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Stochastic FM Models and Non-Linear Time Series AnalysisAuthor(s): D. HuangSource: Advances in Applied Probability, Vol. 29, No. 4 (Dec., 1997), pp. 986-1003Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1427850 .

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Adv.Appl.Prob.29, 986-1003(1997)Printed n N. Ireland) AppliedProbabilityTrust1997STOCHASTIC FM MODELS AND NON-LINEAR TIMESERIES ANALYSIS

D. HUANG,* QueenslandUniversityf TechnologyAbstract

An important model in communications s the stochastic FM signal s,=A cos (twc+ 21k-i mk + 0o), where the messageprocess{m,} s a stochasticprocess.In this paper,we investigate he linear modelsand limitdistributions f FM signals.Firstly,we show that this non-linearmodelin the frequencydomaincan be convertedto an ARMA(2,q + 1) model in the time domainwhen {m,} s a GaussianMA(q)sequence.The spectraldensity of {s,} can then be solved easily for MA messageprocesses.Also, an errorbound is given for an ARMA approximationor moregeneral message processes.Secondly, we show that {s,} is asymptotically trictlystationary f {m,}is a Markovchain satisfyinga certain conditionon its transitionkernel.Also, we find the limit distributionof s, for some message processes{m,}.These resultsshow that a joint methodof probability heory,linearand non-lineartime series analysiscan yield fruitful results.They also have significance or FMmodulationand demodulationn communications.FM SIGNAL IN COMMUNICATIONS;SPECTRAL ANALYSIS; ARMA MODELS; ARMAAPPROXIMATIONAND ERROR BOUND; ERGODIC MARKOV CHAIN; ASYMPTOTICALLYSTRICTLY TATIONARY;LIMITDISTRIBUTION;NON-LINEARTIMESERIESAMS 1991SUBJECTCLASSIFICATION:RIMARY60F05;60G10;60335

0. IntroductionIn recent years, non-linear time series analysis has developed rapidly [1], [12].

Using modern Markov chain theory [13], [14], many types of sufficient and necessaryconditions have been worked out for the existence of ergodic distributions. Theseconditions have been applied to some non-linear time series models to show thatthese models are asymptotically strictly stationary under certain conditions. This is a'probability domain' approach. In contrast, the classical time series analysisconcentrates on the 'time domain' and 'frequency domain'. In the time domain,ARMA models and ARMA estimation methods have been developed. In thefrequency domain, sinusoidal models and the spectra of second-order stationaryprocesses have been investigated. It is well known that the time and frequencydomains are interrelated. New approaches have established the relationship betweentime domain models and probability domain results. For example, thresholdautoregression (TAR) models have been shown to have asymptotically strictstationarity [1].

Received4 October1995;revisionreceived17 June1996.*Postal address:School of Mathematics,QueenslandUniversityof Technology,GPO Box 2434,Brisbane,QLD 4001,Australia.986


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StochasticFM modelsand non-linear imeseriesanalysis

In this paper, combining methods in the three domains, we investigate thefollowingcommunicationsmodel:(0.1) s, = A cos (toc + O,), 0< oc < /r, Ot= Ot-i + pm,, t = 1, 2, 3, *,where A and wc are the amplitudeand carrierfrequencyrespectively,{mj}is themessage process to be transmitted,and f3 is the modulationindex to control thebandwidthof the spectrumof the FM signal.The initialphase 00maybe constantorrandom;we will specify it later in different situations.We call (0.1) a stochasticfrequencymodulation FM) model. This model can be regardedas a non-linear imeseries model since the observation s a non-linear unctionof the processof interest{m,}.The spectral analysisof signalsin communications s very important.As mobileand cellularphone communicationsare developing rapidly,we have to accommod-ate more and more channels in a limited frequency band. This requires us tocalculate the spectraof FM signals accuratelyto guaranteethat the bandwidth neach channel is narrowenough without overlap with others. For example, if themessage processis a pure sinusoidalfunction,the bandwidthof the FM signalscanbe calculatedby Bessel functions. The calculation shows that the bandwidth s afunction of 13,so we can choose the modulation index 3 to satisfy a bandwidthrequirement.This is the method introducedin most textbooks in communications.However, we do not transmit pure sinusoidal messages in real communicationsystems.A moreplausiblemodel of {m,} s that it is a stochasticprocesswithcertainproperties.A thorough studyfor the continuous version of the model (0.1) can befound in [8, ch. 14]. This studyis based on two types of Fourier transforms.Firstly,the autocovariance unction of the signal process {st}can be calculatedusing thecharacteristic unctionthat is the Fouriertransformof the probabilitydistributionofa linear combination of the m,. When the message process {mj}is a Gaussianstationaryprocess, their linear combination s still a Gaussian random variable.Sothe autocovariance unction of {s,}can be representedwith Gaussiancharacteristicfunctions. It turns out that under certain conditions this autocovariance unctiondoes not dependon time t, so {s,}is second-order tationary.Secondly,the spectrumof {s,} can be obtained through the use of the inverse Fourier transform ofautocovariance unctions.

