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O N T H E A V ER A G E N U M B E R O F R E A L R O O T S O F AR A N D O M A L G E B R A I C E Q U A T I O NM. KAC

1. In t roduc t ion . Cons ide r t he a lgeb ra i c equa t ion(1 ) Xo + X xx + X2x2 + + I n - i ^ " 1 = 0,whe re t he X's a re i ndependen t r andom va r i ab l e s a s suming r ea l va lue sonly , and denote by N n = N(X0f , X n - i ) the number o f rea l roo tsof ( 1 ) . W e w an t t o de t e rm ine t he mean va lue (m a them a t i ca l expec t a tion = m.e.) of N n when al l X's have t he s ame no rma l d i s t r i bu t ionwi th dens i t y(2 ) e' 2/* 1*2.This p rob lem was t rea ted by Li t t l ewood and Offord 1 who also cons idered the cases when the X's are un i formly d i s t r ibu ted in ( 1, 1)o r a s sum e on ly t he va lues + 1 and 1 with equa l p robabi l i t i es . L i t t l e -wood and Offord obtain in each case the es t imatem.e. {N n} ^ 25(lg n)2 + 12 lg , n ^ 2000.In our case of normal ly d i s t r ibu ted X's we shal l be able to prove theexac t fo rmula

, , 4 r l [1 - n2[x2(l - x2)/(l - x2n)] 2]1'2(3) m.e. {Nn} = { --^dxTT J 0 1 X2and then ob t a in t he a sympto t i c r e l a t i on(4) m.e. {N n} ~ (2/TT ) lg nand the e s t ima te(5) m.e. {N n} ^ ( 2 / T ) lg n + 1 4 / T , n ^ 2 .In case the X's are no t normal ly d i s t r ibu ted (bu t a l l have the samedis t r ibu t ion w i th s tan da rd dev ia t ion 1) one can s t il l p rov e (4) . T h enecessary l imi t ing processes can then be car r ied ou t by us ing thecen t ra l l imi t theorem of the ca lcu lus o f p robabi l i ty and , as one mayexpec t , t he compu ta t i ons w i l l be qu i t e l eng thy . On the o the r handthey wi l l con t r ibu te re la t ive ly l i t t l e to the genera l p ic ture and , what

Presented to the Society, September 10,1942 ; received by the editors July 2,1942.1 J. London Math. Soc. vol. 13 (1938) pp. 288-295. No proofs are given in thispaper, and the present author was unable to find them anywhere in print.31 4

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A RANDOM ALGEBRAIC EQUATION 31 5

i s worse , they may darken i t by technica l i t i es . We sha l l the re fore as sume in what fo l lows tha t the X's a re no rma l ly d i s t r i bu t ed w i thdens i ty g iven by formula (2) .2 . A formula for the number of real roots. L e t f(x) be a con t inuousfun ction in ( oo, oo) ha vi ng a con tin uo us first de riv at iv e ' (x) a n donly a f ini te number of turning points in each f ini te interval .L e t $t{x) be 1 if e

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3 1 6 M. KAC (April

(l/2) f+Uf(x))\f(x)\dx^ - 0 0

i s equal to the number of real roots of f(x). This is a t r ivia l conseque nce of Le m m a 1 . One m igh t no t ice th a t th e l imi t s of in tegr a t ionare really finite since ^e(f(x)) = 0 for sufficiently larg e \x\.REMARK 3. T he choice of e in Re m ar k 2 obvious ly dep end s on t hecoeff ic ients of th e poly no m ial . H ow eve r , we can el im inate e by res ta t in g th e resu l t of R em ark 2 in the form :

+ 0 0 W / ( # ) ) I f'( x) I dx = number of real roots of f(x).- - - 0 0

(6 ) N n = l im ( l /2 c) f W(x)) | '(*) | dx.

4 - o o(1/2*) I - 0 / .

I f we put nowf(x) = Xo + XiX + + X n_ l*" - \

we ge t

+ 0 0-We sha l l need th i s l emma in what fo l lows .L E M M A 2 . Iff(x) is a polynomial of deg ree n 1 , then for every e > 0

+ 0 0 *( ( * ) ) ! ' ( * ) ! dx3n-5.- 0 0

In this case the set E (see proof of L em m a 1) is a su m of a t m os t2n 3 open i nte rv als . Ind ee d, eac h , co nta ins e i th er a real roo t off(x) or a tu rn ing po in t and there a re a t mos t n 1 rea l roo ts and a tm o s t n 2 t u r n i n g p o i n t s .T h e proof of L em m a 2 fol lows now f rom the obvious rem ark th a t

, f'(x) | dx S 2e(nn + 1) , 1 i 2n - 3 ,nw her e w,- is th e n um be r of tu rn in g p oin ts in ,-.

