J2 e-*= (27T)- 1 / 2 f V* 1 '»' 1 *»

Similar documents
ASYMPTOTIC DISTRIBUTION OF THE MAXIMUM CUMULATIVE SUM OF INDEPENDENT RANDOM VARIABLES

ON THE NUMBER OF POSITIVE SUMS OF INDEPENDENT RANDOM VARIABLES

A CONVERGENCE CRITERION FOR FOURIER SERIES

ON THE MEDIANS OF GAMMA DISTRIBUTIONS AND AN EQUATION OF RAMANUJAN

ON THE AVERAGE NUMBER OF REAL ROOTS OF A RANDOM ALGEBRAIC EQUATION

SOME THEOREMS ON MEROMORPHIC FUNCTIONS

THE POISSON TRANSFORM^)

YALE UNIVERSITY DEPARTMENT OF COMPUTER SCIENCE

SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS

Ht) = - {/(* + t)-f(x-t)}, e(t) = - I^-á«.

N V R T F L F RN P BL T N B ll t n f th D p rt nt f l V l., N., pp NDR. L N, d t r T N P F F L T RTL FR R N. B. P. H. Th t t d n t r n h r d r

n

PR D NT N n TR T F R 6 pr l 8 Th Pr d nt Th h t H h n t n, D D r r. Pr d nt: n J n r f th r d t r v th tr t d rn z t n pr r f th n t d t t. n

Vr Vr

NOTE ON A SERIES OF PRODUCTS OF THREE LEGENDRE POLYNOMIALS

828.^ 2 F r, Br n, nd t h. n, v n lth h th n l nd h d n r d t n v l l n th f v r x t p th l ft. n ll n n n f lt ll th t p n nt r f d pp nt nt nd, th t

Humanistic, and Particularly Classical, Studies as a Preparation for the Law

(3) 6(t) = /(* /(* ~ t), *(0 - /(* + 0 ~ /(* ~t)-l,


HADAMARD DETERMINANTS, MÖBIUS FUNCTIONS, AND THE CHROMATIC NUMBER OF A GRAPH

n r t d n :4 T P bl D n, l d t z d th tr t. r pd l

1 Solution to Homework 4

4 4 N v b r t, 20 xpr n f th ll f th p p l t n p pr d. H ndr d nd th nd f t v L th n n f th pr v n f V ln, r dn nd l r thr n nt pr n, h r th ff r d nd

,. *â â > V>V. â ND * 828.

ON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS

2 K cos nkx + K si" nkx) (1.1)

Upper Bounds for Partitions into k-th Powers Elementary Methods

Th n nt T p n n th V ll f x Th r h l l r r h nd xpl r t n rr d nt ff t b Pr f r ll N v n d r n th r 8 l t p t, n z n l n n th n rth t rn p rt n f th v

238 H. S. WILF [April

On the Equation of the Parabolic Cylinder Functions.

Labor and Capital Before the Law

THE CONTIGUOUS FUNCTION RELATIONS FOR pf q WITH APPLICATIONS TO BATEMAN'S ƒ»«and RICE'S # n (r, p, v)

ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER. Communicated by W. Feller, May 1, 1962

4. cosmt. 6. e tan 1rt 10. cos 2 t

A THEOREM ON THE DERIVATIONS OF JORDAN ALGEBRAS

0 t b r 6, 20 t l nf r nt f th l t th t v t f th th lv, ntr t n t th l l l nd d p rt nt th t f ttr t n th p nt t th r f l nd d tr b t n. R v n n th r