However, the resultantspectrum s describedby an integration.Althoughsomelimitingforms are given in [8] to simplifythis result under some specialconditions,we do not know any error boundbetween the limitingform and the true spectrum.Another spectral study for stochastic FM models is describedin Sections 4.12 and4.13 in [11]. This result is simple-the power spectral density of the FM signalprocess 'is determined by, and has the same form as, the (probability) densityfunction'of the message process.However, the authors of [11] admitted that theirapproach s only 'heuristic'and 'we shall not be able to deduce the spectrumwiththe (same) precision' as when the message process is a deterministicsinusoidal

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function(p. 159in [11]).We shallshow that the second resultmayhave a largeerrorin some cases.Our contribution n this problemis in developing parametricARMA models forsome FM signalsand ARMA approximationsor the spectraof generalFM signals.Firstly, or any MA (q) Gaussianprocess{m,},we show that the signal process{s,}isan exact ARMA (2, q + 1) process. Secondly, we introduce a method that canprovidean errorbound for an ARMA approximationor a generalGaussianprocess{m,}in the spectrum.Examplesshow that this method worksmuchbetter than themethodin [8] even for the case when{mr}s not a MA process.Thus,thenon-linearmodel (0.1) in thefrequencydomain can be converted o or approximated y a linearARMA model in the time domain.These resultscan also be appliedfor FM signal

demodulation, .e. recovering{m,}from noisy observations.Manyexistingmethodshave been developed for this purpose. However, to the author'sknowledge, noexactly optimalresults are available for this non-linear ilteringproblem. Using theARMA representationof FM signals,we can establish a state space model. Let {x,}be the Markovian xtension(see [2]) of the ARMA representationor the FMsignaland{Yt= s, + vj} be the noisy observationsequence.Then(0.2) xt = Gxt,- + u,, yt= [1,0, *- *, 0]xt+ vt.So, the best lineartrackingalgorithm or {s,}can be developed by the Kalman ilter.We will discussthis problemin full detail in a subsequentpaper.In this paper we also study the distributionof {s,} which provies a focus onnon-lineartime series analysis.Firstly,assume that {m,}is a Markovchainprocess,whichmaynot be Gaussian.We consider a vector-valuedMarkovchain(0.3) X, = [ ,,m, t, two+t,(mod 2r), t = 1,2,3, * .Using the classicalresultfor ergodicMarkovchains[3], we show that under certainconditionson the transitionkernelof {m,}, {Xt} s Harrisergodic [13], [14].Then{st}will be asymptoticallytrictlystationary.Usuallyit is difficult o find a closedform ofthe limit distributionor an ergodicMarkovchain.Fortunately, or a verylargeclassof stationaryprocesses{m,},we have

arcsin xIA) 1(0.4) lim Pr{st, x} = +- Ixl A.t--- J~ 2Thisresult shows theprobabilistic haracteristicf the non-linearmodel (0.1). It alsohas significance n FM demodulation.Although the Kalmanfilter can be used forFM demodulationbased on the model (0.2), it is efficientonly when the state andobservationvariablesin that model are Gaussian.The distribution n (0.4) showsthat this is not true. To obtain a more efficient estimator,we have to considerprobabilisticapproaches,such as Bayesian and Gaussiansum methods [10]. Thelimit distribution 0.4) providesa basis for thisapproach.We will continue the studyin this direction n the future.