3 . Interchange of two l imit ing processes . We now reduce the compu ta t i on o f m .e . {N n} t o t he compu ta t i on o f m .e . {$e(f(x))\f'(x)\ }by means o f t h i s l emma .L E M M A 3 . + 0 0 m.e. {Mf(x))\f'(x)\}dx.-

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i943] A RANDOM ALGEBRAIC EQUATION 3 1 7Random var iab les can be cons idered as measurab le func t ions def ined on a set 12 w ith a com plete ly add i t iv e Leb esgu e m easu re . (T hem easure of 12 i s 1 .) M ath em at ic a l ex pec ta t io n (m.e . ) i s no th ing bu t aLebesgue i n t eg ra l w i th r e spec t t o t ha t measu re . Bo th N n a n d^((*)) I'(#) | a r e then m easu rab le func t ions on 12 and the y can berep re sen t ed symbo l i ca l l y a s

Nn(p) an d g9(x, M ),x an d e be ing rea l pa ram ete rs . W e a l so have

m.e. {N n} = I N n(n)dfi9m.e. {g t(x t ti)} = I g e(x, ii)dfx,

J fid/JL ind ic a t ing th a t in teg ra t ion i s be ing per formed wi th respec t to theLeb esgu e me asu re in 12. W e f irst not ice th a t

U +00 \ y +00g*(x, p)dx> = I m.e. [g9(x, n)}dx.This fo l lows f rom Fubin i ' s theorem i f one no t ices tha t the in tegra lwi th respec t to dx i s real ly an integr al betw een f inite l imits (depe nding , of course , on e and /x). Fu r th erm or e , b y Le m m a 2 for every e > 0 +0 0 g(x, ix)dx < 3n 5,- 0 0

and hence , by a wel l known theorem f rom Lebesgue ' s theory ,lim I (1/26) J g t(x,n)dx\dp-*> J Q L J -00 J

= I l im (l/2 c) I g t(x, n)dx dp.J H Le-*0 J oo J

Th is wh en com bined w i th (6) and (7) com ple tes the proof of Le m m a 3 .4 . A formula for m . e . {N n}. In o rde r t o com pu te m .e . {^ t(f(x))\f(x)\}we need the fo l lowing lemma.L E M M A 4. If a0} 0, then the density of thejoint distribution of a0X0 + + a n _ iX w _ i a w d / ? o X 0 + + f t_ iX n _ iis equal to

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3 1 8 M. KAC [April( 1 / T T A 1 ' 2 ) exp { - (pu2 - 2yuv + av2)/A}.

This fac t i s wel l known. 3I t also fol lows from well known facts thatm.e. {\l/*X0 | 2 > * * | }

B ut + o o= l i m ( 1 / 2 6 7 T A 1 / 2 ) \ F(u)du = F ^ / T T A 1 / 2 .

-1-00 | v | e x p { - av 2/A}dv = A/a- - 0 0

and f inal lyl im ( l /2e) m.e. {*.(2>,X X i |} = A l'*/a.

I f we now put a 0 = l, ce i= #, , a n _ i = x n _ 1 , /3o = 0, j8i = l , /32 = 2x, , j8n_i = (w l ) x n ~ 2 , w e o b t a i n b y a n e l e m e n t a r y c o m p u t a t i o nx4n _ ^ 2 x 2 ( n + l ) _|_ 2 ( ^ 2 - l)tf2w - n2X2(n~l) + 1A = (x2 - l ) 4

I t is easy to p rove th a t A > 0 for ev e ry r ea l x and by combin ing thecons ide ra t ions o f th i s sec t ion w i th L emma 3 w e ob ta inm.e. {N n}

(8) 1 r + (xAn - n2x2^+l) + 2(n2 - l)x2n - n2x2^~ l) + l ) 1 / 2T J -o c (#2 - 1 )2 (1 + tf2 + X4 + + X2n~2) dx.

3 See for ins tance S . Berns te in , Sur Vextension du thorme limite du calcul des probabilits aux somme s de quantits dpendantes, Math . Ann. vo l . 97 (1927) pp . 2 -59 , inp a r t i c u l a r , c h a p . 3 ( p p . 4 3 - 5 9 ) . T h i s c h a p t e r c o n t a i n s li m i t t h e o r e m s b y m e a n s ofw h i c h o n e c a n h a n d l e t h e c a s e o f n o t n o r m a l l y d i s t r i b u t e d X's. L e m m a 4 i s b u t as i m p l e c o n s e q u e n c e of t h e m a i n r e s u l t s of t h a t c h a p t e r .

