.4PPLltLl STOCHASTIC MODELS AND DATA ANALYSIS, VOL. 8, 1-5 (1992)

A NOTE ABOUT THE GAUSSIAN PROPERTY OF THE BROWNIAN BRIDGE

RAMON GUTIERREZ AND MARIAN0 J. VALDERRAMA Depurttiient o j Statistics and Operations Research, University of Granada, 18071 Granada, Spain

SUMMARY

The Gaussian property of the Brownian bridge is characterized as an application of Ramachandran's theorem in terms of the independence of the random variables that appear in the Karhunen-Loevc expansion of the process. A reference about the construction of the Brownian bridge by means of functional transformations is also included.

~ t ) \\ OKDS Brownian bridge Wiener-Levy process Karhunen-Loeve expansion Ramachandran's theorem

1 . INTRODUCTION

The Wiener process has usually been taken as a probabilistic model for Brownian motion. Nevertheless, there are other stochastic processes, such as the Ornstein-Uhlenbeck process and the Brownian bridge, that form more approximate models of it. '92*3.4 One of the most important properties of all of them is their Gaussian character. In fact, this consideration allows us to identify them from their first- and second-order moments. Furthermore, Gaussian processes are a subclass of the second-order processes.

The characterization of a second-order process with covariance min ( f , s) as Brownian motion was developed by Pierre. Recently, Gutierrez and Valderrama6 have approached the Gaussian property of the Ornstein-Uhlenbeck process. In the present paper, necessary and sufficient conditions for the process { X ( t ) , 0 ,< t ,< T ) with covariance min { f , SJ - ( t s ) /T to be the Brownian bridge are studied. As a first step we introduce this stochastic process on [0, rl from the Wiener-Levy process, and deduce its Karhunen-Loeve expansion.

2. SOME TOPICS ON THE BROWNIAN BRIDGE

The Brownian bridge may be stated in several ways. Here, we start from the Wiener-Levy process { W ( r ) , t 2 0) and define the Brownian bridge on [0, r ] as follows: B ( f ) = W ( r ) - ( t / T ) W ( T ) . Then its covariance function is given by

8755-OO24/92/010001-05$07.50 01992 by John Wiley & Sons, Ltd.

Received 6 December 1989 Revised I5 May 1991

2 R. GUTIERREZ A N D M . J . VALDERRAMA

or in an equivalent form

min ( f , s ) [ T - max ( t , s)] T RB(f . S) =

(Let us observe that T < + co because otherwise we have the Wiener-Levy process.) The Brownian bridge is quadratic mean (q.m.) continuous because for h > 0 we have

Likewise, the Brownian bridge is a q.m. integrable process because its covariance is a continuous function on [0, T]x[O, T], but it is not q.m. derivable since

that diverges when h --* 0.

as a denumerable series of uncorrelated random variables in the following way: Nevertheless, on the basis of the q.m. continuity, the Brownian bridge can be represented

m

B ( t ) = C G n ( t ) U n I l = l

(9.m. convergence uniformly in t ) where Un is a q.m. integral defined as T

U n = 1 +n(t )X(t )dt 0

Such a representation is called the Karhunen-Loeve expansion of the process. The sequence ( r # ~ ~ ( t ) ] is an orthogonal eigenfunction system of the Fredholm integral equation:

By differentiating twice with respect to t , equation (3) becomes the boundary problem:

A4”(t) + 4(t) = 0, 4(0) = 0, 4 ( T ) = 0

4 ( t ) = C . sin(r/,iX)

and its solution is given by

where 772

An = 1 n2x2’ n = 1,2, ...

so that, once the eigenfunctions have been normalized, we have

A more general transformation:

where ( W ( t ) , t 2 0)

( 2 J 2 . (n;t) & ( t ) = - sin ~ , n = 1,2, ...

derivation of the Brownian bride can be

(4)

( 5 )

made by exploiting the

two continuous functions

GAUSSIAN PROPERTY OF BROWNIAN BRIDGE 3

on [0, m ) so that r( * ) is non-decreasing. Thus, by operating we arrive to the equality

R d t . s) = f(f)f(s)(min (W) , Us)] ) and by identification we conclude that

so that the Brownian bridge on [0, r ] is given by

Nevertheless, the first approach will be appropriate for our purposes.

3. THE GAUSSIAN CHARACTERIZATION OF THE BROWNIAN BRIDGE

Let us consider the expansion (2a, b) over two intervals [0, T I ] and [O, Tz] with TI < Tz m

~ ( r ) = C d n ( f , k ) u n ( k ) , k = 1 9 2

n = I Observe that

71 m

u n ( l ) = S &(t , 1)B(t)dt = C um(2)Amn(2,1) 0 m = I

where 71

0 ~ m 1 1 ( 2 , 1 ) = j + n ( t , 1)4m(t, 2)dt

By replacing expression ( 5 ) in (7) we have

(7)

Previously to state the characterization theorem we are going to prove the following operative lemma.