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Stochastic FM models and non-linear time series analysis

Then, sinceE exp (i3 2 mj) =exp [-132(k)],

(1.3) holds. Finally,for any 8 > e > 0, when k is largeenough,it follows from (1.5)that +(k) - (8 - E)k.So Rk tends to zero geometrically.Thus, (1.4) is true.Remark1.2. All FM signalswith 'low-pass(or all-pass)modulation' 4] satisfythecondition (1.2). However, many FM signalswith 'high-passor band-passmodula-tion' also satisfy(1.2). Althoughtheoretically he transfer unctionof idealhigh-passor band-passfilters should be zero at frequencyzero, most real digital filtersarenot. For example, any {m,}generated by AR (all-pole) models satisfies (1.2), nomatterin which frequencyband the peaks of the AR spectraldensityare located.Also, if (1.2) is not true but 0ois uniformlydistributedon [a - 7r,a + 7r],then bothE[exp (iOo)]andE[exp (i20@)]anish. So {s,}will be exactlysecond-order tationaryand (1.3) holds.When{m,} s an MA (q) sequence,we have that rn= 0 for all Inl> q. So it followsfrom (1.5) that when k > q, o(k) is a linear function of k, i.e.

(1.6) (k) =k rn- Inlrn, forall k - q.n=-q 2n=-qThis gives us the clue to simplifying 1.4) for MA messageprocesses.For an MAprocess{m,},we denote {rn}, +(k)} and {Rk} as above andput, for all k # 0,(1.7) m, = f3kE-k, 3o= 1, E{?2} = 2, E{e,+k} =O,k-O(1.8) 8 = (t) a=e-^2 k=O /

2 2(1.9) ao = 1, a, = -2a cos wo, a2 = a2, y, = akajR,+kk=Oj=O

Theorem1.3. Assume that {m,} is a GaussianMA(q) process definedby (1.7)-(1.9).(i) If ,=oPkO00, for any initial phase 0o that is constant or random butindependent f {m,, {s,}is asymptotically purelynon-deterministicRMA(2, q + 1)process satisfying

(1.10) s,- 2a cosocs,t- + a2s,_t2 = bkt-k,k=0where{u,}is a whitenoise sequencewithvarianceone, {bk}beingtheuniquesolutionof theequation

q+l-1 l1(1.11) 2 bkbk+j = , j=0,1, 2,--, q +1, bkzk forall iz < 1.k-0 k=O

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Thespectraldensity unctionof {s,}is

(1.12) 2 7teiu q bkeufs=kt -q-1 k=Of(A) =2ir 11 2a cos wceiA + a2ei2AI2 2nr11 2a cos wce'i + a2ei2A12(ii) If the sumof the1k equalszeroand the initialphase 00is uniformlydistributedon [a - n, a + Ir], with a a real number,{s,} has a mixed spectrum.The purelynon-deterministicomponentof {s,}is an MA (q - 1) processwhile the deterministiccomponents the sinusoidal unctionwith carrierrequency w.Proof. When{m,} s an MA (q) process,(1.6) is trueand

1 t ( 1f) 2(3k) = 8.2n=-q k=-O

So, let b = 2 q Inl n;we have c(k)= 8k - b. Then, according o Remark1.2,under eithercondition n (i) or (ii) we haveRK= ?A2exp [p2(b - 8k)] cos kwo= ?A2exp (bp2)ak cos kwc, k - q.

Since cos kwoc 2 cos oc cos (k - l)oc + cos (k - 2)oc = 0, we have

Rk- 2a cosocRk-l + a2Rk-2 = 0, for all k _ q + 2.Thus,{s,}must be an ARMA (2, q + 1) processand (1.11) has a uniquesolution{bk} satisfying 2?,ty- = ' bkeiA2, (see, for example, Theorems 3.9 and3.10 in [2]).Finally,when the sum of the fk does not equalzero, we have that a < 1, so theroots of the polynomial z2 - 2a cos wcz+ a2 are inside the unit circle; otherwise theroots are on the unit circle.Then the resultsin Theorem 1.3 follow directlyfromTheorem3.9 in [2].UsingTheorem1.3, we can constructexamplesto show that the statement n [11]maynot be correct.In fact,we canfinddifferentGaussianMA processessharing hesame mean and variance but having a different spectral density, for example,{m, = E,+ 0.9E,-_}and {mt = E,- 0.9E,-1}. According to Section 4.12 and 4.13 in [11],

the two correspondingspectral density functions should have the same shape.However, using (1.8) we find differentparameters8 and a for them. So they aredifferentARMA(2,2) processes.We plot them in Figure1 to show the difference.

