GENERALIZATION OF A THEOREM OF CHUNG AND FELLER MIRIAM LIPSCHUTZ

GENERALIZATION OF A THEOREM OF CHUNG AND FELLER MIRIAM LIPSCHUTZ 1. Introduction. Chung and Feller [l] have obtained the following result on fluctuations in coin tossing. Let Xn be a sequence of independent random variables, each assuming the values +1 with probability 1/2. Let Sn = Xi+ - +Xn and let Nn denote the number of those terms in the sequence Si, S2,, Sn which either are positive, or which are zero but follow a positive term. With the familiar notation for conditional probabilities the Chung-Feller theorem states that for each integer r ^ «(1) P{iVr2 ^2r 52 = 0} = -- M + 1 In other words, under the hypothesis 52n = 0 the variable N2n becomes uniformly distributed. This is in contrast to the unconditional distribution for A^n which is given by the arc sine law. The aim of the present paper is to generalize this result to arbitrary lattice variables.1 In this case the result of the form (1) will have only an asymptotic character. We shall prove the following. Theorem. Let Xk be a sequence of independent random variables with a common distribution F(x) such that : (A) The r.v. have mean zero, variance 1, and finite fourth moment; (B) The r.v. have a lattice distribution such that the minimum distance between jumps is one unit. (The r.v. Xk thus assume integer values only.) Then, fa»]+ 1 (2) P(Nn = an \ Sn = 0) = -+ g(n), Oá^l, «+ 1 where g(n) = 0(-I \ n1'30 ) if the r.v. have third moment zero, and / log n \ g(n) =01-1 if the r.v. have third moment differing from zero. Received by the editors April 7, 1951 and, in revised form, September 23, 1951. 11 wish to thank Professor K. L. Chung for suggesting this problem to me. 659

660 MIRIAM LIPSCHUTZ [August 2. Method of proof. The proof is carried out in three steps. The first step adapts the Erdös-Kac invariance method [2; 3] to conditional probabilities of the form P(A7n^a«S = 0). This leads IjO (10). The next step involves the application of a result of Esseen [4] on the multi-dimensional central limit theorem for lattice distributions to the upper and lower bounds in this inequality (10). In order to do so we use Euler's formula to extend Esseen's theorem to regions of summation appropriate to our case. This gives (18) which may be of independent interest. Having thus obtained bounds, with errors which tend to zero in the limit, for P(Nnúcm, S = 0), we note that these bounds are independent of the distribution of the individual Xk. We complete the proof by substituting for these the known limiting value in the special case of Chung-Feller. We add that, unfortunately, the order of magnitude of the error term in the final approximation to the uniform distribution is still very large. 3. Generalization of the invariance principle of Erdös-Kac. Let and note that Let now k = nl'\ w - {J (1 if Sr > 5,- r < n,2 (O otherwise, V(Nn < an, Sn = 0) = P f <t>(sr) < a). \n r_i / e = k-1'3, «= *-«,, rini r(* "(i - 1/2)»' i/2)»i»,= -, Wi =- ÎOT i = 1, 2,, k, and put We have Dn = -( 0(5,) - (m - n'i-msnù). n \ _i,«i / E( ö. )^-Z Z E( *(SBj)- <t>(sr) ) n,=i r»<_i'+i 1 Note the convention made in the introduction concerning the positiveness of the sums 5r.

1952] GENERALIZATION OF A THEOREM OF CHUNG AND FELLER 661 and we now wish to estimate E( 0(Sr) <j>(sni)\ ) for «/-i^rá«,'. Notice that and that for e>0 E( *(Sr) - < >(Sn,) = P(S»,. > Sr < Sn = 0) + P(S, < Sr> Sn = 0) P(5.,. > Sr < Sn = 0) = P(e«r > Sni >Sr< Sn = 0) 1/2 + P(5ni > m-,sr<sn = 0) (4) 1/2 = P(e«> Sni >Sn = 0) We have, for «4'_i<rgwt-, P(5n - Sr > mt, Sn = 0) + P(Snj -Sr> en]'2, Sn = 0). = Z P(5 - Sr = y, Sn = 0) V>«W>'2 (5) = P(5,4 - Sr = y, Sn - (Sn,. ~ Sr) = ~ J») = Z P(Sni - Sr = y)p(sn_(n,._r) = - v), whereas for»,<rá»/i P(Snj - Sr > e«,1/2, Sn = 0) = Z P(Sr - S,. = - y)p(s _(r_n,.) = y). Now, since in both cases considered we have 0^» r\ ^n/2k and «_«(\m r\)-=n n/2k, it follows by the central limit theorem that (7) P(Sn_,._r = y) <- ' y (n - n/2ky>2 n1'2 By Chebycheff's inequality we then obtain ni r c (5) < i2«,- «1/2 r ni (6)< - c We consider next

