8/8/2019 1427094

1/4

Asymptotics of Two Integrals from Optimization Theory and Geometric ProbabilityAuthor(s): D. J. GatesSource: Advances in Applied Probability, Vol. 17, No. 4 (Dec., 1985), pp. 908-910Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427094

Accessed: 05/11/2010 04:51

Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at

http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless

you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you

may use content in the JSTOR archive only for your personal, non-commercial use.

Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at

http://www.jstor.org/action/showPublisher?publisherCode=apt.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed

page of such transmission.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of

content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact support@jstor.org.

Applied Probability Trustis collaborating with JSTOR to digitize, preserve and extend access toAdvances in

Applied Probability.

http://www.jstor.org

http://www.jstor.org/action/showPublisher?publisherCode=apthttp://www.jstor.org/stable/1427094?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/action/showPublisher?publisherCode=apthttp://www.jstor.org/action/showPublisher?publisherCode=apthttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/1427094?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=apt

8/8/2019 1427094

2/4

Adv. Appl.Prob.17, 908-910 (1985)Printed n N. Ireland? AppliedProbabilityTrust1985

ASYMPTOTICS OF TWO INTEGRALS FROM OPTIMIZATIONTHEORY AND GEOMETRIC PROBABILITY

D. J. GATES,* CSIRO Division of Mathematics and StatisticsAbstract

Asymptotic series are derived for two integrals using a Gaussianidentity and Laplace's method, demonstrating an improvement overearlier methods.LAPLACE'S METHOD; OPTIMIZATION

Anderssen et al. (1976) obtain various bounds and approximations for the expecteddistance(1) mk + + X ) dxl dXkfrom the origin of a point uniformly distributed in the cube [0, 1]k. They evaluate m1,m2 and m3 exactly. Otherwise their computationally most efficient formula, by far, is theasymptotic series(2) mk= (k/3)?(1- 1/10k- 13/280k2- 101/2800k3-37533/1232000k4)+ O(k )as k -> oo.Terms up to k-3 give, for example, n4 accurate to five figures, m0loaccurateto six figures and n20 accurate to seven figures. Their derivation of (2) is, however,cumbersome. We give a simple derivation based on Laplace's method.The authors also study the expected interpoint distances(3) Mk= -.. {x,- y+)2 +(xyk)2}1dx idy. dxk dYk,but do not give an asymptotic series like (2), presumably because of the work requiredusing their method. We give a simple derivation of such a series, again using Laplace'smethod.Since(4) AX=(2/V)i)Ak ds exp (- 1s2)we can write(5) mk = (2/rr)kJ f'(- s2)(- S2k-1 ds,

Received 28 August 1985.* Postal address: CSIRO Division of Mathematics and Statistics, GPO Box 1965, CanberraACT 2601, Australia.908

8/8/2019 1427094

3/4

Letters o theeditor 909where

(6) f(t) = fexp (tx2)dx.Since f(t) has a maximum at t = 0, and f'(0)= 3, we write

(7) f(t)= exp (t/3) {1+ t(x2-)+t2(2-)2 .} dx= exp(t/3)(1+ 2t2/45+ ?? ).Similarly(8) f'(t) = exp (t/3)( + 4t/45 + * ).Finally

mk = (2/,rr)kJ exp (-ks2/6)( - s2+-ks+ ...) ds(9)")(1- 1/10k + ).

Turning now to (3) we have, similarly,(10) Mk= (2/w)kg'(-s2)g(-2k1 S,

where(11) g(t) = J1exp (t(x - y)2) dxdy,which has a maximum at t = 0, where g'= g. Thus we write(12) g(t)= exp(t/6) t"In.=Owhere

In= - I{(x-yy--}dxdy,and(13) g'(t) = exp (t/6) t"J,,n=Owhere

J, = In/6+ (n + 1)In,1.Then Io=1, I1=0, 12=7/360, 13= 11/5670, Jo= 1/6, J1=7/180 and J2= 137/15120.Now putting (12) and (13) in (10) and using standard formulae for moments of anormal density gives(14) Mk = (k/6)A(1-7/40k -65/896k2+ .).Anderssen et al. (1976) compute M1, M2 exactly and M3,'" , Mlo by a slowly

8/8/2019 1427094

4/4

910 Letters o the editorconvergent series method. They also obtain an upper bound(15) Mk < (k/6)1[{1+ 2(1- 3/5k)'}/3]1.The table lists the M1, ? , Ml,, from Anderssen et al. and their deviations from (14) (asshown) and (15) denoted (14)-Mk and (15)-Mk respectively. This illustrates theaccuracy of (14), for k not too small, while its efficiency is obvious.

k Mk (14)- Mk (15) - Mk1 0-33333 -0-026 0-0212 0-52141 -0-005 0-0243 0-66167 -0-001 0-0204 0-77766 -0-0006 0-0175 0-87853 -0-0003 0-0156 0-96895 -0-0001 0-0147 1-05159 -0-00007 0-0138 1-12817 -0-00004 0-0129 1-19985 -0-00002 0.011

10 1-26748 -0-00001 0.010

ReferencesANDERSSEN, R. S., BRENT, R. P., DALEY,D. J. AND MORAN,P. A. P. (1976) Concerning

f1... (x0 + .. + x)dxdx1 dXk and a Taylor series method. SIAM J. Appl. Math. 30, 22-30.