A general expression for the distribution of the maximum of a Gaussian field and the approximation of the tail

0.33 MB·English

by  Jean-Marc Azaïs Mario Wschebor

A general expression for the distribution of the maximum of a
Gaussian field and the approximation of the tail
∗
Jean-Marc Aza¨ıs †, [email protected] Mario Wschebor ‡, [email protected]
7
0 February 2, 2008
0
2

n
a
J AMS subject classification: Primary 60G70 Secondary 60G15

8 Short Title: Distribution of the Maximum.

Key words and phrases: Gaussian fields, Rice Formula, Euler-Poincar´e Characteristic, Dis-
]
R tribution of the Maximum, Density of the Maximum, Random Matrices.
P
. Abstract
h
t
a We study the probability distribution F(u) of the maximum of smooth Gaussian fields
m defined on compact subsets of Rd having some geometric regularity.
[ Our main result is a general expression for the density of F. Even though this is an

implicit formula, one can deduce from it explicit bounds for the density, hence for the
2
distribution, as well as improved expansions for 1 F(u) for large values of u.
v −
1 The main tool is the Rice formula for the moments of the number of roots of a random
4 system of equations over the reals.
0 ThismethodenablesalsotostudysecondorderpropertiesoftheexpectedEulerCharac-
7
teristic approximationusing only elementary argumentsand to extend these kind of results
0
to some interesting classes of Gaussian fields. We obtain more precise results for the ”di-
6
0 rect method” to compute the distribution of the maximum, using spectral theory of GOE
/ random matrices.
h
t
a
m 1 Introduction and notations
:
v
Let = X(t) : t S be a real-valued random field defined on some parameter set S and
i
X X { ∈ }
M := sup X(t) its supremum.
t∈S
r
a The study of the probability distribution of the random variable M, i.e. the function
F (u) := P M u isaclassical probleminprobabilitytheory. WhentheprocessisGaussian,
M
{ ≤ }
general inequalities allow to give bounds on 1 F (u) = P M > u as well as asymptotic
M
− { }
results for u + . A partial account of this well established theory, since the founding paper
→ ∞
by Landau and Shepp [20] should contain - among a long list of contributors - the works of
Marcus and Shepp [24], Sudakov and Tsirelson [30], Borell [13] [14], Fernique [17], Ledoux and
Talagrand [22], Berman [11] [12], Adler[2], Talagrand [32] and Ledoux[21].
During the last fifteen years, several methods have been introduced with the aim of ob-
taining more precise results than those arising from the classical theory, at least under certain
restrictions on the process , which are interesting from the point of view of the mathematical
X
theory as well as in many significant applications. These restrictions include the requirement
∗This work was supported by ECOS program U03E01.
†Laboratoire de Statistique et Probabilit´es. UMR-CNRS C5583 Universit´e Paul Sabatier. 118, route de
Narbonne. 31062 Toulouse Cedex 4. France.
‡Centro de Matem´atica. Facultad de Ciencias. Universidad de la Repu´blica. Calle Igua 4225. 11400 Monte-
video. Uruguay.
1
the domain S to have certain finite-dimensional geometrical structure and the paths of the
random field to have a certain regularity.
Some examples of these contributions are the double sum method by Piterbarg [28]; the
Euler-Poincar´e Characteristic (EPC) approximation, Taylor, Takemura and Adler [34], Adler
and Taylor [3]; the tube method, Sun [31] and the well- known Rice method, revisited by Aza¨ıs
and Delmas [5], Aza¨ıs and Wschebor [6]. See also Rychlik [29] for numerical computations.
TheresultsinthepresentpaperarebaseduponTheorem3whichisanextensionofTheorem
3.1 in Aza¨ıs and Wschebor [8] allowing to express the density p of F by means of a general
M M
formula. Even though this is an exact formula, it is only implicit as an expression for the
density, since the relevant random variable M appears in the right-hand side. However, it can
be usefully employed for various purposes.
First, one can use Theorem 3 to obtain bounds for p (u) and thus for P M > u for
M
{ }
every u by means of replacing some indicator function in (4) by the condition that the normal
derivative is ”extended outward” (see below for the precise meaning). This will be called the
”direct method”. Of course, this may be interesting whenever the expression one obtains can
be handled, which is the actual situation when the random field has a law which is stationary
and isotropic. Our method relies on the application of some known results on the spectrum of
random matrices.
