Computing the covariance of two Brownian area integrals

Transcription

1 Statistica Neerlandica () Vol. 56, nr.,. ±9 Comuting the covariance of two Brownian area integrals J. A. Wellner* University of Washington, Statistics, Box 3543, Seattle, Washington , U.S.A. R. T. Smythe Deartment of Statistics, Oregon State University, Corvallis, Oregon , U.S.A. We comute the exected roduct of two correlated Brownian area integrals, a roblem that arises in the analysis of a oular sorting algorithm. Alongthewaywe ndthreedifferentformulasfortheexectation of the roduct of the absolute values of two standard normal random variables with correlation h. These two formulas are found: (a) via conditioning and the non-central chi-square distribution; (b) via Mehler's formula; (c) by reresenting the correlated normal random variables in terms of indeendent normal's and integration using olar coordinates. Key Words and Phrases: bivariate normal distribution, Brownian bridge, correlation, exectation, Mehler's formula, non-central chi-square, roduct of absolute values. The roblem Suose that B j; j ; ; 3 are indeendent Brownian bridge rocesses on [, ]; recall that a Brownian bridge rocess B is a mean zero Gaussian rocess on [, ] with covariance Cov B s ; B t s ^ t st; s; t ;Š: De ne two random variables A and A by A jb t B t j dt; A jb t B 3 t j dt: The following question arose in the course of trying to analyze the behavior of a certain sorting algorithm; see SMYTHE and WELLNER (999): *jaw@stat.washington.edu smythe@stat.orst.edu Ó VVS,. Published by Blackwell Publishers, 8 Cowley Road, Oxford OX4 JF, UK and 35 Main Street, Malden, MA 48, USA.

2 J. A. Wellner and R. T. Smythe Problem. What are the numbers E A A ; Cov A ; A, and Correl A ; A? PROPOSITION. E A A :545...; Cov A ; A :8796; and Correl A ; A.378. PROOF, PART. Note that by Fubini's theorem we have E A A EfjB s B s jjb t B 3 t jg ds dt Efj s jjz t jg ds dt where s B s B s ;Z t B t B 3 t have a bivariate normal distribution with zero means, variances s s and t t resectively, and correlation Corr s ; Z t = s=t t = s if s O t h s;t : = t=s s = t if t O s Thus, by standardizing s and Z t, the right side of () is equal to s s t t Efj~ s j j~z t jg ds dt where ~ s and ~Z t have a joint normal distribution with means, variances, and correlation given by (). Thus we need to comute E h jxj jyj as a function of h where (X, Y) have a bivariate normal distribution with means, variances, and correlation h. 3 Formulas for E h jxj jyj Here we rove the following roosition giving two di erent in nite series reresentations of the function E h jxj jyj : PROPOSITION. If (X, Y) have a bivariate normal distribution with means, variances, and correlation h, then E h jx jjyj h X 3= h k C k 4 C k = k 8 " 4 X X # 9 < k h k k! j k j k j! = j! j 5 : k j! ; Ó VVS, fharcsin h j h g: 6

3 Covariance of two Brownian area integrals 3 PROOF. We will show that (4), (5), and (6) are all valid formulas reresenting the exectation E h jxjjyj as a function of h. The rst series formula is obtained by conditioning and using the series reresentation for the non-central chi-square distribution, while the second series formula is obtained by using Mehler's formula for the Radon±Nikodym derivative of the bivariate normal distribution with correlation h with resect to the normal distribution with correlation (indeendence). The third exression is obtained by reresenting (X, Y) in terms of indeendent N(, ) random variables, and integrating via use of olar coordinates. To rove (4), rst write E jxjjyj E jxje jyj X where the conditional distribution of Y given X is N hx; h. Thus the distribution of Y= h conditionally on X is N hx= h ;. Hence it follows (see e.g. GRAYBILL 976, 5±6) that s s jy j Y h h v h X h where Z v d has density f Z z X k k d= g z; k =;= ; here k d= is the Poisson d= robability given by k d= e d= d= k ; k! and g k z; k =; = is the Gamma density with arameters (k + )/ and / given by Thus z= g k z; k =; = k C k = ex z= : E= X k X k d= k z = z= k ex z= dz C k = C k k d= C k = : Hence we nd that ( " #) E h jx jjy j h jy j E jx je jx h Ó VVS,