Lemma

Let us suppose that iT1 # jT2 for any positive integer numbers i and j . Then the sequence {&,;/Al,,,, ni = 1 , 2 , . .. is bounded and [ A , ; A , , m = 1,2, .. .) keeps uniformly different from zero.

Pro0 f Let 11s observe from expression (8) that

Thus, for any fixed couple ( i , j ) E N x N, the sequence

m2T: - i2T$ [m’T: - j 2 T : ]

4 R. GUTIERREZ AND M. J . VALDERRAMA

converges towards one when m - + 00 , and therefore ( A m i / A , j ) is bounded. Furthermore, A,iA,,,j # 0 for all rn = 1,2, . . . since sin(rnrTl/ T z ) = 0 if and only if t n r ( T ~ / Tz) = j r for some integer j (i.e. TI =jT2), which contradicts the hypotheses.

The main goal of this paper is as follows.

Characterization theorem

Let ( X ( r ) , 0 < t < T ) be a second-order stochastic process with covariance min If, s) - (ts)/T, and consider two representations of the Karhunen-Loeve type for it in terms of uncorrelated random variables over two intervals [O, T I ] and [0, Tz] , where TI < Tz. Suppose that iTI # jT2 for any positive integers i and j . Then ( X ( t ) ) is a Gaussian process if and only if (un (k ) ) , k = 1,2, are sequences of independent random variables.

Pro0 f The necessary condition is obvious since the Brownian bridge is a Gaussian process; the

uncorrelation is equivalent to independence. In order to prove sufficiency let us observe that the Karhunen-Lotve expansion converges in probability because it is a q.m. convergent series. Besides, as an application of the lemma, the subsequences (Ami/A,?ljJ are bounded for any positive integers i and j , and all rn so that A m i A m j # 0. Finally, from the fact that variables ( u n ( l ) ) are independent, ( ~ ~ ( 2 ) ) are Gaussian variables as a consequence of the Ramachandran’s theorem, and therefore the process is also Gaussian.

4. APPLICATIONS

The ability to identify a stochastic process as Gaussian on the basis of the independence of the random coefficients of its orthogonal representation is a very useful tool in stochastic prediction theory. Hence it can be proved that Gaussian processes are the only ones having mutually independent Nyquist samples inside the class of stationary band-limited white noise processes and, therefore, the hypothesis of mutual independence is a very restrictive one on non-parametric and robust techniques. lo

On the other hand, in many physical and economics applications we are concerned only with the harmonic coefficients of the transformed Karhunen expansion, which have been isolated by means of suitable filters, and we need to deduce some struci:ural properties about the complete process (for example, its Gaussian character). Then, the procedure described in Section 3 may be available for solving such a problem.

Finally, let us indicate that linear representations of Gaussian processes are usually employed in studying differentiability properties of probability measures. In fact some results about absolute continuity with respect to the Wiener measure described by Wong and Hajek I 2

can be extended by using ‘characterization theorem’ of Section 3, as well as its application to the problem of detecting a Brownian bridge signal in Gaussian noise.

REFERENCES

1. J. L . Doob, ‘Heuristic approach to the Kolmogorov-Smirnov theorems’, Ann. Math. Slat., 20, 393-402 (1949). 2. G. E. Uhlenbeck and L. S. Ornstein, ‘On the theory of Brownian motion’, Pbys. Rev., 36, 823-841 (1930).

GAUSSIAN PROPERTY OF BROWNIAN BRIDGE 5

3 . K . J . Adler, A n Introduction to Continuity, Extrema, and Related Topics for General Gaussian Processes, I .M.S. l~ecture Norez-Monograph Series, Vol. 12, Hayward, 1990, pp. 131-139.

-1. K . N. Bhattasharga and E . C . Waymire, Stochastic Processes with Applications, Wiley, New York, 1990,

5 . P. A . Pierre. ‘On the independence of linear functionals of linear processes’, SlAM J . Appl. Math., 17, 624-637 (1969).

6. K. Gutierrez and hl. J . Valderrama, ‘On the Gaussian characterization of certain second order processes with \pecified covariances’, l€EE Trans. Inform. Theory, 35(1), 210-21 1 (1989).

7. bl. Loeve, Probabr/it.y Theory, Vol. 2, Springer-Verlag, New York, 1978. 8. K. Gutierrez and hl. J . Valderrama, ‘On the Karhunen-Loeve transformed expansion’, Trali. Estad.. 2. 81-90

9. B. Rarnachandran, Advanced Theory of Characteristics Functions, Statistical Publishing Society, Calcutta, 1967. 10. P. A . Pierre, ‘Characteristics of Gaussian random processes by representation in terms of independent random

I I. A . Papoulis, Probabilizy, Random Variables and Stochastic Processes, McGraw-Hill, New York, 1984. 12. E. Wong and K. B. Hajek, Stochastic Processes in Engineering Systems, Springer-Verlag, New York , 1985, pp.

pp. 35-39.

(1987).

\ariables’, I € € € Trans. Inform. Theory, 15(6), 648-658 (1969).

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