Second, one can use Theorem 3 to study the asymptotics of P M > u as u + . More
{ } → ∞
precisely, one wants to write, whenever it is possible
1u2
P M > u = A(u) exp + B(u) (1)
{ } − 2σ2
(cid:0) (cid:1)
where A(u) is a known function having polynomially bounded growth as u + , σ2 =
→ ∞
sup Var(X(t)) and B(u) is an error bounded by a centered Gaussian density with variance
t∈S
σ2, σ2 < σ2. We will call the first (respectively the second) term in the right-hand side of (1)
1 1
the ”first (resp second) order approximation of P M > u .”
{ }
First order approximation has been considered in [3] [34] by means of the expectation of the
EPC of the excursion set E := t S :X(t) > u . This works for large values of u. The same
u
{ ∈ }
authors have considered the second order approximation, that is, how fast does the difference
between P M > u and the expected EPC tend to zero when u + .
{ } → ∞
We will address the same question both for the direct method and the EPC approxima-
tion method. Our results on the second order approximation only speak about the size of the
variance of the Gaussian bound. More precise results are only known to the authors in the
special case where S is a compact interval of the real line, the Gaussian process is stationary
X
andsatisfiesacertainnumberofadditionalrequirements(seePiterbarg[28]andAza¨ısetal. [4]).
Theorem 5 is our first result in this direction. It gives a rough bound for the error B(u) as
u + , in the case the maximum variance is attained at some strict subset of the face in S
→ ∞
having the largest dimension. We are not aware of the existence of other known results under
similar conditions.
In Theorem 6 we consider processes with constant variance. This is close to Theorem 4.3
in [34]. Notice that Theorem 6 has some interest only in case sup κ < , that is, when
t∈S t ∞
one can assure that σ2 < σ2 in (1). This is the reason for the introduction of the additional
1
hypothesis κ(S) < on the geometry of S, (see below (64) for the definition of κ(S)), which
∞
is verified in some relevant situations (see the discussion before the statement of Theorem 6).
In Theorem 7, S is convex and the process stationary and isotropic. We compute the exact
asymptotic rate for the second order approximation as u + corresponding to the direct
→ ∞
2
method.
In all cases, the second order approximation for the direct method provides an upper bound
for the one arising from the EPC method.
Our proofs use almost no differential geometry, except for some elementary notions in Eu-
clidean space. Let us remark also that we have separated the conditions on the law of the
process from the conditions on the geometry of the parameter set.
Third, Theorem 3 and related results in this paper, in fact refer to the density p of
M
the maximum. On integration, they imply immediately a certain number of properties of the
probability distribution F , such as the behaviour of the tail as u + .
M
→ ∞
Theorem 3 implies that F has a density and we have an implicit expression for it. The
M
proof of this fact here appears to be simpler than previous ones (see Aza¨ıs and Wschebor [8])
even in the case the process has 1-dimensional parameter (Aza¨ıs and Wschebor [7]). Let us
remark that Theorem 3 holds true for non-Gaussian processes under appropriate conditions
allowing to apply Rice formula.
Our method can be exploited to study higher order differentiability of F (as it has been
M
done in [7] for one-parameter processes) but we will not pursue this subject here.
This paper is organized as follows:
Section 2 includes an extension of Rice Formula which gives an integral expression for the
expectation of the weighted number of roots of a random system of d equations with d real
unknowns. A complete proof of this formula in a form which is adapted to our needs in this
paper, can be found in [9]. There is an extensive literature on Rice formula in various contexts
(see for example Belayiev [10] , Cram´er-Leadbetter [15], Marcus [23], Adler [1], Wschebor [35].
In Section 3, we obtain the exact expression for the distribution of the maximum as a conse-
quence of the Rice-like formula of the previous section. This immediately implies the existence
of the density and gives the implicit formula for it. The proof avoids unnecessary technicalities
that we have used in previous work, even in cases that are much simpler than the ones consid-
ered here.
In Section 4, we compute (Theorem 4) the first order approximation in the direct method
for stationary isotropic processes defined on a polyhedron, from which a new upper bound for
P M > u for all real u follows.
