\eta ^{ - 1}}1}},{y_{{\eta ^{ - 1}}2}}, \cdots ,{y_{{\eta ^{ - 1}}d}})\prime \in \eta S_{jk}^ \ast \). But πx = ηy would simply that \({x_{{\pi ^{ - 1}}m}} = {y_{{\eta ^{ - 1}}m}}\), m = 1, …, d, which in tern implies

But

and

Now substituting (5.2) in (5.4) provides that

which is impossible, because of (5.3), unless π = η and k = l. Finally

where \(\bigcup\nolimits_\pi {} \pi S_{jk}^ \ast \) is the union of all distinct permutations of \(S_{jk}^ \ast \).

A2. A Modification of the Theorem of Byczkwski, Nolan and Rajput (1993) Let us now introduce some notions and notations that were employed in Byczkwski, Nolan and Rajput (1993). Let (Y, ρ) be a metric space, let S be a compact subset of Y, and let M(ε; S) = minimal number of closed ρ—balls of radius ε needed to cover S. For a function f from S to C, a uniform models of continuity is a function δ(ε) such that, ρ(x, y) < δ(ε) implies ∣f(x) − f(y)∣ < ε.ε For a family \({\cal F}\) of functions from S to C, a modulus of continuity is a single δ(.) that simultaneously is a uniform modulus of continuity for every \(f \in {\cal F}\).

Lemma 5.2 Let \({\cal F}\) be a family of continuous functions mapping S_d−1 into C. Suppose for ε > 0, there is a modulus of continuity δ(ε) for \({\cal F}\). If Γ is a finite exchangeable Borel measure on S_d−1, then for any ε > 0 there is a discrete exchangeable spectral measure Γ* on S_d−1 for which

Proof Fix ε, set M = M(δ(ε/Γ(S_d−1)); S_d−1) and let \({\cal B} = \{ {B_1}, \cdots ,{B_M}\} \) be closed ρ-balls of radius δ(ε/Γ(S_d−1)) that cover S_d−1. Now form the class \({\cal G} = \{ B \cap E\;\;:\;\;B \in {\cal B},\;\;E \in {{\cal M}_{d - 1}}\} \), where \({{\cal M}_{d - 1}}\) a minimal exchangeable generator for S_d−1. The class \({\cal G}\) is a finite class, \({\cal G} = \{ {G_1}, \cdots ,{G_N}\} \). Now define \(E_1^\ast = {G_1},\;\;E_j^\ast = {G_j} - ({ \cup _{k < j}}{G_k}),\;\;j = 2, \cdots ,N\) and let \({{\cal E}^ \ast } = \{ E_1^ \ast , \cdots ,E_N^ \ast \} \). An argument similar to the one given in A1 reveals that \({\cal E} = \{ \pi {E^ \ast },\;\;{E^ \ast } \in {{\cal E}^ \ast },\;\;\forall \pi \} \) is an exchangeable partition of S_d−1. Now corresponding to each E* ∈ ε* fixed an s* ∈ E* such that ρ(s*, s) < 2δ, s ∈ E*. Now define Γ*(.) such that for each π to be supported by {πs*, s* ∈ ∀π}, and

Then the finite measure Γ*(.) is exchangeable. For any \(f \in {\cal F}\),

where \({{\bf{s}}_E} = {\bf{s}}_j^ \ast \), if \(E = \pi E_j^ \ast \).

Remark 5.1 When the appropriate mass points in {s: s₁ ≤ … ≤ s_d} ∩ S_d−1 are chosen, then a discrete exchangeable Γ* is constructed by permuting the selected point masses. A FORTRAN subroutine (DESM) is written to perform this process (it can be downloaded from the author’s homepage).

In the next theorem without loss of generality we assume that the location parameter is zero.

Theorem 5.1 Let X be a truly d-dimensional ESRV (d ≥ 2, 0 < α < 2), with spectral measure Γ and density p(x). Let ε > 0.

1. (i)

   There is a discrete exchangeable spectral measure Γ* with corresponding exchangeable stable density p*(x) satisfying

   $$\mathop {\sup }\limits_{x \in {{\bf{R}}^d}} \vert{p}({\bf{x}}) - {p^ \ast }({\bf{x}})\vert \le \varepsilon .$$
2. (ii)

   There is a discrete exchangeable spectral measure Γ** with corresponding ESRV X** which satisfies

   $$\mathop {\sup }\limits_{E \in Borel({{\bf{R}}^d})} \vert{P}({\bf{X}} \in E) - P({{\bf{X}}^{ \ast \ast }} \in E)\vert\; \le \varepsilon $$

Proof The result follows by using Lemma 5.2 and an argument very similar to the one given by Byczkwski, Nolan and Rajput (1993).

A3. Proof of Theorem 3.3 Follows by using Theorem 5.1 instead of Theorem 1 in Byczkwski, Nolan and Rajput (1993), see also Lemma 2 in Modarres and Nolan (1994).

A4. Proof of Lemma 3.2 Let \(S_{d - 1}^\Gamma \) be the support of Γ. Now define:

• \(S_1^ \ast = \{ {\bf{s}}\;:\;{s_1} < {s_2} < \cdots < {s_d}\} \cap S_{d - 1}^\Gamma \),

• \(S_2^ \ast = \{ {\bf{s}}\;:\;{s_1} = {s_2} \ne {s_3} \ne \cdots \ne {s_d}\} \cap S_{d - 1}^\Gamma \), and for j = 3, …, d − 1,

• \(S_j^ \ast = \{ {\bf{s}}\;:\;{s_1} = {s_2} = \cdots = {s_j} \ne {s_{j + 1}} \ne {s_{j + 2}} \ne \cdots \ne {s_d}\} \cap S_{d - 1}^\Gamma \),

• \(S_d^ \ast = \{ {\bf{s}}\;:\;{s_1} = {s_2} = \cdots = {s_d}\} \cap S_{d - 1}^\Gamma \). Note that \(S_j^ \ast \cap S_k^ \ast = \emptyset ,\;\;j \ne k\), and since Γ is exchangeable it follows from Lemma 3.1 that

  $$\bigcup\limits_j {} \bigcup\limits_\pi {} \pi (S_j^ \ast ) = S_{d - 1}^\Gamma ,$$

  where \(\bigcup\nolimits_\pi {} \;\;\pi (S_j^ \ast )\) denotes union over all different permutation of the \(S_j^ \ast \). Therefore we can write

  $$n = \# (S_{d - 1}^\Gamma ) = \# \left( {\bigcup\limits_j {} \bigcup\limits_\pi {} \pi (S_j^ * )} \right) = \sum\limits_j {} \# \left( {\bigcup\limits_\pi {} \pi (S_j^ \ast )} \right)$$

Now let \({k_j} = \# (S_j^ \ast )\), then \({k_j} = \# \pi (S_j^ \ast )\), for each π, j = 1, …, d − 1, and since there are d!/j! distinct \(\pi (S_j^ \ast )\),

where k_d ∈ {0, 1, 2}. Define \(k = \sum\nolimits_{j = 1}^{d - 1} {} {{(d - 1)!} \over {j!}}{k_j}\) to conclude the result.

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