4 4 J. A. Wellner and R. T. Smythe q (! ) h E jx j X ex h X h X = h k C k k h k! C k = 8 q!! 9 X h C k < k k!c k = E jx jex h X h X = : h h ; k h 3=X h k C k C k = ; k after comuting the exectation under the sum and nding that it equals = h k h C k : Thus (4) holds. (We call (4) Perlman's formula since this method was suggested to us by Michael Perlman). Now for (5). Mehler's formula (see e.g. MEHLER, 866, BUJA, 99, and KLAASSEN and WELLNER, 997, 74) says that the bivariate normal density / h of (X, Y) can be written as ( ) / h x; y X h k Hk x H k y / x / y ; here / x = ex x = is the standard normal density, and fh kg are the Hermite olynomials normalized to be orthonormal with resect to /. Since EjXj EjYj =, by interchanging the sum and the integration we nd that E h jx jjy j X h k b k where Now Z Z b k jxjhk x / x dx k! H k x k! X k=š j j! j k j! xk j ; j jxjh k x ex x = dx k! c k: see e.g. ABRAMOWITZ and STEGUN (97), 774, where their He n is our H n. Hence we comute c k for k odd, and for k even Ó VVS, c k k! X k=š j j! j k j! j k! X k=š j k= j j! j k= j! k j! : jxjx k j e x = dx

5 Covariance of two Brownian area integrals 5 For examle, Mathematica yields c ; c 4 ; c 6 6; c 8 3; c ;... Combining the above yields 8 ( E h jx jjy j 4 X X ) 9 < k h k k! j k j k j! = : j! j k j! ; j ( ) 4 X h k c k : k! Thus (5) holds. (We call (5) Wellner's formula because the rst author obtained it via Mehler's formula in the course of this study.) To rove (6), note that if X and Z are indeendent N(, ), then de ning Y hx h Z; it follows that (X, Y) has a bivariate normal distribution with zero means, unit variances, and correlation h. Thus it follows by using olar coordinates for the joint distribution of (X, Z), that EjXY j Z jhr cos / h r cos / sin /jre r = dr d/ Z r 3 e r = dr cos /jh h tan /j d/ fh arcsin h h g where the last equality follows by noting that the function / 7! h h tan / is ositive for / ;= [ / h ;3= [ / h ; ; negative for / =;/ h [ 3=; / h, where / h arctan h= h =; ; / h / h ; and then doing a careful integration for the searate ieces. (We call (6) the Janson± Smythe formula because it was derived by the second author after a suggestion by S. Janson.) ( Fig.. Plot of E h jx jjyj: Ó VVS,

6 6 J. A. Wellner and R. T. Smythe In the above lot, the grah descending raidly near is the lot via Perlman's formula; the grah which increases monotonically to is from Wellner's formula and the Janson±Smythe formula. The divergence near is due to truncation of the series for g at 5 terms. The lot was roduced by Mathematica code available from the rst author. Added in roof: Steve Stigler has ointed out that (6) (and other absolute moments of bivariate and trivariate normal distributions) were derived by NABEYA (95) and KAMAT (953); also see JOHNSON and KOTZ (), age 9 and (3) on age 9. In articular, our (6) is given by KAMAT (953), age 6, in his formula for (,,). 3 Comletion; combining the arts PROOF OF PROPOSITION, PART. Now we can use the series (4) and (5) of Proosition to comute the exectation inside the integral in (3). By the series in (4) Efj~ s jj~z t jg h X 3= s; t h k C k s; t C k = k where h s; t is given by (). Plugging this in and changing the order of summation and integration we nd that E A A is equal to 4 s s t t h s;t X k X 4 k C k C k = C k C k = b k X 3= h k C k s;t C k = k s s t t h s;t 3= h k s;t Comuting the integrals numerically in Mathematica yields Table. Ó VVS, Table. Values of the integrals b k k ) ) ) ) ) ) )9 b k