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10-5-O A1 1I. 1 1-1 -0.5 0 0.5 1

Figure 1. FM spectral density functions. Two MA(1) models, m, = e, + 0.9e,-_ and m, = e, - 0.9e,_1, areused for modulating processes, where {e,} is a Gaussian white process with mean zero and variance one.Both models have the same mean variance. Using (1.16) we can find the corresponding spectral densityfunction for {s,}. They are totally different. The x-axis in the graph is for frequency with unit n.Wc= 0.25x, p = 0.1, a = 10 and PI = 0.9 (top) and -0-9 (bottom)

Now we consider the error bound of ARMA approximationsor general FMsignals.We denote +(k), {R},,ak, a and{yj}as above and let2

c, = ajR,_j = R, - 2a cos wocRt_+ a2Rt-2,j-o

(1.13) P(A) = yte' + 2cq+icos (q + 1)At=-q+ 2Cq+2[cos (q + 2)A + a1 cos (q + 1)A],

C = 1 (1 - a2)2sin2w+ 4na2[max (o, 1 + a2 cos C )1]22 lal o-Theorem 1.4. Assume that the messagesequence{m,} is a Gaussianstationary

sequence and that (1.2) holds. Then the spectral densityfunction f,(A) of {st}satisfies(1.14) f(A) -P(A)/(2r l akei 2)c C i/(ir akeA ) C Ic,k-O t=:q+3 k-O t=q+3

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Further, if the partial sum of rkconverges, we have

Ic,I A2p2e-(') 2 Ern max 1, exp (_2 rn(1.15) n+ r,_-+ 2 rn max 1,exp(-32(r,l + 2 r

Proof Under condition(1.2) it followsfromLemma 1.1 thatxc 2 2

R,etA 2 akeikA(1.16) f(A)k=O + P(A) 2.6 fsW ^2 2 2 2+'2r X ake k 2r akekA0k=O k=OwhereA is the conjugateof A and

A =-[ [eitA - cq+lei(q+l - (1 + ale -A)C+2e +2)A 2 I akeFrom(1.9) and (1.13),we have

oc cx 2 2 cyt-te = akt+ ke = ak c,e i(-k)At=q+l t=q+l k-O k-O t=q+l+k

2= 2 ake-*A cte"i + (1 + ale-i)cq+2ei(q+2)A + c+le(q+l)A.k=O t=q+3

Thus,|A|= /( a \)Ce itA 2It akeikAt=q+3 ( k-O

and the firstinequality n (1.14) follows from(1.16).Now, we evaluate the denominator n A. Manipulating rigonometricoperations,we have11 2a cos w,ceiA a2e2AI2= 11 aei(A+)J2)l11aei-(A-w)12

22A2[ \\1+a2 12=(1- a2)2sin2wo+ 4a2[ os A 2 c ].Note hatcsThe secondinequality n (1.14) is trueNote that IcosAl -1. The secondinequality n (1.14) is true.


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Finally, we evaluate the c,. Since cos tow= 2 cos wC os (t - 1)o, - cos (t - 2)0,, itfollowsfrom(1.13)and (1.3) thatc, = ?A2e-O(')2[costwc- 2 cos ,ccos (t - 1)wcet[(t)-(t'-1)-_8]02

+ Cos (t - 2)woce[4()0-(t-2)-2]2]= ?A2e-*(')){2 cos wt cos (t - l)wc[l - e[1(')-('-1)-18]2]

- cos (t - 2)0c[1 - e[O(')-O('-2)-26]2]}.Note that r, = r_,. Since{r,} is summable, t follows from(1.5) that

1~ 110 1 t-1 1 t-2o(t) - C(t- 1)-- , r. - (t-lnl)rn - (t - 1 Inl)rn2n 2 nlt n=2-t

1- E rn = rnn=-oo n=t

1 t-1 1 t-32)- rn (t -nl)rn - (t-2- nl)rnn= -cc 2 n=l-t 2 n=3-t- nn=-r,-,l-2 rn-

n=-ox n=tThen,usingthe inequality

11 eXlI e'dt _ lxlmax 1,e),we have that (1.15)holds.