{ }
In Section 5, we consider second order approximation, both for the direct method and the
EPC approximation method. This is the content of Theorems 5, 6 and 7.
Section 6 contains some examples.
Assumptions and notations
= X(t) : t S denotes a real-valued Gaussian field defined on the parameter set S. We
X { ∈ }
assume that S satisfies the hypothesis A1
A1 :
S is a compact subset of Rd
•
3
S isthedisjointunionofS ,S ...,S ,whereS isanorientableC3 manifoldofdimension
d d−1 0 j
•
j without boundary. The S ’s will be called faces. Let S , d d be the non empty face
j d0 0 ≤
having largest dimension.
We will assume that each S has an atlas such that the second derivatives of the inverse
j
•
functions of all charts (viewed as diffeomorphisms from an open set in Rj to S ) are
j
bounded by a fixed constant. For t S , we denote L the maximum curvature of S at
j t j
∈
the point t. It follows that L is bounded for t S.
t
∈
Notice that the decomposition S = S ... S is not unique.
d 0
∪ ∪
Concerning the random field we make the following assumptions A2-A5
A2 : is in fact defined on an open set containing S and has 2 paths
X C
A3 : for every t S the distribution of X(t),X′(t) does not degenerate; for every s,t S,
∈ ∈
s = t, the distribution of X(s),X(t)(cid:0) does not d(cid:1)egenerate.
6
(cid:0) (cid:1)
A4 : Almost surely the maximum of X(t) on S is attained at a single point.
For t S , X′(t) X′ (t) denote respectively the derivative along S and the normal deriva-
∈ j j j,N j
tive. Both quantities are viewed as vectors in Rd, and the density of their distribution will be
expressed respectively with respect to an orthonormal basis of the tangent space T of S at
t,j j
the point t, or its orthogonal complement N . X′′(t) will denote the second derivative of X
t,j j
along S , at the point t S and will be viewed as a matrix expressed in an orthogonal basis
j j
∈
of T . Similar notations will be used for any function defined on S .
t,j j
A5 : Almost surely, for every j = 0,1,...,d there is no point t in S such that X′(t) = 0,
j j
det(X′′(t)) = 0
j
Other notations and conventions will be as follows :
σ is the geometric measure on S .
j j
•
m(t) := E(X(t)), r(s,t) = Cov(X(s),X(t)) denote respectively the expectation and co-
•
variance of the process ; r (s,t), r (s,t) are the first and the second derivatives of r
0,1 0,2
X
with respect to t. Analogous notations will be used for other derivatives without further
reference.
If η is a random variable taking values in some Euclidean space, p (x) will denote the
η
•
density of its probability distribution with respect to the Lebesgue measure, whenever it
exists.
ϕ(x) = (2π)−1/2exp( x2/2) is the standard Gaussian density ; Φ(x) := x ϕ(y)dy.
• − −∞
R
Assume that the random vectors ξ,η have a joint Gaussian distribution, where η has
•
values in some finite dimensional Euclidean space. When it is well defined,
E(f(ξ)/η = x)
is the version of the conditional expectation obtained using Gaussian regression.
E := t S : X(t) > u is the excursion set above u of the function X(.) and A :=
u u
• { ∈ }
M u is the event that the maximum is not larger than u.
{ ≤ }
, , , denote respectively inner product and norm in a finite-dimensional real Euclidean
• h i kk
space; λ is the Lebesgue measure on Rd; d−1 is the unit sphere ; Ac is the complement
d
S
of the set A. If M is a real square matrix, M 0 denotes that it is positive definite.
≻
4
If g : D C is a function and u C, we denote
• → ∈
Ng(D) := ♯ t D :g(t) = u
u { ∈ }
which may be finite or infinite.
Some remarks on the hypotheses
One can give simple sufficient additional conditions on the process so that A4 and A5 hold
X
true.
If weassumethat for each pairj,k = 0,...,d andeach pair of distinct pointss,t, s S ,t
j
∈ ∈
S , the distribution of the triplet
k
X(t) X(s),X′(s),X′(t))
− j k
(cid:0) (cid:1)
does not degenerate in R Rj Rk, then A4 holds true.