THE REPRESENTATION OF INTEGERS BY THREE POSITIVE SQUARES E. GROSSWALD, A. CALLOWAY AND J. CALLOWAY

arxiv: v1 [math.gm] 31 Dec 2015

PERIODIC SOLUTIONS OF SYSTEMS OF DIFFERENTIAL EQUATIONS1 IRVING J. EPSTEIN


SOME FINITE IDENTITIES CONNECTED WITH POISSON'S SUMMATION FORMULA

ON RESTRICTED SYSTEMS OF HIGHER INDETERMINATE EQUATIONS * E. T. BELL

22 t b r 2, 20 h r, th xp t d bl n nd t fr th b rd r t t. f r r z r t l n l th h r t rl T l t n b rd n n l h d, nd n nh rd f pp t t f r n. H v v d n f

a» [r^ivrrwr'c.)*(-) (,-)X*r

ON THE CONVERGENCE OF APPROXIMATE SOLUTIONS OF

A Chebyshev Polynomial Rate-of-Convergence Theorem for Stieltjes Functions

46 D b r 4, 20 : p t n f r n b P l h tr p, pl t z r f r n. nd n th t n t d f t n th tr ht r t b f l n t, nd th ff r n b ttl t th r p rf l pp n nt n th

ƒ f(x)dx ~ X) ^i,nf(%i,n) -1 *=1 are the zeros of P«(#) and where the num

ON THE DIVERGENCE OF FOURIER SERIES

(4) cn = f f(x)ux)dx (n = 0, 1, 2, ).

Calculation of the Ramanujan t-dirichlet. By Robert Spira

AN ARCSINE LAW FOR MARKOV CHAINS

I ƒii - [ Jj ƒ(*) \'dzj'.

T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r )

ON THE EXTENSION OF A FUNCTIONAL INEQUALITY OF S. BERNSTEIN TO NON-ANALYTIC FUNCTIONS 1


Introduction Derivation General formula Example 1 List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)

lim Prob. < «_ - arc sin «1 / 2, 0 <_ «< 1.

Introduction Derivation General formula List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)

Colby College Catalogue

Math 113 (Calculus 2) Exam 4

The polar coordinates

ON THE HURWITZ ZETA-FUNCTION

NORMS OF PRODUCTS OF SINES. A trigonometric polynomial of degree n is an expression of the form. c k e ikt, c k C.

Notes on uniform convergence


Colby College Catalogue

SOME STEADY STATE PROPERTIES OF. (fj(t)dt)/f(x) VIVIENNE MAYES

MATH 251 Examination II April 3, 2017 FORM A. Name: Student Number: Section:

4 8 N v btr 20, 20 th r l f ff nt f l t. r t pl n f r th n tr t n f h h v lr d b n r d t, rd n t h h th t b t f l rd n t f th rld ll b n tr t d n R th

NEW GENERATING FUNCTIONS FOR CLASSICAL POLYNOMIALS1. J. w. brown

Exhibit 2-9/30/15 Invoice Filing Page 1841 of Page 3660 Docket No

ON CONVERSE GAP THEOREMS

Colby College Catalogue

Colby College Catalogue

Solutions for Math 217 Assignment #3

TWO NEW PROOFS OF LERCH'S FUNCTIONAL EQUATION

252 P. ERDÖS [December sequence of integers then for some m, g(m) >_ 1. Theorem 1 would follow from u,(n) = 0(n/(logn) 1/2 ). THEOREM 2. u 2 <<(n) < c

ON FUNCTIONS WITH BOUNDED DERIVATIVES

1981] 209 Let A have property P 2 then [7(n) is the number of primes not exceeding r.] (1) Tr(n) + c l n 2/3 (log n) -2 < max k < ir(n) + c 2 n 2/3 (l

A NOTE ON FUNCTIONS OF EXPONENTIAL TYPE 1. An entire function ƒ(z) is said to be of exponential type at most T if

{~}) n -'R 1 {n, 3, X)RU, k, A) = {J (" J. j =0

z E z *" I»! HI UJ LU Q t i G < Q UJ > UJ >- C/J o> o C/) X X UJ 5 UJ 0) te : < C/) < 2 H CD O O) </> UJ Ü QC < 4* P? K ll I I <% "fei 'Q f

RESEARCH ANNOUNCEMENTS

APPH 4200 Physics of Fluids

R e p u b lic o f th e P h ilip p in e s. R e g io n V II, C e n tra l V isa y a s. C ity o f T a g b ila ran

THE DIFFERENCE BETWEEN CONSECUTIVE PRIME NUMBERS. IV. / = lim inf ^11-tl.