Example.We now apply the results in Theorem 1.4 to the Gaussian Markovprocess(1.17) m, = pmt-1+ , pl< 1,where {E,} is a Gaussian i.i.d. withzero mean and variance a2 and {e,} is independentof 00. It is easy to check that the spectraldensityfunction

0.2fmr(A) = 2r 11 peiA12

is continuousat A= 0 and 8 = fm(0)> 0. Further,we have(1.18) rk =1- p2 P, a = exp (2r( _2)) (k)=8 k- p )Let Co = exp (28pf32/(1 - p2)) and hk = exp (-28pp2pk/(1 - p2)); then

e-(k)2 = Cohkak, Rk = A2Cohk cos kwcak/2.Assume thatp > 0. Then0 < hk< 1 and we have that IRkl A2Coak/2.

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So, when we use a truncatedFourier series ]'k=-t RkeikA/(2;r) o approximatehespectraldensityfunction s(A)in (1.4), a tighterrorboundisRkl/r C-A2Coa'+1/[2r(1l- a)].k=t+l

Then,given a toleranceA, the smallestnumbert satisfyingA2Coat+'/[2l(1 - a)]


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(a) Fourierwith45 terms, ErrorBound:2.824

W-, I I

MA& fv-N i

-1 -0.5 0 0.5 1Fourierwith448 terms, ErrorBound: 9.581e-006 ARMA(2,43),ErrorBound:8.579e-006

(b) Fourierwith45 terms, ErrorBound: 241.986-I I42-

-2-1 -0.5 0 0.5 1

1Fourierwith 100 terms, ErrorBound: 225.81510'

5

-5-1 -0.5 0 0.5 1Fourierwith 13648 terms, ErrorBound: 9.977e-006 ARMA(2,48),ErrorBound: 9.927e-006150

100-

50-

n

15(

10(

5{

0 I

0n-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1

Figure2. FM spectraldensityfunctionapproximation y truncatedFourierseries and ARMA(2,q + 2)processfor an AR (1) modulatingprocess.(a) B= 0.05;(b) 3= 0.01

420

R I

-7

Fourier with 100 terms, ErrorBound: 0.5062v

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K(Y IV) such that K(y Iv) =0 for all y and v outside a boundedinterval[a,b]. Wedenote I1= [0, 2,rIX [a, b], I2 = I x II, andK2(x,yIu,v)= E K(y1x-u-2w,-y+2k7r)

(2.1) k=-oXK(x-u-2co -y+2krjv), (x,Iy, u, v) E 12.(2.2) f(2n(X,Y Iu, V) = (x, Y I t)&-2(S, t u, v) dsdt, n >1.

nIf there is an integer n such that k2n(x, y Iu, v) is positive almost surely in 112and iscontinuouson fl2, then the process{X,}definedby (0.3) is a Harrisergodic processand the corresponding FM signal s, has a limit distribution.

Proof. First, to define the transform in modulo 2ir uniquely, we define

qi(x) = C (x -2kr)I{2kirwx


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StochasticFMmodelsand non-linear ime seriesanalysis

A class of message processes satisfying the conditions in Theorem 2.1 is given inthe following result.Corollary 2.2. Let {mj} be a Markov chain with a continuous transition densitykernelK(y Iv) on R2and that K(yI v)>O for (y, v)e [a, b] x [a, b], -00


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a neighbourhood of (x + 2k7r,y) with positive probability. Note that 0-, q(tw, +8,). We can conclude that the kernel R21(x, y Iu, v) >0 for all (x, y, u, v) e H2.Thus,Theorem2.1 is validin our case and this corollaryholds.

Next, we develop throughthe followinglemma some probabilitydomain resultswithout the assumption hat {m,} s a Markovchain.Lemma 2.3. Let {8,, t = 1, 2, 3, } be a set of randomvariableswithprobability

density functions {f,(u), t = 1, 2, 3, . respectively.Assume the following.(i) There exists an integer n for which all functions f,(u) are n-piecewisemonotonic, i.e. for eachfuinction,(u) we canfind a partition-00 = ao(t)< a,(t)


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Thus,since (arcsinx)' = 1/Vl -x2,F(x =x 1 2 [f(2kr + arcsinx

- v)V X2 J-o k=-oo(2.7) + f (2kir lr- arcsinx - v)] dC,(v).Further, or each given integerk and real numberv, we have

sup ft(u - v) ft(2k4r+ arcsinx - v) > inf ft(u - v),12k*-u sr/2 12kxr-ul inf f(u - v).I2kx-ujI|r/2 7t J2kf-rn/2 12krn-ulinr/2