× ×
This is well-known and follows easily from the next lemma (called Bulinskaya ’s lemma)
that we state without proof, for completeness.
Lemma 1 Let Z(t) be a stochastic process defined on some neighborhood of a set T embedded
in some Euclidean space. Assume that the Hausdorff dimension of T is smaller or equal than
the integer m and that the values of Z lie in Rm+k for some positive integer k . Suppose, in
addition, that Z has 1 paths and that the density p (v) is bounded for t T and v in some
Z(t)
C ∈
neighborhood of u Rm+k. Then, a. s. there is no point t T such that Z(t) = u.
∈ ∈
With respect to A5, one has the following sufficient conditions: Assume A1, A2, A3 and as
additional hypotheses one of the following two:
t X(t) is of class 3
• C
•
sup P det X′′(t) < δ/X′(t) = x′ 0, as δ 0,
| | → →
t∈S,x′∈V(0) (cid:0) (cid:0) (cid:1) (cid:1)
where V(0) is some neighborhood of zero.
Then A5 holds true. This follows from Proposition 2.1 of [8] and [16].
2 Rice formula for the number of weighted roots of random
fields
In this section we review Rice formula for the expectation of the number of roots of a random
system of equations. For proofs, see for example [8], or [9], where a simpler one is given.
Theorem 1 (Rice formula) Let Z : U Rd be a random field, U an open subset of Rd and
→
u Rd a fixed point in the codomain. Assume that:
∈
(i) Z is Gaussian,
(ii) almost surely the function t Z(t) is of class 1,
C
(iii) for each t U, Z(t) has a non degenerate distribution (i.e. Var Z(t) 0),
∈ ≻
(iv) P t U,Z(t) = u,det Z′(t) = 0 = 0 (cid:0) (cid:1)
{∃ ∈ }
Then, for every Borel se(cid:0)t B co(cid:1)ntained in U, one has
E NZ(B) = E det(Z′(t))/Z(t) = u p (u)dt. (2)
u Z | | Z(t)
(cid:0) (cid:1) B (cid:0) (cid:1)
If B is compact, then both sides in (2) are finite.
5
Theorem 2 Let Z be a random field that verifies the hypotheses of Theorem 1. Assume that
for each t U one has another random field Yt : W Rd′, where W is some topological space,
∈ →
verifying the following conditions:
a) Yt(w) is a measurable function of (ω,t,w) and almost surely, (t,w) Yt(w) is continu-
ous.
b) For each t U the random process (s,w) Z(s),Yt(w) defined on U W is Gaussian.
∈ ×
(cid:0) (cid:1)
Moreover, assume that g :U (W,Rd′) R is a bounded function, which is continuous when
one puts on (W,Rd′) the to×poClogy of un→iform convergence on compact sets. Then, for each
C
compact subset I of U, one has
E g(t,Yt) = E det(Z′(t)g(t,Yt)/Z(t) = u).p (u)dt. (3)
Z(t)
Z | |
(cid:0)t∈I,XZ(t)=u (cid:1) I (cid:0)
Remarks:
1. We have already mentioned in theprevious section sufficient conditions implyinghypoth-
esis (iv) in Theorem 1.
2. With the hypotheses of Theorem 1 it follows easily that if J is a subset of U, λ (J) = 0,
d
then P NZ(J) = 0 = 1 for each u Rd.
{ u } ∈
3 The implicit formula for the density of the maximum
Theorem 3 Under assumptions A1 to A5, the distribution of M has the density
p (x) = E 1I /X(t) = x p (x)
M Ax X(t)
tX∈S0 (cid:0) (cid:1)
d
+ E det(X′′(t)) 1I /X(t) = x,X′(t) = 0 p (x,0)σ (dt), (4)
Z | j | Ax j X(t),Xj′(t) j
Xj=1 Sj (cid:0) (cid:1)
Remark: Onecan replace det(X′′(t)) in the conditional expectation by ( 1)jdet(X′′(t)),
| j | − j
since under the conditioning and whenever M x holds true, X′′(t) is negative semi-definite.