SOME INFINITE SERIES. H. f. SANDHAM. 1. We sum eight general infinite series [l], particular cases of which are. Is 33 53

INTEGRABILITY ALONG A LINE FOR A CLASS

ON THE CONVERGENCE OF DOUBLE SERIES

JUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson

ON A CONVERGENCE THEOREM FOR THE LAGRANGE INTERPOLATION POLYNOMIALS

UNIFORM INTEGRABILITY AND THE POINTWTSE ERGODIC THEOREM

ON EQUIVALENCE OF ANALYTIC FUNCTIONS TO RATIONAL REGULAR FUNCTIONS

LA PRISE DE CALAIS. çoys, çoys, har - dis. çoys, dis. tons, mantz, tons, Gas. c est. à ce. C est à ce. coup, c est à ce

ON THE GIBBS PHENOMENON FOR HARMONIC MEANS FU CHENG HSIANG. 1. Let a sequence of functions {fn(x)} converge to a function/(se)

ON REARRANGEMENTS OF SERIES H. M. SENGUPTA

ADVANCED PROBLEMS AND SOLUTIONS Edited by RAYMOND E.WHITNEY Lock Haven State College, Lock Haven, Pennsylvania

A note on a scaling limit of successive approximation for a differential equation with moving singularity.

A NEW METHOD OF INVERSION OF THE LAPLACE TRANSFORM* ATHANASIOS PAPOULIS. Polytechnic Institute of Brooklyn

Transcription:

THE NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION AND A PROOF OF A CONJECTURE OF RAMANUJAN 1 TSENG TUNG CHENG 1. Summary. The Poisson distribution with parameter X is given by (1.1) F(x) = 23 p r where p r = er\ n = [x], for xào. It is well known that F(x) converges to the normal distribution as X» oo. We shall prove the following theorem. THEOREM I. Let *=X- 1/2 (w-x+l/2). Then J2 e-*= (27T)- 1 / 2 f V* 1 '»' 1 *» (1.2) - rl J - + (1/6) (2TTX)- 1 / 2 (1 - x 2 )e-w* 2 + Ô, where 8 satisfies the inequality (1.3) [ Ô <.076X- 1 +.043X- 8 / 2 +.13X" 2. This formula is analogous to Uspensky's [l] 2 estimate of the error term in the normal approximation to the binormal distribution and our proof consists in an adaptation of Uspensky's method. At the same time we verify the following conjecture of Ramanujan: for every positive integer n the value of 6 which satisfies the equation ( n n n ~ l n n \ (1.4) \ 1 + - +... + + 0 - U - = 1/2 I 1! («1)1 n\) lies between 1/2 and 1/3, and tends to 1/3 as n *<x>. A proof of the above statement was given by Szegö [3]. We give a more elementary proof, using only standard tools, for the following theorem. THEOREM II. For n^7.37 and 1/3. the root of the equation (1.4) in 0 lies between Finally, we shall obtain the following asymptotic expansion of 0: Received by the editors March 15, 1948. 1 This paper contains the results of a master's thesis done under the direction of Professor W. Feller and accepted by Cornell University in September, 1947. 2 Numbers in brackets refer to the references cited at the end of the paper. 396

NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 397 (1.5) 0 = 1/3 + (4/135)»- 1 + (8/2835)w- 2 + 0(rr*). 2. Proof of Theorem I. From the characteristic function we get oo X p r e irt = «*< **-», r«=0 (2.1) f *-«" *&*-»dt = 2x^r. From (2.1) it follows that i> r = (27T)- 1 f'csin^)- 1 r=0 J 0 exp {X(cos * - 1)} sin [(» + 1/2)/ - X sin /]<Ö + (27T)- 1 f (sin //2)- 1 exp {x(cos t - 1)} sin (X sin / + */2)(ft. The second integral is independent of X, since its derivative with respect to X vanishes, as may easily be verified. Putting X = 0 we see the value of the integral is 1/2. Hence we have =(2T)-* f'(siii//2)- rto r\ 2 I Jo (2.2) -exp {X(cos / - 1)} sin [(» + 1/2); - X sin *]<# = /. To estimate the integral I we introduce Then A = (2T)-* f *-*«1 /*(</2)- 1 sin [X»/*a* + X(< - sin 0]<«. / - h\ ^ (2X)- 1 f e-^l\t/2)- l dt (2 3) r T + (27T)- 1 I {r*- 1 ** «/2( sin j/2)-i - er^«1 /«(//2)-i}^ / o = ƒ i + /«+ /a, where the integrals J* are defined and estimated as follows:

398 T. T. CHENG [April /i = (2a-)- 1 f {<r 2X 8in2 " 2 (sin t/2)~ l cos */2 - e-^i % {t/2)- 1 }dt (2.4) T = - (27T)- 1 f e-***(t/2)- l dt < 0; (2.5) J, = (2x)- x f e-^i\t/2)- l dt < (32*-)- 1 f Pr^^dt; j t = (27T)- 1 f «-» * " 2 (sin</2)- 1 (l - cos */2)<Ö «/ o < (2*-)- 1 f e~ 2X 8i» 2 " 2 - (sin //2)- 1 (2-6), (sin 2 1/2 cos */2 1 < + (1 - cos tl2) sin 2 t/2\dt < {Sir)- 1 f te-^iht + (32a-)- 1 f *V»«, ''\tt. It follows from (2.3)-(2.6) that I I - /il < (Sir)- 1 f fer** 1 /**» + (327T)- 1 f t*(r**i^dt (2.7) Jo Jo Now we write = (87TX)" 1 + (7T 3 /256)\- 2 < (ZTTX)- 1 +.13X~ 2. I x = TT" 1 f r^- x<2 / 2 -sin X^atf-cos X(* - sin t)dt + TT" 1 f r^- x<2 /2-cos X^atf-sin X(/ - sin t)dt (2.8) = K- 1 f rv^«f ' 2 -sinx 1 '»*tó* +/A + JTT- 1 f r^-^'/^cos X 1 ' 2 */- dt + ƒ 5} where = (27T)- 1 / 2 f e~?m + (2TTX)- 1 / 2 (1 - tf 2 )*-* 2 ' 2 + J A + J B, 6 j 4 = Tr-i f r^- x * 2 / 2 [cos X(/ - sin t) - l]<tt,

1949] NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 399 (2.9) / 5 = TT-i f t-h-*' 2 ' 2 cos \ l ' 2 xt- sin X(* - sin /) \dt. Upper limits for / 4 and J 5 are obtained as follows: 2 \ 2 0 - sin t) 2 / 4 <7T" 1 f / 0 2 X ƒ 2 / 6 ri e -^t 2 /2 dt=z (97TX)- 1 ; 0 72 I / 5 i < Tr- 1 f r 1^' 2 ' 2 j[" - X(* - sin *)1 (2.10) + fx(/ sin t) sin \{t sin t) 1 > dt L ^ } V yj j o L120 6 4 J = (2TT)- 1 / 2 X- 3 / 2 [1/40 + 35/432] <.043X- 3 ' 2. From (2.7)-(2.10) and the inequality (8X)- 1 + (9TT)- 1 <.076 the required result follows immediately. 3. Proof of Theorem II. Solving (1.4) for 0 and using the formula (2.2), we get (3.1) 26-1 = - r/(n n e~ n /nty, where ƒ' = (27T)- 1 f (sin ] -exp {w(cos t - 1)} < sin f n H ] / w sin / + sin In 1 / n sin / >dt = 7T" 1 I (sin 1 -exp {n(co$ t 1)} -cos sin n(t sin /)d/ = 27T- 1 f r 1 * «''«-sin n 2 arc sin t(\ J 1*. By Maclaurin expansion, 2 arc sin t/2 - /(l - * 2 /4) 1/2 = / 3 /6 + t*/8q + 3* 7 /1792 + t*f(t),

