La Municipalidad Provincial de Cajamarca

We begin by introducing some notation and by considering some preliminaries on the exponential map and operator self-similarity that are used throughout the paper.

4.2.1 Some notation

In this paper, the notation and terminology for finite-dimensional operator theory will be prevalent over their matrix analogues. However, whenever convenient the latter will be used.

All with respect to the fieldR,M(n) orM(n,R) is the vector space of alln×noperators (endomorphisms), GL(n) or GL(n,R) is the general linear group (invertible operators, or automorphisms), O(n) is the orthogonal group of operators O such that OO∗ _{=I =O}∗_O

(i.e., the adjoint operator is the inverse), SO(n) ⊆ O(n) is the special orthogonal group of operators with determinant equal to 1, andso(n) is the vector space of skew-symmetric operators (i.e., A∗ _{=−A). The sign * always indicates the adjoint operator, regardless of}

whether the underlying field isRorC. Matrix-wise, it should be interpreted as transposition or Hermitian transposition, accordingly. Otherwise, the notation will indicate the change to the fieldC. For instance,M(n,C) is the vector space of complex endomorphisms. Whenever it is said that A ∈ M(n) has a complex eigenvalue or eigenspace, one is considering the operator embeddingM(n),→M(n,C). We will say that two endomorphismsA, B∈M(n) are conjugate (or similar) when there exists P ∈ GL(n) such that A = P BP−1_{. In this}

case, P is called a conjugacy. The expression diag(λ1, ..., λn) denotes the operator whose

matrix expression has the valuesλ1, ..., λnon the diagonal and zeros elsewhere. An operator

U ∈ M(n,C) is said to be unitary when U U∗ _{= U}∗_{U =I. An operator A ∈ M(n,C) is}

said to be normal if it commutes with its adjoint, that is, AA∗ _{= A}∗_{A. By the Spectral}

Theorem, an operator A ∈ M(n,C) is normal if and only if there exists an orthonormal basis of eigenvectors ofAfor the underlying vector space. If the normal operatorA∈M(n) is self-adjoint, then such basis can be written with purely real coordinates.

4.2.2 The exponential map

The meaning of the expression cH _{in (4.1) withc >0 and H ∈M(n) is given through}

the notion of exponential map by settingcH _{:= exp(log(c)H), where}

exp(A) = ∞ X k=0 Ak k!,

and this infinite series converges for all A∈M(n) (see also Hausner and Schwartz (1968), pp. 59-60). A few remarks about the exponential map are of importance here.

(R1) Loosely speaking, an exponential map

exp :g→G

takes a vector space of operators g ⊆ M(n) into a closed subgroup G ⊆ GL(n) of operators. In this sense, it de-linearizes the vector space. For example,

exp(M(n))⊆GL(n), exp(so(n)) =SO(n). (4.2)

In other words, the exponential of any operator is invertible, and the exponential of any skew-symmetric operator is an orthogonal operator with det = 1 (and vice-versa). Whenever well-defined (as in (4.2)), the inverse of the exponential map, appropriately called log map, may be considered.

(R2) More precisely, let G be a closed (sub)group of operators. Denote by g = T(G) the tangent space of G, i.e., the set ofA∈M(n) such that

A= lim

n→∞

Gn−I

dn , for some{Gn} ⊆Gand some 0< dn→0.

In this sense, g is, in fact, a linearization of Gin a vicinity ofI.

It can be shown (Jurek and Mason (1993), pp. 15-16) that the exp map takes g

into G. The relation between G and g may be pictured as a hyperplane (the latter) touching a manifold (the former) at I. The group operations onGL(n) are infinitely

differentiable, so GL(n) is a Lie group. The tangent space M(n) endowed with the Lie Bracket [A, B] =AB−BAis a Lie Algebra.

(R3) It is not true in general that exp(A+B) = exp(A) exp(B). This relation holds if

A and B commute; however, commutativity is not a necessary condition (Horn and Johnson (1991), p. 435).

(R4) It is easily seen that, for invertibleP,eP AP−1

=P eA_P−1_. 4.2.3 Operator self-similar processes

The definition of operator self-similarity is as follows.

Definition 4.2.1. A stochastic process {X(t)}_t∈R on a finite-dimensional vector space V

(typically, Rn_{) is said to be operator self-similar (o.s.s.) if it is continuous in law at each}

t6= 0 and if for every c >0 there exists a linear operatorA(c) onV and a vectora(c) in V

such that

{X(ct)}t∈R=d {A(c)X(t) +a(c)}t∈R. (4.3)

Throughout the paper, we will assume all processes to be proper, i.e., for each t the distribution is not contained in a proper subspace ofV. Furthermore, we will only consider what is called strictly o.s.s. processes, in the sense that a(c)≡0 (see Corollary 3, Hudson and Mason (1982)).

Theorems 1, 2 and 3 in Hudson and Mason (1982) give the general relation between

A(c) in (4.3) and an (operator) exponent H for the o.s.s. process X. They provide the conditions for the existence, the non-uniqueness and the restrictions on such operator H. For the reader’s convenience, we will state and briefly relate them here.

The first theorem says that, just like in the univariate case, A(c) in (4.3) can be inter- preted in terms of a scaling law.

Theorem 4.2.1. (Hudson and Mason (1982): Existence ofH) Let {X(t)}t∈R be a proper

o.s.s. process. Then, there exists an operator H such that, for eachc >0, (4.1) holds.

An operator H that satisfies (4.1) is called an exponent of the process X, and the set of all suchH is denoted byE(X).

The non-uniqueness of H satisfying (4.1) depends on the symmetry group G1 of X,

which is defined as follows.

Definition 4.2.2. The symmetry group of an o.s.s. process X is the set G1 of operators

A∈GL(n) such that

{X(t)}t∈R=d {AX(t)}t∈R (4.4)

Theorem 4.2.2. (Hudson and Mason (1982): Non-uniqueness of H) Let {X(t)}_t∈R be a proper o.s.s. process. Then, for any H∈ E(X),

E(X) =H+T(G1), (4.5)

where T(G1) =WL0W−1 for some positive-definite operator W and some subspace L0 of

so(n). Consequently, X has a unique exponent if and only ifG1 is finite.

It turns out that the symmetry group G₁ is always compact, which implies that there exists a positive definite self-adjoint operator W and a closed subgroup O₀ of O(n) such thatG1=WO0W−1 (see, for instance, Hudson and Mason (1982) pp. 285, 289). A process

X that has maximal symmetry, i.e., such that G1 = W O(n)W−1, is called elliptically symmetric.

Theorem 4.2.3. (Hudson and Mason (1982): Admissibility ofH) H∈M(n) is an expo- nent for some o.s.s. process X if and only if

(i) every eigenvalue ofH has non-negative real part;

(ii) every eigenvalue ofH having null real part is a simple root of the minimal polynomial of H.

If H∈M(n) satisfies the conditions in Theorem 4.2.3, it is calledadmissible.