≤ j
Proof of Theorem 3
Let N (u),j = 0,...,d be the number of global maxima of X(.) on S that belong to S and are
j j
larger than u. From the hypotheses it follows that a.s. N (u) is equal to 0 or 1, so
j=0,...,d j
that P
P M > u = P N (u) = 1 = E(N (u)). (5)
j j
{ } { }
j=X0,...,d j=X0,...,d
The proof will be finished as soon as we show that each term in (5) is the integral over (u,+ )
∞
of the corresponding term in (4).
This is self-evident for j = 0. Let us consider the term j = d. We apply the weighted Rice
formula of Section 2 as follows :
Z is the random field X′ defined on S .
d
•
For each t S , put W = S and Yt :S R2 defined as:
d
• ∈ →
Yt(w) := X(w) X(t),X(t) .
−
(cid:0) (cid:1)
Notice that the second coordinate in the definition of Yt does not depend on w.
6
In the place of the function g, we take for each n = 1,2,... the function g defined as
n
•
follows:
g (t,f ,f ) = g (f ,f )= 1 (supf (w)) . 1 (u f (w)) ,
n 1 2 n 1 2 n 1 n 2
−F −F −
(cid:0) w∈S (cid:1) (cid:0) (cid:1)
where w is any point in W and for n a positive integer and x 0, we define :
≥
(x):= (nx) ; with (x) = 0 if 0 x 1/2 , (x) = 1 if x 1 , (6)
n
F F F ≤ ≤ F ≥
and monotone non-decreasing and continuous.
F
It is easy to check that all the requirements in Theorem 2 are satisfied, so that, for the value 0
instead of u in formula (3) we get:
E g (Yt) = E det(X′′(t)g (Yt)/X′(t) = 0).p (0)λ (dt). (7)
n n X′(t) d
Z | |
(cid:0)t∈SdX,X′(t)=0 (cid:1) Sd (cid:0)
Notice that the formula holds true for each compact subset of S in the place of S , hence for
d d
S itself by monotone convergence.
d
Let now n in (7). Clearly g (Yt) 1I . 1I . The passage to the limit
n X(s)−X(t)≤0,∀s∈S X(t)≥u
→ ∞ ↓
doesnotpresentanydifficultysince0 g (Yt) 1andthesumintheleft-handsideisbounded
n
by the random variable NX′(S ), whi≤ch is in L≤1 because of Rice Formula. We get
0 d
E(N (u)) = E det(X′′(t) 1I 1I /X′(t) = 0).p (0)λ (dt)
d X(s)−X(t)≤0,∀s∈S X(t)≥u X′(t) d
Z | |
Sd (cid:0)
Conditioning on the value of X(t), we obtain the desired formula for j = d.
The proof for 1 j d 1 is essentially the same, but one must take care of the parame-
≤ ≤ −
terization of the manifold S . One can first establish locally the formula on a chart of S , using
j j
local coordinates.
It can be proved as in [8], Proposition 2.2 (the only modification is due to the term 1I )
Ax
that the quantity written in some chart as
E det(Y′′(s)) 1I /Y(s)= x,Y′(s)= 0 p (x,0)ds,
Ax Y(s),Yj′(s)
(cid:0) (cid:1)
where the process Y(s) is the process X written in some chart of S ,
j
(Y(s) = X(φ−1(s))), defines a j-form. By a j-form we mean a mesure on S that does not
j
depend on the parameterization and which has a density with respect to the Lebesgue measure
ds in every chart. It can be proved also that the integral of this j-form on S gives the
j
expectation of N (u).
j
To get formula (2) it suffices to consider locally around a precise point t S the chart φ
j
∈
given by the projection on the tangent space at t. In this case we obtain that at t
ds is in fact σ (dt)
j
•
Y′(s) is isometric to X′(t)
• j
where s = φ(t). (cid:3)
The first consequence of Theorem 3 is the next corollary. For the statement, we need to
introduce some further notations.
For t inS , j d we define as theclosed convex conegenerated by theset of directions:
j 0 t,j
≤ C
t s
λ Rd : λ =1 ; s S,(n = 1,2,...) such that s t, − n λ as n + ,
n n
{ ∈ k k ∃ ∈ → t s → → ∞}
n
k − k
7
whenever this set is non-empty and = 0 if it is empty. We will denote by the dual
t,j t,j
C { } C
cone of , that is:
Ct,j b
:= z Rd : z,λ 0 for all λ .
t,j t,j
C { ∈ h i ≥ ∈ C }
Notice that these definitiobns easily imply that T C and N . Remark also that for
t,j t,j t,j t,j
⊂ C ⊂
j = d , = N .