662 MIRIAM LIPSCHUTZ [August V(m]'2 > Sn; > Sn = 0) = 22 P(Sn,- = y, S = 0) 1/2 0<l/<en, = P(Sn = y,sn Sn< = y) 0<i/<«ni1/2 = Z PCS.«= y)p(s -,. = - y) 0<y<tn 1/2 We operate in exactly the same way with the second term of (3) and obtain,. 1 * i(ni n'i-i)(ni «Li 1) + (»,' «,-)(»i»< 1) c E( > ) = 2^ S- M,=1 I. 2»,- W1'* / / 1/2 1/2 C I + («,-»i_i)p( mi <Sni<mi )-- -> (» «)1/2; Now for large «, «i n'i_1 = n/2k+dl and «< «= «/2 +0/', where \dl and 0/' are less than 1. Thus the first term in (8) is equal to Since J_ [(n/2k) + e'j][(n/2k) + 6Ï] = j «2 i=i íw/ 1*1 1 + log /fe = Z- + KIX fo2 _i i ke2 1/2 1/2 1 Cl / /2\ ÖÖ P(-é»< < S, < e»; ) = - I exp {-J dt -\-, where3 \d\ <1 and Ç 1S a function of the distribution of X only, the second term in (8) equals 1 TV ' ' M7 öö 1 i n ti \ (my>2 ) (n - my2 Finally we obtain e * 1 C = - + II = I + II =-h 0(1). * i («-»,.)1/l W1/2 3 The value of these constants need not be the same each time these symbols appear.

1952] GENERALIZATION OF A THEOREM OF CHUNG AND FELLER 663, 1 + log k C Ce (9) E( Dn\ )<--f- + = R(n, e, *). kt2 nlli m1'2 From here on we proceed as in Kac-Erdös to conclude that fl ',, ) R(n, e, k) P \- Z (m- nli)4>(sno < «- 4 - -lj~j-l \n i~i ) o (10) = V(Nn < an) ( 1 ^, / 1 R(n, e, k) ^ P \~ Z («.'- nli)4>(sni) < a + 5} + -1-L-L-L. \n,-=i J 5 4. Application of the theorem of Esseen. We now proceed to evaluate Write p( 22 (n'i- nli)<b(sni) <a). \n <_i / vt = Z *,-. i=i,---,k, j=p mod (ni),)sni (î>) (P) Ft+i = F* + XBl+p, and consider the (& + l)-dimensional We have ni vector f(p) = (v p), vt, -., vt, víl\). 22 V(p) = (Snv, S t, Sn) ~ (S 1? S 2,, S h, Sn). p-1 (The ~ on the sums are necessary since Z"=i V p) may not be exactly equal to Sn,.. For this sum equals X1 + X2 + + Xni+ni + -Anl+l + + X '1+ 1 + A 2+i + l -A nt-_i+ni 1 l A-m i+ni but ní 2<«i+«i^«i, «2 2<«i +«i «2,, «< 2<«,_!+«! = «,. Thus Sni = Snj Ti where 7,l^2Si and i^k. Since we are however within the domain of the central limit theorem, the sum Ti is negligible for large «.) The V(p) are identically distributed r.v. in Rk+i with mass concentrated in the lattice points Zfíi1 *&* The at are (&+l)-dimensional vectors with components a, equal to 5,y and the p,- = ±1, ±2,.

664 MIRIAM LIPSCHUTZ [August The correlation matrix A/2k-l = (ßij)/2k~1 and its inverse A-1 = (A,-y)/ A are given by A = 1,1,1, 1,3,3, 1, O, J, 1,3,5,7, 1 3 5 7 1,3,5,7,- U.3,5,7, 3/2,-1/2, 0-1/2, 1, -1/2, 0-1/2, 1, -1/2-1/2, 1 2k- 1,2* 2k - 1,2* -1/2, 0 0 0 0 o, o, o, The determinant A of A equals 2*-1.4 We write for all p = l,, «i, -1/2, 1, -1/2,0-1/2, 3/2, -1 1, 1 (ID a] = j(vt)'dp, and note that k (12) a) = (2j - 1)% and a) - Ck* if y = E(X*) ^ 0. j=i We introduce the following transformation of variables for all P = l,,«!, Y(P) = (yt, <PK, * k+l) with Y v) = 22)t\ CijVJp), and the c{j so determined that 4 This is easily seen by noting that in the expansion of a by the first row, only the first and second terms differ from zero since all the other determinants in the expansion have at least one column which is a multiple of the first column. The same is again true for the new determinant obtained, etc.