7 Covariance of two Brownian area integrals 7 Using the numerical values of b k to comute the series, we nd that E A A :545. Alternatively, by the series in (5), Efj~ s jj~z t jg ( ) 4 X h k s; t c k ; k! and lugging this in and changing the order of summation and integration, we nd that E A A is equal to Z ( ) s s t t 4 X h k s; t c k k! 8 s s t t ds dt X c k s s t t h k s; t ds dt ( k! 8 X ) c k 8 k! d k 6 X c k k! d k: Evaluating the series numerically we nd again (with only four terms) that E A A :545...: Using the third formula (6), we nd that E A A s s t t q fh s;t arcsin h s; t h s; t g ds dt 8 Z t s s t t ( r arcsin r! r ) t s t s 4 t s 4 Z t s t arcsin r! t s 8 Z r t s s t t ds dt 4 s t 8 I II : Now by letting x s= s and y t= t, Ó VVS,

8 8 J. A. Wellner and R. T. Smythe Table. Values of the integrals c k and d k k c k d k ) ) ) )5 6 ) ) )7 8 ) )7 II Z y y u = r s s t t ds dt 4 s t s xy x 3 y 3 x dx dy 4 y y 3 u = yu 3 u=4 u=4 du dy y uy 3 dy du 3 y On the other hand, Z t I s t arcsin r! t s Z t s t t t arcsin r! t s Z y x y 3 x 3arcsin x=y dxdy y u y 3 yu 3 arcsin u dudy Z uarcsin y u y 3 dy du: 3 yu Combining these ieces yields E A A 8 uarcsin u = u u=4 k u du where y k u y 3 yu 3 dy log =u fu 4u g 3u 3 u 5 Ó VVS,

9 Covariance of two Brownian area integrals 9 satis es lim u! k u =3. Evaluating these integrals numerically (by taking k u =3 for :995OuO), yields E A A :543. It follows from results of SHEPP (99) (see also PERMAN and WELLNER, 996, 7) that E A E A =4 :443...and Var A Var A 7=3 =6 : This results in Cov A ; A :8795, and Correl A ; A :378. ( Acknowledgements Michael Perlman roduced the rst series reresentation given in Proositon, and this lead to the second series reresentation using Mehler's formula. Svante Janson suggested the indeendent normals reresentation used in the third aroach to Smythe. Moreover, we owe thanks to Steve Stigler for directing our attention to Johnson and Kotz, Nabeya, and Kamat. Research suorted in art by National Science Foundation grant DMS and NIAID grant RO AI References ABRAMOWITZ, M. and I. A. STEGUN (97), Handbook of mathematical functions (9th ed.), Dover, New York. BUJA, A. (99), Remarks on functional canonical variates, alternating least squares methods, and ACE, Annals of Statistics 8, 3±69. GRAYBILL, F. A. (976), Theory and alication of the linear model, Wadsworth, Belmont. JOHNSON, N. L. and S. KOTZ (), Distributions in statistics: continuous multivariate distributions. Wiley, New York. KAMAT, A. R. (953), Incomlete and absolute moments of the multivariate normal distribution and some alications, Biometrika 4, ±34. KLAASSEN, C. A. J. and J. A. WELLNER (997), E cient estimation in the bivariate normal coula model: normal margins are least favorable, Bernoulli 3, 55±77. MEHLER, F. G. (866), Ueber die Entwicklung einer Funktion von beliebeg vielen Variablen nach Lalaceschen Funktionen Hoeherer Ordnung. J. Reine Angew. Math. 66, 6±76. NABEYA, S. (95), Absolute moments in -dimensional normal distributions, Annals of the Institute of Statistical Mathematics 3, ±6. PERMAN, M. and J. A. WELLNER (996), On the distribution of Brownian areas, Annals of Alied Probability 6, 9±. SHEPP, L. A. (98), On the integral of the absolute value of the inned Wiener rocess, Annals of Probability, 34±39. SMYTHE, R. T. and J. A. WELLNER (999), Stochastic analysis of shell sort, Technical Reort htt:// Received: December 999. Revised: Setember. Ó VVS,