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1943) A RANDOM ALGEBRAIC EQUATION 3 1 95. Estimates and the asymptotic formula. By an e l emen ta ry t r ans fo rma t ion we ge t

4 HJ-x))1'2 Jm.e. \Nn\ = I dxy.e . {N n\ = -IT J o 1 X 2w h e r e

nxn~x(\ x2)hn(x)1 - X2I t should be ment ioned tha t the va lue of the in tegrand a t x = l is(n*-l/12)u*. W e h a v e

1 - X2n = (! _ x)(l + x + X2 + . . . + X 2n-1) < 2 w ( l - * )and there fore

hn{x) > xn-l(l + *)/21 ~ hl(x) < [1 - x w - l ( l + * ) / 2 ] l l + * " ^ ( 1 + * ) / 2 ]

< 2 - ^ " K l + *)Us ing t he mean va lue t heo rem we ge t

2 - xn- l(l + x) = (1 - a) [0" - 1 + ( - 1 )0"-2(1 + (9)]< (In - 1 )(1 - a) , s < 0 < 1.

F ina l l y ,(1 - ^ ( x ) ) 1 / 2 / ( l - x2) < (In - l ) 1 / 2 /( l - ^)1 /2 > 0 * < 1.

On the o the r hand2 1 / 2 2 2

(1 - *,( *)) ' / ( i - * ) g 1/1 - * , 0 ^ * < 1,and we can wr i t e

f i ( l - hl(x))m f1-1'" dx r l ( 2 w - l ) 1 / 2I dx < I h I d#J o 1 - * 2 J o 1 - * 2 J i - i / (1 - * ) I / 2= (1 /2 ) lg + (1 /2 ) lg (2 - l / i) + 2(2 - 1/n)1 '* < (1/2) lg n + 3-5 .F ina l l y ,(9) m.e . {# } < (2 /x) lg n + 1 4 / T , | 2 .In o rder to ob ta in a lower es t im ate le t e an d S be a rb i t r a r y pos i t ivenu m ber s l es s t ha n 1 . W e ha ve

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320 M. KAC

dx. / 0 1 ~ X2 JQ 1 ~ X2But for O^x^l-n*-1

hn(x) < nxn~ l ^ n(\ nh~ l)n"1and th i s l as t express ion can be made smal le r than e 1 /2 if n is suffic ient ly large. Hence, for suff ic ient ly large n f

ax > iJo 1 - x* Jo 1 - x2(1 - lg n.2

Using (9) and the fac t tha t e a n d S can be made a rb i t ra r i ly smal l weo b t a i n t h e a s y m p t o t i c f o rm u l a(10) m.e. {N n} ~ ( 2 / T ) l g .L e t us ad d th a t it fol lows f rom (9) th a t th e probab i l i ty th a t (1 ) hasm o r e t h a n 2/ir.

6. Final remarks. I t i s qu i te c lea r tha t the average number o f rea lroots of (1) fa l l ing in the interval (a , b) is give n by-f orm ula (8) if onerep laces and + b y a an d 6, resp ect iv ely. T h e proof of th iss ta tement does no t d i f fe r f rom the one g iven above and one mus t on lyno t i ce t ha t t he p robab i l i t y t ha t e i t he r a or b is a root of (1) is 0.

One can a l so see a lmos t immedia te ly tha t i f ( a , b) does no t conta in e i the r 1 or 1 th e avera ge nu m be r of real roo ts of (1) fa l lingwi th in (a, b) i s 0 (1) . This means , roughly speak ing , tha t mos t o f therea l roo ts o f mos t o f the equa t ions c lus te r a round 1 and 1. T h eproblem of the exac t de te rmina t ion of the average d i s t r ibu t ion of rea lroots of (1) on the real axis wil l , of course , depend on a more del icatet rea tment o f the in tegra l (8 ) . I t would a l so be in te res t ing to knowthe h ighe r momen t s o f N n* The presen t l ine of a t tack would lead tovery compl ica ted in tegra l s , bu t i t may be hoped tha t some o ther approa ch wi ll fu rn ish m ore in form at ion a bo ut the d i s t r ib u t ion of thenumber of rea l roo ts o f equa t ion (1) .

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