1952] GENERALIZATION OF A THEOREM OF CHUNG AND FELLER 665 (F-PV = 1 and E(yTyT) =0 for i fi j. It then follows that c,7 =1/ A V» and (13) Z (Yi)2 = (^i/à)vivi 1*>1 i, 3=1 where we have written Ft- for Y v> since the Y p) are also identically distributed. The mass of the Y(p) will be concentrated in the points Zi-i1 vi i where the c are (*+l)-dimensional vectors with components en and where Vi = +1, +2,.Then (14) where P(F? = i)i, V? = v2,---,v? = Vk, V?+i = 0) / k k k \ P ( Fi = 2_i cwii, Y2 = j turn,, F*+i = 2_, Ck+i.iVi 1, \ =1 =1 i-l / 22yT = y7, 22v? = v7. p=i p=i We now quote the following theorem of Esseen [4]: Theorem. Let F(1), F(2),, F(n), be a sequence of independent, identically distributed r.v. in Rk with the lattice distribution defined above for the Y{p). Let an arbitrary r.v. of the sequence have the properties: (1) the mean values are equal to zero, (2) the dispersions are equal to one, (3) the mixed moments of first order are zero, (4) the fourth moments are finite. Then the probability distribution Pn(E) of (F(1)+F(2,+ +F(n))/«1/2 is also a lattice distribution and the probability mass qn(íi, e2,, et) at a discontinuity point («i, «2,, e*) of Pn(E) is expressed &y5».<».*) - ( ) ' I ft, I { exp ( - ( l+ "2+e*)) 1 / d d d y ( «i-1- «2-h + ak-) óm^v 3ei ae2 ÓW -e.(-(^^))}+0(^). We also recall the following facts derived from the Euler summation formula; let f(x) be continuous and have continuous first and second derivatives and /( 00 ) =/'( 00 ) =/"( 00 ) =/'"( 00 ) = 0. Then 6 Note the definition (11).

666 MIRIAM LIPSCHUTZ [August Z /(' ) = ("AW - rf(t)pi(t)dt (,->0 J 1/2 J 1/2 (15) = fjw - P2 (^jf (y) + j Kp2(t)f"(t)dt = jj(t)dt - P2(-~)f(j) - jys(t)f'"(t)dt where and Pi(t) = [t]-t+ 1/2, MO = Z cos 2rvt 2(vvy " sin 2rvt Pz(t) = Z '-,±í 2M2 ' P2(l/2) < P2(0) =, P3(l/2) - 0. We apply the theorem of Esseen to the expression in (14). We define 770 = 0. From (13) and the fact that r k+i = 0 we have that (14) equals ( \(*+d/2 1 í r * * ~i ^) i«5{[-5 ««-«*] «* 3 s3 r * * -i) <16) -«S "^e,pl-ss("-"")!j/ Let now + o( \ f(vu, Vk) = exp - 22 (vi - -m+i)2 L 4«,-_o J Note that - 2^ «ptt exp --l(i i- i7i+i)2 6* p_! Ô7?p L 4«t_o J

1952] GENERALIZATION OF A THEOREM OF CHUNG AND FELLER 667 d fill, >Vk) drip d2 2 f(vu, Vk) dr d3 TjfiVhdyP Ik / k\ (2r)p ijp-j - ijp+i)/(»?i,,1ft) + [ )> 2 n ( k k2 \ / k\2 = S-h y (2r p - rip-i - Vp+iy}f(vi, - -, Vk) +0 I I, ( n ml ) \n /,r)k) = iy( ) (2'p ~ 'J*-1 _ vp+ù + {~) ffiiu ' i*)«let y" be a fe-dimensional vector with y = 0 or 1 for any i and such that 22*-i (y?/k) <ct. Denote by Fa the set of all such vectors and by Eo and Ei the set of all integers less than zero and greater than zero, respectively. We have for any vector y'e:fa / * \(H-»/ï 1 +o(-l-) (fc \(fc+l)/2 1 ^-) or«, un Z Z Z /0?l,,Vk) 2irn/ 2<*-o/2 Sir,.' v «6^ + / i \ o(-i-y fi^^i^' Vi^^v2' vic*=-evi Using formula (15) above and remembering that the regions EViare either (1, o) or ( «o, 1), we obtain Z ÄVi,,Vk) = I / On,, vk)dr i + hi, where Similarly l*i < (k/24n)(l - V2)f(l/2, V2,,,).

