LEVEL CROSSINGS OF A RANDOM POLYNOMIAL WITH HYPERBOLIC ELEMENTS K. FARAHMAND. (Communicated by Richard T. Durrett)

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1 proceedigs of the america mathematical society Volume 123, Number 6, Jue 1995 LEVEL CROSSINGS OF A RANDOM POLYNOMIAL WITH HYPERBOLIC ELEMENTS K. FARAHMAND (Commuicated by Richard T. Durrett) Abstract. This paper provides a asymptotic estimate for the expected umber of AMevel crossigs of a radom hyperbolic polyomial gi cosh x + g2 cosh 2x + + g cosh x, where g (j = 1, 2,..., ) are idepedet ormally distributed radom variables with mea zero, variace oe ad K is ay costat idepedet of x. It is show that the result for K = 0 remais valid as log as K = K = 0{s/). 1. Itroductio Let (SI, A, Pr) be a fixed probability space ad let {gj(co)}"=l be a sequece of idepedet idetically distributed radom variables defied o 2. Although there has bee cosiderable attetio give to algebraic ad trigoometric polyomials with coefficiets g/s, very little is kow about the behaviour of the radom hyperbolic polyomial, (LI) P(x) = P(x, co) = J2 gj(co) cosh jx. 7=1 Deote by NK(a, ß) the umber of real roots of the equatio P(x) = K i the iterval (a, ß) ad by EN ((a, ß) its expected value. The oly literature that this author could fid cocerig EN is a report by Bharucha-Reid ad Sambadham [1, p. 110] o a upublished result of Das [4], where it is stated that for K 0 ad idepedet ormally distributed coefficiets with mea zero ad variace oe ENQ(-oo, oo) is asymptotic to (l/^)log«. This is iterestig as it shows that ENo for radom hyperbolic polyomials does ot correspod with that of the radom algebraic polyomial (1.2) F(x) = F (x, co) = J2gj(co)xj, j=i Received by the editors October 6, Mathematics Subject Classificatio. Primary 60H99; Secodary 42BXX. Key words ad phrases. Gaussia process, umber of real roots, Kac-Rice formula, algebraic polyomials, trigoometric polyomials, fixed probability space America Mathematical Society /95 $1.00+ $.25 per page

2 1888 K. FARAHMAND or with that of the radom trigoometric polyomial (1.3) T(x) = T (x,co) = ^2 gj(œ) C0*JX i=x From Kac [7] or Wilkis [10] we kow that for the algebraic polyomial (1.2), ENq(-oo, oo) ~ (2/^)log«is twice that of the hyperbolic case reported by Bharucha-Reid ad Sambadham [1], while for the trigoometric case (1.3), ENQ(0, 2) ~ (2/v/3) (see Das [3] ad Wilkis [9]). Therefore it is of special iterest to establish for the hyperbolic case which of the kow patters, if ay, ENK, for K ^ 0, will follow. Oe ca expect that, because of the similarity of order of EN0, the AMevel crossig would be similar to that of the algebraic case. I Farahmad [6] it is show that ENK for the equatio F(x) = K is asymptotically reduced to (l/7r)log(/x2) i the iterval (-1, 1) while its remais the same as K = 0 i the iterval (-00, 1) U (1, oo) as log as K = K = 0(y/h~). For the trigoometric equatio T(x) = K, however, Farahmad [5] shows ENK(0, 2) remais asymptotic to (2/V3). Our result here uexpectedly shows that the AMevel crossig of the hyperbolic polyomial is similar to that of the trigoometric oe. If oe classifies the oscillatio of differet types of polyomials accordig to the behaviour of their real zeros, viz. the algebraic types with ENo = 0(log«) ad the trigoometric types with ENo O(), it seems iterestig to ote that although radom hyperbolic polyomials will fall ito the first category, their properties of ÄMevel crossigs follow the secod. We prove the followig: Theorem. For ay sequece of costats K=K suchthat {A^2/(«log)} teds to zero as teds to ifiity the mathematical expectatio of the umber of real roots of the equatio P(x) = K satisfies ENK(-oo, oo) ~ (l/7t)log«. Let 2. A FORMULA FOR THE EXPECTED NUMBER OF CROSSINGS <D(0 = (27t)~1/2 / J OO exp(-y2/2) / ad <p(t) = d<t>(t)/dt = (27T)-1/2 exp(-i2/2) ; the by usig the expected umber of level crossigs give by Cramer ad Leadbetter [2, p. 285] for our equatio P(x) - K = 0 we ca obtai (2 EN(a, ß) = jß (B/A)(l - C2/A2B2)x/2y>(-K/A).[2<p() + {2<S>()-l)]dx,