0 Ct,j t,j b
We will say that the function X(.) has an ”extended outward” derivative at the point t in
b
S , j d if X′ (t) .
j ≤ 0 j,N ∈ Ct,j
b
Corollary 1 Under assumptions A1 to A5, one has :
(a) p (x) p(x) where
M
≤
p(x) := E 1IX′(t)∈Cbt,0/X(t) =x pX(t)(x)+
tX∈S0 (cid:0) (cid:1)
d0
Xj=1ZSj E(cid:0)|det(Xj′′(t))| 1IXj′,N(t)∈Cbt,j/X(t) = x,Xj′(t) =0(cid:1)pX(t),Xj′(t)(x,0)σj(dt). (8)
+∞
(b) P M > u p(x)dx.
{ } ≤ Z
u
Proof
(a) follows from Theorem 3 and the observation that if t S , one has
j
∈
M X(t) X′ (t) . (b) is an obvious consequence of (a). (cid:3)
{ ≤ } ⊂ { j,N ∈ Ct,j}
b
The actual interest of this Corollary depends on the feasibility of computing p(x). It turns
out that it can be done in some relevant cases, as we will see in the remaining of this section.
Our result can be compared with the approximation of P M > u by means of +∞pE(x)dx
{ } u
given by [3], [34] where R
pE(x) := E 1IX′(t)∈Cbt,0/X(t) = x pX(t)(x)
tX∈S0 (cid:0) (cid:1)
d0
+Xj=1(−1)jZSj E(cid:0)det(Xj′′(t)) 1IXj′,N(t)∈Cbt,j/X(t) = x,Xj′(t) =0(cid:1)pX(t),Xj′(t)(x,0)σj(dt). (9)
Under certain conditions , +∞pE(x)dx is the expected value of the EPC of the excursion set
u
E (see [3]). The advantagRe of pE(x) over p(x) is that one can have nice expressions for it in
u
quite general situations. Conversely p(x) has the obvious advantage that it is an upper-bound
of the true density p (x) and hence provides upon integrating once, an upper-bound for the
M
tail probability, for every u value. It is not known whether a similar inequality holds true for
pE(x).
On the other hand, under additional conditions, both provide good first order approximations
for p (x) as x as we will see in the next section. In the special case in which the process
M
→ ∞
is centered and has a law that is invariant under isometries and translations, we describe
X
below a procedure to compute p(x).
8
4 Computing p(x) for stationary isotropic Gaussian fields
For one-parameter centered Gaussian process having constant variance and satisfying certain
regularity conditions, a general bound for p (x) has been computed in [8], pp.75-77. In the
M
two parameter case, Mercadier [26] has shown a bound for P M > u , obtained by means of a
{ }
method especially suited to dimension 2. When the parameter is one or two-dimensional, these
bounds are sharper than the ones below which, on the other hand, apply to any dimension but
to a more restricted context. We will assume now that the process is centered Gaussian,
X
with a covariance function that can be written as
E X(s).X(t) = ρ s t 2 , (10)
k − k
(cid:0) (cid:1) (cid:0) (cid:1)
where ρ : R+ R is of class 4 . Without loss of generality, we assume that ρ(0) = 1.
→ C
Assumption (10) is equivalent to saying that the law of is invariant under isometries (i.e.
X
linear transformations that preserve the scalar product) and translations of the underlying
parameter space Rd.
We will also assume that the set S is a polyhedron. More precisely we assume that each
S (j =1,...,d) is a union of subsets of affine manifolds of dimension j in Rd.
j
The next lemma contains some auxiliary computations which are elementary and left to the
reader. We use the abridged notation : ρ′ := ρ′(0), ρ′′ := ρ′′(0)
Lemma 2 Under the conditions above, for each t U, i,i′,k,k′,j = 1,...,d:
∈
1. E ∂X(t).X(t) = 0,
∂ti
(cid:0) (cid:1)
2. E ∂X(t).∂X(t) = 2ρ′δ and ρ′ < 0,
∂ti ∂tk − ik
(cid:0) (cid:1)
3. E ∂2X (t).X(t) = 2ρ′δ ,E ∂2X (t).∂X(t) = 0
∂ti∂tk ik ∂ti∂tk ∂tj
(cid:0) (cid:1) (cid:0) (cid:1)
4. E ∂2X (t). ∂2X (t) = 24ρ′′ δ .δ +δ .δ +δ δ ,
∂ti∂tk ∂ti′∂tk′ ii′ kk′ i′k ik′ ik i′k′
(cid:0) (cid:1) (cid:2) (cid:3)
5. ρ′′ ρ′2 0
− ≥
6. If t S , the conditional distribution of X′′(t) given X(t) = x,X′(t) = 0 is the same as
∈ j j j
the unconditional distribution of the random matrix
Z +2ρ′xI ,
j
where Z = (Z : i,k = 1,...,j) is a symmetric j j matrix with centered Gaussian
ik
×
entries, independent of the pair X(t),X′(t) such that, for i k, i′ k′ one has :
≤ ≤
(cid:0) (cid:1)
E(Z Z ) =4 2ρ′′δ +(ρ′′ ρ′2) δ δ +4ρ′′δ .δ (1 δ ) .
ik i′k′ ii′ ik i′k′ ii′ kk′ ik
− −
(cid:2) (cid:3)
Let us introduce some additional notations:
H (x),n = 0,1,... are the standard Hermite polynomials, i.e.
n
•
H (x) := ex2 ∂ ne−x2
n
− ∂x
(cid:0) (cid:1)
For the properties of the Hermite polynomials we refer to Mehta [25].
H (x),n = 0,1,... are the modified Hermite polynomials, defined as:
n
•
H (x) := ex2/2 ∂ ne−x2/2
n
− ∂x
(cid:0) (cid:1)
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We will use the following result:
Lemma 3 Let
+∞
J (x) := e−y2/2H (ν)dy, n= 0,1,2,... (11)
n n
Z
−∞
where ν stands for the linear form ν = ay+bx where a,b are some real parameters that satisfy
a2+b2 = 1/2. Then
J (x) := (2b)n√2π H (x).
n n
Proof :
It is clear that J is a polynomial having degree n. Differentiating in (11) under the integral
n
sign, we get:
+∞ +∞
J′(x) = b e−y2/2H′(ν)dy = 2nb e−y2/2H (ν)dy = 2n b J (x) (12)
n n n−1 n−1
Z Z
−∞ −∞
Also:
+∞
J (0) = e−y2/2H (ay)dy,
n n
Z
−∞
so that J (0) = 0 if n is odd.
n
If n is even, n 2, using the standard recurrence relations for Hermite polynomials, we have:
≥
+∞
J (0) = e−y2/2 2ayH (ay) 2(n 1)H (ay) dy
n n−1 n−2
Z − −
−∞ (cid:2) (cid:3)
+∞
= 2a2 e−y2/2H′ (ay)dy 2(n 1)J (0)
Z n−1 − − n−2
−∞
= 4b2(n 1)J (0). (13)
n−2
− −
Equality (13) plus J (x) =√2π for all x R, imply that:
0
∈
(2p)!
J (0) = ( 1)p(2b)2p(2p 1)!!√2π = ( 2b2)p √2π. (14)
2p
− − − p!
Now we can go back to (12) and integrate successively for n = 1,2,... on the interval [0,x]
using the initial value given by (14) when n = 2p and J (0) = 0 when n is odd, obtaining :
n
J (x) = (2b)n√2πQ (x),
n n
where the sequence of polynomials Q ,n = 0,1,2,... verifies the conditions:
n
Q (x) = 1 (15)
0
Q′ (x) = nQ (x) (16)
n n
Q (0) = 0 if n is odd (17)
n
Q (0) = ( 1)n/2(n 1)!! if n is even. (18)
n
− −
It is now easy to show that in fact Q (x) = H (x) , n = 0,1,2,... using for example that:
n n
x
H (x) = 2n/2H .
n n
√2
(cid:0) (cid:1)
(cid:3)
The integrals
+∞
I (v) = e−t2/2H (t)dt,
n n
Z
v
will appear in our computations. They are computed in the next Lemma, which can be proved
easily, using the standard properties of Hermite polynomials.
10