668 MIRIAM LIPSCHUTZ [August Z Z fivi,, Vk) = I I f(vi, -, Vk)driidr>k IjG^j' Vl^Ey^ J EyJ " Ey2' where and + I hidr 2 + I h2dr i + hï2, J Ey J EVl> h2 < (*/24«)(l Tii r,3)f(r,i, 1/2, r,3,, Vk) A < (k/12n)f(l/2,l/2,v3,...,vk). We proceed in this way and obtain after integration terms of the error (17) and rl-jni t: I^Vi', ' ' ', ^n t P'Vk'l >n = O) / * y*+t>/«1 ç ç *2 / 1 \ + 6Q+01-) n V \«<*+W 1 JL Pi Z («'- «1-1)0(5»,) < a) \n,=i / 2-, P(Sn, G»,',, S t G E»i', S = 0) -era (Wl'.l/j'. -.»l')g^a * - (18) ^-j i,*j (iji vk) /*V'2 * 3) k2 + (-) «ez«,> +-«e \ n / p~i J n / 1\1/2 But for large «= 2«' * *(F«) + g(n, k) S $ >,i(f ) + (»', *). 5. The limiting distribution. We now refer back to the introduc-

1952] GENERALIZATION OF A THEOREM OF CHUNG AND FELLER 669 tion of our paper, where we stated that in the case P(X = 1) = P(X= -1) = 1/2, we have (1). From this it follows that for «= 2«' and for any a, 0 = a = 1, We utilize the result \n'a] + 1 F(Nn < na, S = 0) = P(S = 0)» + 1 (10) to obtain [(a - S)n'] + 1 c R(2n', e, *) g(n', k) «' + 1 (2m')1'2 5 1 Ú-4V,*(F0) (2«')1'2 ^ [(a + h)n'] + 1 c R(2n', e, k) g(n', k) From (11) and (18) it follows that,1/2 n'+l (2n'y>2 Ô n"2 eok * a k2 9Qk k2 g(», k)=-^22»p + «Ö = C# + 6Q. n1'2 p=i n1" nll n1'2 We recall that fe = «1/5, = *_1/3, ô = *_1/6; thus we have R(n,, *) _ /log * k 1 \ c If the third moments \ k1'* à1'6/»1'2 /log«1'6 1 \ c ~ \ w1'30 n1'30) n112 y equal zero, this gives g(n, k) R(n, e, k),1/2 (log»1/s\ 1130 )nll2 Whereas, if the third moments differ from zero, we stipulate instead that *=«1/12, e = *-1/s, h = k~m and obtain g(n, k) R(n, t, k) / log nxi12\ c c Thus and w1'2 \ n1172 )ñ"2 [an'] + 1 / log w1/s \ $n'.k(fa) =- + 0[- -J in the first case, A «1/3 ) «' + 1

670 MIRIAM LIPSCHUTZ [an'] + 1 /log w1'12 \ $n'.k(fa) =-h 01-1 in the second case. n' + 1 \ m1'72 / We use (10) once more and conclude that for any sequence of r.v. satisfying the conditions of the theorem or Q.E.D. [an] + 1 / log«\ V(Nn Ú an, Sn = 0)= ' + ol-f ) «+ 1 \ W1'30 / \an] + 1 / log«\ V(Nn Û an, Sn = 0) = * + O(-f-). «+ 1 \ nllli / References 1. K. L. Chung and W. Feller, On fluctuations in coin-tossing, Proc. Nat. Acad. Sei. U.S.A. vol. 35 (1949) p. 60S. 2. P. Erdös and M. Kac, On the number of positive sums of independent random variables, Bull. Amer. Math. Soc. vol. 53 (1947) p. 1011. 3. K. L. Chung, An estimate concerning the Kolmogoroff limit distribution, Trans. Amer. Math. Soc. vol. 67 (1949) p. 36. 4. G. G. Esseen, Fourier analysis of distribution functions, Acta Math. vol. 77 (1945) p. 1. University of Maryland