3 where LEVEL CROSSINGS OF A RANDOM POLYNOMIAL 1889 (2.2) A2 = var{p(x) - K} = ^ cosh2 jx, 7=1 (2.3) B2 = var{p'(x)} = j2 sih2 ;x, 7=1 (2.4) C = cov[{p(x) - /C"}, P'(x)] = ^ ;'sih/x cosh/x, ad?/ = -C7:/^(^2y52-C2)1/2. Let A2 = A2B2 - C2 ad erf(x) = /* exp(-i2) dt ; the from (2.1) we ca write the extesio of a formula obtaied by Kac [7] ad Rice [8] for the case of K = 0 as (2.5) EN(a,ß) = Ii(a,ß) + I2(a,ß), where rß (2.6) h(a,ß)= (A/A2)exp(-B2K2/2A2)dx Ja ad rß (2.7) I2(a,ß)= (V2/7i)KCA-3exp(-K2/2A2)eTf(KC/AA\/2)dx. Ja We remark that although we are iterested i x (-oo, oo) it is sufficiet to restrict our attetio to the umber of real roots for positive x oly; sice to each root of P(x, co) = K i (0, oo) there correspods a root of P(x, co) K i (-oo, 0), ad coversely. Therefore EN(-oo, oo) = 2EN(0, oo). From (2.2)-(2.4) we obtai the followig relatios: (2.8) A2 = (2-1 )/4 + sih(2«+ 1 )x/4 sih x, B2 = -(+ l)(2 + l)/l2 + (2+ l)2sih(2 + l)x/16sihx (2.9) - (2 + 1 ) cosh(2«+ 1 )x cosh x/8 sih2 x + (2 cosh2 x/ sih2 x - 1 ) sih(2«+ 1 )x/16 sih x, (2.10) ad therefore C = (2 + l)cosh(2«+ l)x/8sihx - sih(2«+ 1 )x cosh x/8 sih2 x, A2 = sih2(2«+ l)x/64sih4x + (2-l) sih(2«+ 1 )x( 1 + cosh2 x)/64 sih3 x (2.11) -(42- l)cosh(2+ l)x cosh x/32 sih2 x + (2«+ 1 )(8« ) sih(2 + 1 )x/192 sih x - (2«+ l)2/64sih2x - ( + l)(42 - l)/48.

4 1890 K. FARAHMAND 3. Proof of the theorem First we let x be the iterval ((log«)1/2//?, 1). As it turs out this iterval will make the mai cotributio to the umber of real roots. To fid the domiat terms i (2.2)-(2.4) we observe that i this iterval cothx < e/x < «(log«) -1/2 ad therefore the derivative of f,p(x) = (sihx)(sihx)-'' p = 1, 2, 3, ad for sufficietly large. Hece f,p(x) > sih{(log)1/2}[sih{(log«)1/2/«}]^ > sih{(log)1/2}[(/4)(log/i)-'/2f. is positive for Use has bee made of the fact that sihx < 4x i (0,1). Sice (log«)1/2/«is a decreasig fuctio of «, (3.1) will remai valid for «replaced by 2«+ 1. Hece for all «> 49 ad p = 1, 2, 3 from (3.1) we ca obtai (3.2) f2+i,p(x) > («p/3.254)(log«)-^2exp{2(log«)1/2}. Now from (2.8) (2.11) ad (3.2), for all sufficietly large «, ad sice sihx > x/4 i (0, 1) we ca show (3.3) A2 = [{sih(2«+ l)x}/4sihx]{l + 0(log «)"'}, (3.4) B2 = [(2«+ l)2{sih(2«+ l)x}/16sihx]{l + 0(log)~xl2}, (3.5) C = [(2«+ l){cosh(2«+ l)x}/8sihx]{l + 0(log«)_1/2}, ad (3.6) A2B2 - C2 = [{sih(2«+ l)x}/8sih2x]2{l + 0(log)~{}. I the followig for «sufficietly large we evaluate 7i((log«)'/2/«, ed from (2.6) ad (3.3)-(3.6) we have Ix((log)x'2l,l) = (2)-l[l + 0(log«)-'] 1). To this / (cschx)exp{-2#2(2«+l)2sih3x/ J(log)'/2/ (3.7) sih(2«+ l)x[l + 0(log«rI/2]} x = (27r)-'[l + 0(log«)-'] / (cschx)<7x J{log)ll2/ K2(2 + l)2 f {sih2x/sih(2«+ l)x}dx J(\ogyi2l /(log)'/2/i The first term appearig o the right-had side of (3.7) ca be evaluated as (27T)-'[1 + 0(log«)-1]{log(tah 1/2) - log[tah{(log«)1/2/2«}]} = (2yx[l + 0(log«)-']{0(l) + ^{«(log«)-'/2} -log[l + 0(«-2log«)]} = (2^)-1{log/i-(l/2)log(logM) + 0(l)}.

