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ADAM JAKUBOWSKIA Non-Central Functional Limit Theorem for QuadraticForms in Martingale Difference SequencesPublications de l’Institut de recherche mathématiques de Rennes, 1989-1990, fascicule 1« Probabilités », p. 69-73<http://www.numdam.org/item?id=PSMIR_1989-1990___1_69_0>

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69

A Non-Central Functional Limit Theorem for Quadratic Forms

in Martingale Difference Sequences

Adam Jakubowski* Uniwersytet Mikolaja Kopernika, Instytut Matematyki

Torun, Poland.

Abstract

For quadratic forms with nulls on the diagonal the partial-sum process is both a martingale and a stochastic integral. Using corresponding tools we derive a result on the convergence to the double Wiener-It o integral.

Let { A n = ( a J j ) } n € N be a sequence of infinite matrices with nulls on the diagonals:

< t = 0, ¿ = 1 ,2 , . . . , n G N . (1)

Let { X j } j € N be a martingale difference sequence with respect to some filtration {^/}jei\iu{o} such that

E{X2

j\T^) = \, i = 1 ,2 , . . . . (2)

Then for each n 6 N the process

Snlk= £ ¿ = 1 ,2 , . . . , (3)

l<t\j<* j < k \ i < j )

is both a square integrable martingale and a stochastic integral with respect to {Fj}-

Using the martingale structure, a Functional Central Limit Theorem for suitably scaled

{£n,A:}fceN w & s proved in [JaMe90]. In the present paper we state a non-central functional

limit theorem based on limit theorems for stochastic integrals given in [JMP89]. The limit

is identified with a double Wiener-Ito stochastic integral (see e.g. [Ito51] or [Maj81]). In

some sense it is not surprising: such limits arise, for example, in limit theory for quadratic

forms in stationary gaussian sequences exhibiting long-range dependence (see [Ros79], also

[SuHo86]).

* Supported by CPBP 01.02

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For each n € N, let A n be the symmetrization of A 7 1, i.e. the matrix with entries

(alj + ali)/2' F o r A " ' d e f i n e i t s representation in Z,2([0, T ] 2 ) by the formula

AS(ti,t;) = af n t t ] f [ n v j for T > u,t> > 0. (4)

Finally, let

Xn(t) = £ X, (5) l<j<[nt]

y-(t) = n" 1 j f a&X.-X,-. (6) l<ij<[nt]

Theorem Suppose A n 's satisfy (1) and {Xj} is a martingale difference sequence such

that (2) holds.

If f\n converges in L 2 ([0, T ] 2 ) to some function A, then

Yn V F ' ( 7 )

where

Y(t) = J jA(u,v)I[0tt?(u,v)dWudWv (8)

is the classical Wiener-Ito integral P R O O F . For each e > 0 there are continuous functions I = 1,2, . . . , ra and

numbers a x , . . . , am such that

*(M) = JLa' 9

satisfies

| | A - * | | 2 < c . (9)

Hence for n > n x , ||A" — $| | 2 < 2s and by continuity of $ ||A" - $„ | | 2 < 3e for n > n 2 , where $n(s,t) = $([ns]/n, [n*]/n).

If

Zn(t) = n-1 ¿2 9(i/n,j/n)XiXjt (10)

then for n > n 2

£ sup \Yn(t) - Zn{t)\2 < AE\Yn{T) - Zn{T)\2 < 4||S* - $ n | | ' < 4 • 9e 2 . (11) l < t < T

Let

Z(*) = / J$(u,v)I[0,t?{u,v)dWudWv. (12)

By the isometry property for Wiener-Ito integrals

E sup \Y( t ) - Z(t)\2 < 4E\Y(T) - Z(T)\2 = 4\\A - $ | | 2 < 4e 2 . (13)

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Hence it is enough to prove that

z n v z (14) on the space D ( [ 0 , T ] ) . But

Zn(t) = 2U-1 ¿2 ^{i/nJ/^XiXj l<i<j<[nt]

m f XX

1=1 \i<.<i<H v n v n

+ £ ^ ( i / n J M j / n ) ^ ) i<i<j<M VnVnJ

= X > (jf Mv) (jQ

v~ Hu) dx^j dx:+

v p a i (jf (jf~ ̂ /(w) ^ +

+jf m») (jf ^(«) <w«) ^ )

where the convergence in distribution holds by [JMP89, Theorem 2.6] •

Example 1. Let / : [0, l ] 2 -+ R 1 . Define quadratic forms

"?j = f

If gn{u,v) = f([nu],[nv]) —v L i f(u,v), then

n _ 1 £ -T f fl{u,v)dWudWv, X<ij<[nt\ V J ° J °

where / is the symmetrization of / .

Example 2. Let

Co = 0, C X = 6i, C 2 = ¿2, • • • , Q = Q + l = &1> c d + 2 = ¿2, • . •

Define a matrix A by ai,i = c\i-j\-

In this case A" does not converge, so we cannot apply directly our theorem.

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Take n = N - d and define a rearranging of coordinates:

t\ i—» ei

erf h» e(d_i)Ar+i

(The general formula is of the form e^d+i *-* e^i^+k+i if 0 < k < N — 1, 1 < z < d). Under this rearranging, the distribution of Xn remains unchanged, while A n transforms

to the form D n = ( d p ^ ) , where for p = (i - 1)N + k + 1 and q = (j - 1)N + 1 + 1

{ 0 if j = z, 1 = k

bd if j = + k &y_t-| if j > z, l> k or j < z, I <k bd~\j-i\ if j > z, / < or j < z, / > A:.

It is easy to see that now D n —* D in £ 2 ( [0 , l ] 2 ) , where 0 if u = u 6d if [d • u] = [d • v] , u + v b\[d.u]-[d.v]\ if [d • u] > [d • v] and { d • u} > {d •

r>/ \ I o r

^ " [d. u) < [d • v] and {d • ix} < { d • &cH[*"M<HI if [d-u]> [d • v] and {d • u} < {d • i?}

or [d • u] < [d • v] and {d • u} > {d - v}

Here [x] is the integer part of x and {x} = x — [#]. The convergence is easily extendible from subsequence N • d to the whole N. But we

do not have the functional convergence, since

l<ij<[nt]

and

l<«,i<[n«]

have nothing common except for t = 1, where they coincide.

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References

[Ito51] Itô, K. Multiple Wiener integral, J. Math. Soc. Japan 3 (1951), pp. 157-169.

[JaMé90] Jakubowski, A. & Mémin, J. A functional central limit theorem for quadratic forms in independent random variables, preprint (1990).

[JMP89] Jakubowski, A., Mémin, J. & Pages, G. Convergence en loi des suites d'integrales stochastiques sur l'espace $1 de Skorokhod Probab. Th. Rel. Fields 81 (1990), pp. 111-137.

[Maj81] Major, P. Multiple Wiener-Ito Integrals, Lecture Notes in Math., 849, Springer 1981.

[Ros79] Rosenblatt, M. Some limit theorems for partial sums of quadratic forms in stationary gaussian variables Z. Wahrsch. verw. Gebiete 49 (1979), pp. 125-132.

[SuHo86] Sun, T.C. & Ho, H.C. On central and non-central limit theorems for nonlinear functions of a stationary Gaussian process, in: Eberlein, E. & Taqqu, M., Eds, Dependence in Probability and Statistics, Birkhäuser, Boston 1986.

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