So, 1 r2kn+yr/2f (2kr+ arcsinx - v) - - f,(u- v)duf J2kn-n 2sup f,(u - v)- inf ft(u - v).12kn-u lcn/2 12kn-ulr1l2

Similarly, 1 r2k,r-r+n/2ft(2kr - r- arcsinx) - f(u - v) dut 2kn-rn-r/2C sup f(u - v) - inf ft(u - v).12kx-nx-uj-12 12k- n--u\in/2

Thus,since for any given real numberv,w rkx + x/2 1, p ^f,(u v) duv)du - d=1,k=- kx-x/2 -

oc2 [f(2ktr+ arcsinx- v) + f(2kr - r- arcsinx- v)]k=--

_ 1 rknc+rk/2- - - f,(u -v)duk=-at Jklrx- x/2= | [f(2kir + arcsinx - v) +ft(2kir- - arcsinx- v)]k=-oo cr

i [ sup ft(u- v) - inf ft(u- v)].k=-oo LIkx-u/Sr/2 Ikn-ulin/2However,ft(u) is monotonicon each interval[aj_l(t),aj(t)]. Let ft(?o) = 0. We

haveEo [ sup f(u)- inf 122 If(aj- (t))- f(aj(t))I.k=-c LIkx-ul|/2 IkY-uj S/2J j=l

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Note that fJo, dC,(v)= 1. It follows from (2.8) and above that

SE [f,(2kr + arcsin x - v) + f(2kir - r - arcsin x - v)] dC,(v) - -|-ok=-=| 2~ [f,(2kir + arcsinx - v) + f,(2kr - - arcsinx - v)]- dC,(v).c1 =-o2 If,(a_(t)) -f,(a(t))l dCt(v)cc j=l

n=2 E If,(aj-,(t))-f(aj(t)).j=lUndercondition(ii) we have that, as t-- oouniformly or all x,

n2 If(aj_,(t))-f,(aj(t)) -nx sup{ft(u);uE(- ',c)}IF;(x)IrV'I -x2= as t - oo.V'I x2Then, (2.6a) followsfromF(y)- (arcsiny+ 1 [F(x) -

1x2

c 2n x sup {f,(u); u E (-oo, oo)}1J- V' -x2 X=2n(arcsin y + i)) sup {ft(u); u E (- 00,c)}-. 0 as t- oo.

The same arguments an be used to prove (2.6b).Theorem2.4. For the FM signalprocess definedby (0.1), if the messageprocess{m} is a Gaussian tationary rocessandsatisfies he condition 1.2), then(0.4) holds.Proof. Under these conditions, 0, is Gaussian with zero mean and variance

2,324(t). So, all density functions f,(u) of the 0, are increasing on (- o, 0) and aredecreasing on (0, oo). Also, since under condition (1.2) cf(t)-- oo as t- oo (see theproofof Proposition1.1), then

sup {f(u); -oo < < oo}= 1/(2V/r(t)B3)--0 as t oo.Thus,all conditions n Lemma2.3 hold and (0.4) followsfrom(2.6b)with w,= W.

In fact, for Gaussianr.v.'s with zero mean and variancesgreaterthan two, theformula(2.4) is very accurateas shown in Table 1. The values in the table arecalculatedwithin tolerance 0.00001. In the last column, F(x)= arcsinx/rx+0.5 is

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StochasticFMmodelsandnon-linear imeseriesanalysisTABLE.Cumulative distribution of SIN NORMAL. F,(x)=

2k- _=-Pr{2kir- Xr arcsin x _ , -- 2k,r + arcsinx}, 0, - N(0, t2)x F1(x) F2(x) F(x) F(x)

-0.9 0.1098 0.1435 0.1436 0.1436-0.8 0.1635 0.2047 0.2048 0.2048-0.7 0.2101 0.2531 0.2532 0.2532-0.6 0.2538 0.2951 0.2952 0.2952-0.5 0.2960 0.3332 0.3333 0.3333-0.4 0.3374 0.3689 0.3690 0.3690-0.3 0.3783 0.4030 0.4030 0.4030-0.2 0.4190 0.4359 0.4359 0.4359-0.1 0.4595 0.4681 0.4681 0.4681

0 0.5 0.5 0.5 0.5

the limitingdistribution.One can confirm hat, for standarddeviationgreaterthantwo, F,(x) and F(x) are very close.Acknowledgments

The author is gratefulto the editor and referee for their valuable comments andsuggestions.References

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