5 LEVEL CROSSINGS OF A RANDOM POLYNOMIAL 1891 Now we show that the secod term appearig o the right-had side of (3.7) is small compared with the value obtaied i (3.8). To this ed we write this term as (3.9) O K2(2 + l)2 f 7(log)'/2/(«+l/2) = 0 K2(2 + l) r2+\ = 0[K2(2 + l)~1]. /2(log) /2 {x2csch(2«+ l)x}dx csch udu Therefore, sice K =»(«log«)1/2, from (3.7) (3.9) we have (27t)-1 log as the asymptotic value for 7i((log«)'/2/«, 1). Now we show 7i(0, (log«)1/2/«) is small compared with this asymptotic value. For this rage of x the domiat term for A2, B2 ad C caot be foud. However, sice from (2.8) A2 > {sih(2«+ l)x}/4sihx ad sice u coth u is a icreasig fuctio of u we ca have (B2/A2)< «(«+ l) + (l/2)cothx{cothx-(2«+ l)coth(2«+ l)x} (3.10) - {«(«+ 1)(2«+ l)sihx}/{3sih(2«+ l)x} <«(«+ l)<(«+l/2)2. Therefore for x > 0 ad for all «> 2 from (3.10) we ca obtai (3.11) (A/A2)<(B/A)< + l/2. Therefore from (2.6), (3.11) ad for all sufficietly large «we ca obtai (3.12) h(0, (log«)'/2/«) = O((log«)'/2). To estimate a upper limit for 72 we ote that sice C = (l/2)d(a2)/dx (2.7) we ca write (3.13) / OO 72(0, oo) < (27T)"1/2 / \KC\A-3dx Jo /»OO = \K\(2)-xl2 H JV = 0(K/l/2).,-1/2 / A-2 A- da Now it oly remais to cosider the case of x > 1 for 7i (2.11) ad for sufficietly large «we have A2 < sih2(2«+ ad A2 > sih(2«+ l)x/4sihx. Therefore for all positive x, (3.14) (A/^2)<cschx. Hece from (2.6) ad (3.14) we obtai l)x/16sih4x from From (2.8) ad (3.15) /oo /OO (A/A2) dx cschxdx = (Ä)'log{coth(l/2)}.

6 1892 K. FARAHMAND Fially from (3.8), (3.9), (3.12), (3.13), ad (3.15) we have proof of the theorem. 4. Remark By lookig at the proof it is apparet that although i the iterval of (-1, 1) the hyperbolic polyomial has asymptotically as may roots as the algebraic polyomial, outside this iterval, ulike the algebraic case, the hyperbolic polyomial does ot possess ay sizeable roots. Perhaps this is caused by (expoetially) fast icreases (decreases) of the terms i (-oo, -1) u (1, oo) which makes the cacellatio i this type of polyomial difficult. Ackowledgmet The author wishes to thak the referee for his detailed commets which improved the earlier versio of this paper. It is a pleasure to thak Professor Sambadham for discussio o the earlier result of this paper. Refereces 1. A. T. Bharucha-Reid ad M. Sambadham, Radom polyomials, Academic Press, New York, H. Cramer ad M. R. Leadbetter, Statioary ad related stochastic processes, Wiley, New York, M. Das, The average umber of real zeros of a radom trigoometric polyomial, Proc. Cambridge Philos. Soc. 64 (1968), _, O the real zeros of a radom polyomial with hyperbolic elemets, Ph.D. dissertatio, upublished. 5. K. Farahmad, Level crossigs of a radom trigoometric polyomial, Proc. Amer. Math. Soc. Ill (1991), _, Real zeros of radom algebraic polyomial, Proc. Amer. Math. Soc. 113 (1991), M. Kac, O the average umber of real roots of a radom algebraic equatio, Bull. Amer. Math. Soc. 49 (1943), S. O. Rice, Mathematical theory of radom oise, Bell. System Tech. J. 25 (1945), J. E. Wilkis, Jr., A asymptotic expasio for the expected umber of real zeros of a radom polyomial, Proc. Amer. Math. Soc. 103 (1988), _, Mea umber of real zeros of a radom trigoometric polyomial, Proc. Amer. Math. Soc. Ill (1991), Departmet of Mathematics, Uiversity of Ulster, Jordastow, Co. Atrim BT37 0QB, Uited Kigdom