400 T. T. CHENG [April where ƒ(t) is an increasing function in the interval 0g/^2. Hence Assume n^7. therefore (3.2) sin n[2 arc sin t/2 - /(l - / 2 /4) 1/2 ] ^ sin (w/ 3 /6 + nt*/80) + 3nf/1792 + nt 9 f(2) <> (nt z /6 + nt 5 /SO) - (nt z /6 + nfi/80)*/6 + (nt*/6 + nfi/soy/120 + Snt 7 /1792 +.0024w/ 9. Then the last expression is positive when 2>2, and nt z nt b 1 / nt z nt b \* o I 6 80 6\ 6 80/ + ( 1 H f-.oolint'} dt 120\ 6 80/ 1792 ) < (2 1 rw)- 1 ' 2 {l/3 -.087W- 1 +.012w- 2 + 1.87ir-«+ 4-1»- 4 + 6w~ 6 + 5n-«+ In-- 1 ). In obtaining the last inequality we have, as before, used the familiar relation: 2(2*)- 1 '» f fi*g-vm*dt =1-3 (2* 1). Taking in (3.2) wï; 7 we get (3.3) I' < (2iro) _1/2 (l/3 -.03»- 1 ). On the other hand we can write 2 arc sin t/2 - t(l - P/i) 1 ' 2 = t*/6 + t*g(t) because g(t) increases and g(2) <.057. Hence à t 3 /6 +.0571\ I' > 2X- 1 f r-v-' 1 '""' 2 jsin - -.051nÀdt Therefore /.(tit* 1 / nt*\* Ï rv-a/i>»i»j / j _.057nt 6 \dt. (3.4) /' > (2TTW)- 1 / 2 (1/3) -.S05W- 1 ) >.26(2xw)- 1 ' 2, when» è 7. From (3.1), (3.3), (3.4), and Stirling's formula,

i 9 49] NORMAL APPROXIMATION TO THE POISSON DISTRIBUTION 401 (2irn) ll2 n n e- n < nl< (2Tfi) 1 f 2 n n e- n+1/12n ) it follows that 20-1< -.26, or 0<.37; and 26-1 > - e 1 / 12w -(27rw) 1 / 2 -(27T^)- 1 / 2 (l/3 -.03/») and therefore 6> 1/3. > - (1 +.09ft- 1 )(1/3 -.03»- 1 ) = - 1/3+.0027»- 2 > - 1/3, 4. Proof of the equation (l.s). It can be verified without difficulty that V = (2TT»)- 1 / 2 {1/3-47/540»- 1 + 71/6048»" 2 } + 0(»" 7 / 2 ). From a modified form of Stirling's formula due to Feller [2], namely, for w^4, ( 1 1 + Ô).. n\ = (2 )*/*»» exp { -» + - - }, 8 < 1/6, I 12» 360» 3 ; it follows that 20-1 = ~ [1/3-47/540»- 1 + 71/6048»" 2 + ] [1 + 1/12»- 1 + 1/(2.12»)»-* + ] = - 1/3 + 8/135»" 1 + 16/2835»- 2 + 0(»~ 3 ). An obvious reduction then gives the equation (l.s). REFERENCES 1. J. V. Uspensky, Introduction to mathematical probability, New York, 1937. 2. W. Feller, On the normal approximation to the binormal distribution, Ann. Math. Statist, vol. 16 (1945) pp. 319-329. 3. G. Szegö, Über einige von S. Ramanujan gestellte Aufgaben, J. London Math. Soc. vol. 3 (1928) pp. 225-232. CORNELL UNIVERSITY

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback