TRU Mathematics 12−2〔1976〕 t
ON THE NUMBER OF RANDCM SI…TS
G正畑RATED BY A RANDCM FIELD
Kbsei IWASE 〔Received December 25, 1976〕 INrRO皿ICrI㎝.1£t Z伝、 y)be a rea1−valued homogeneous(遁ussian層rand㎝ field defined fbr(x, y)in a two−dimens iona1正泊clidean space R2. We shall want to characterize and relate the mean values of certain geometric measures of roughness of realisations of Z rx3 y). For exa㎎Ple, problems similar to ones of the crossings of level arise naturally in studying of random fields. Hbwever general situation here is considerably.more complex than in the case of the crossings of a leve1. Let us assume that Z rx. y/ satisfy the fbllowing regUlarity conditions. the sa珂ple f㎞ctions of Z(x3 y) have continuous partial derivatives of up to second order, with finite variance, in D・And Z rx, y) has fillite spectral moments of up to and including order 6, and has a spectram with a continuous component. We shall call a two−dirnensional homogeneous Gaussian random field Z(x、 y), (x3 y)∈D=[O」?]x[O, T],suitab Zy?eguZar. If Z(x。 y) is a suitably regular random field, it may be defined 8ω={(x。y月z (x. yノ逗.(x。y)・D}. Lω={rx, y)lzrx.〃已. rx. y)∈D} fbr any fixed leve1 α. Then, in this paper, 1et us consider the fbllowing㎜bers;
v「(a) andλ.ra)=the number of comected comPonents of S(a) and L(αJ respectively, vωandλω ・the number of comected colrponents of 5ωand L ra) respectively W血ich do not joint the bound of D. In the following section we shall show the expectati㎝of loweT boymds of v(a) and λrとx) and of哨)per bounds of v rαノ and λ ra). 1.EXPECTATION・OF LOn]R AND UPPEy DS OF v(a) ANDλ ra).Under the regular conditions of Z(x診y), we may define follow血g n血bers;・㌔。誕・㎜…fp・血・・i…ati・f声g・r剣属Z。r・・内
Z“⑭〉⑳dZ㏄⑭〉°}・ 45、 fie ta, (2) E[Arq。μ刀=一■2
(・) 聯,川∴ζ
, 〔4) 、 E[C(a3uJ}= 一夕2 ・ −T2. 碗θ㌘θ了r∂.w。 s。 t) isZnv「¢・yル ..
∼ 46 K. IWASE 、: ’ ・チーf・.・ノー{・輌・.・fp。int・血D・ati・fying・Zrx・yノ・a・Z。 (x・y)一・u・ Z許・yノ〉°and Zdr(x・〃2<°}S 」.一□1て.』1v㌔。・戸伍迦厩・fpi・9・m・・sat助i皿9多働ノPt・Z許・ピ・
Z“⑭〈°and Zue(x.y)’〉°}・ ’
v−一(a3 uJ={nu血ber of points in 1)satisfying l (xl Y)⇒α3 Zx(x, y)=us ・Z“⑭〈°and Zdr(x.y)〈°}.: . fbr arbitrary rea1αand u. And let us define A(α▲uノ,BCa,u/ and C(a3 u, as follows,..i魂。ノ」。㌔.ul.。秤・。.u・;.. ,「、、「
ttt.・.Br。。山=f。㌔.u)+v’一(。、。〃/23 . .・ ..『 ”・垣パー・⑭w+ra・・ノ・. ’.”.
甑en炉0. we. have −
蹴1.L・t Z(・,y)ヵ… ui1励ヱy・・9Uヱe・r h・卿・・θα娼鋤8砲力励d・mfield.垢π .. .一 ド
(1) A(α30)≦v(a)≦β (a30ノ≦λ(aノ≦9 Ca,0/3. ’ − .f・・…』鋤.・了「砺〃L 『:..『: ’1二...1
PROOF. It ea,ily fb11。W・仕㎝th・d・f血iti㎝・・f・升r・。0ノ. V” rd.の。v−+ra。0/and v−「 rd。の. . . 』. .・ ・
LEbMA 2. Z>et.z (¢ち y) カeαsuitabZyア.egttZqr homクgeneou●Cimtse伽random力励・声r.鋤伽卿α唖鞠 ,
・.[ム
㏄ ゜・`︷
8 ㏄ 。・?i⇒o﹁﹂エ
S 8 0 ・rガ(ら.・・旬4・dち 栖一W
カ=一◎。 ・ 0 . . か=−oc ec 8ザ伝.u。8. tJdsdt カ=0∫
∫1・1ザ・・…8ぷち
∫・ザ…聯・ゐ批
∫
砺ゴ・伽鋤W鋤・伽・了(Z鋤ノ・Z。⑭・Zン旬・
∼ ︶− ON THE NUMBER OF RANDOM SErS 47 PROOF. Adler and Hasofer I1] proved the form肛1a (2) in the case of炉0. This theor㎝is a generalisat’ion tg ulO of counting 1. technique in炉O used毎 them and is proved in a si皿ilar way. The essentiql parts of the proof are as fbllows. We consider the fOrmula (3〕. For a real isation of Z rx3 yノ, 1et∂rεノ i den。t・th・di・blrlraf・y)一・・Z党・y)−u)1.くεin theぬge s』・D・fi・i・g・ ㎞・ti・・δ。rZ鋤ハ・・Z〆・・y)一・ノ。n R2 whi・h i・c・n・t・nt・鴨・dr・ノ・nd・e・・ ・1・ewher・・and f・・曲i・h fdr。)δ。々励旬・it fb11㎝・th・t 、 βrα.ω4仇B rα..u2 ε→⇔ ε ㌘ T
一駕‘∫。∫b…(・…21H(・h・ rx,y)・δ…一・・}・ゐ吻
NDw, fbr.fixed ε, we have coE[・、…川・S2∫∫∬1・1・・。・・一α・蛸…⊇・・一・ 『
ロ も The integral interchange is justified by FUbini,s theoreln. Then, using the dominated c㎝verg㎝ce theorem, we have ec O・[・・偽噸……月=一設.㏄な...1・1ザ・一・・…
Using completely analogous argu㎜ts we碑y also show the formulae (2) and (4〕. ’ REMARK. In the paper [1L Adler and Hasofer showed,’the natural generalisation of number of up−crossings of aσne−dimensional stochastic pτ㏄ess to r鋤d㎝fields”. But their results is essentially the. e)Φectation of a lower bound of v rα). Nbw let K(C.ζ) denote the covariance f㎞cti㎝of Z(x♪yノ. Thβns・computing the integrals of (2), (3) and 〔4), we may easily obtain THEORIN・1. L・t Z・(x.・y)加α・uitabZy・・9・la・・ h・m・g・・・…α・us・伽r⑳id・m ず’i e Zd toith E[Z(x. yノ】=03 ガ2θη3 for a17カitrCti・Y αCtncl Us .(・)・[・一・…戸V2
ナ、【・rk22・、、一・紬ユ/2鴫1、/・k22k・・−kl…li]
㌢噺三一嚇・・イ・㊨・]⊇6・る]・
48 K.IWASE
(・)軌u月
ー衰1≡ii ヱ/ち吟、㍑、22乏il
禰rkS・戸/・・Φrrk;・、・/・・月
k22k33一陽〆㌔ たヱヱ㌦一㌦〆㌦ [友鷲;;3rl、22ii:、;,嬬i]・碍友22ξ;;3・’/2、122−、22欝弓、戸/2、122〃⊇i吻・
(・)−4一仁1・考}、li・μ・9:1’ilfll22akE:sii2ili22k33ks・ノヱ/2.
[づ罎]・・ヱ/2・・〃k221Φ・Q1/玩〃・・ヱ川r・UP[一躍i] .+R’/2・u〃・22〃・一〆・…一・Rヱ/2・u//5、)川…ap[−IQr・・/k、、ノ] −Qヱ/2・・/〆k、、)(ヱーぼ・・Q’/2・・/v’T・、7)))reup[一弘r・・/k22ノ] 一縮%/k22・Φ・−R’/2・u//k22川f・r (kiik44一弓2ノ「瓦22㌔3一弓3ノ+0・』・ ・ 「
k・ユーK… ・!, k22−一芸子・9・・ll、.,’ k・ゴー茸・・…ζ。∂ ・44寺・9・の9.。’ k23=一、「語・9…q=、.ジ瞬・/・㌔・44−kS・1・ ≠、/rk22k,、一弓、・−a・d・ω一・㌔即Fl・・]dt. _OD From the lenvna 2 the proof is straightfomard and is amitted. .7ぼ{EOR正羽2舎 こJ%der the scnne oonditions of the theorem 13 we have 〔8) E[Ara30ノ】≦Elvraノ]≦E[B(a.0ノ]≦β[λrαノ]≦宮[C(a.0ノ] 1’ CN THE NUMBER OF RANDCM SETS 49 、 fbr Ctrbita,ctr7y・α、励θrθE[A Cα、0月。 E[B rα,0ノ]αη〈ヱE[o rα。0月αrθg加θη痴(5), 〔6) and (7) ヱ7espeotive Zy. ’ The proof of this theorem fbllows easily f士om the lemma l and the ・ theorem 1. ’ 』P・ace・we c・n・id・r・’r・)…d・’ω…t’NTr・)・b・th・・㎜ber・f up−crossings of a leve1 α by a saエ哩)1e fUnction of Z (x,0/ in O≦¢≦T. Then the expect・ti…f侮「・ノi・giv・n by[2] 卿・月一1・/…〃(k22/・、ノ/2・xp[−1・2/k、,]・ Using this, we Inay have THEOREM 3.〃ndθr the 8amθeondiカion$ of the thθo「θm 13 we havθ (9) E[Arα30)]−n Ca)≦E[vF(αノ]望[B(a30」]・Fn ra)3 、 (10) E[B (a,0ノ]−2n ra)≦罫[λ「rαノ]≦写[C ra. O)]≠2n ra) ゴb㌘arbitrct?yα♪ where ・ω一・r〆疋、・〆q、)/r西、剛一la2/k、ヱ]・ EXAMP凪When Z (x. y/is the isotropic homgeneous Gaussian random field such that E[・r副一・・K(9,・4)一σ2・ap[−1α2rξ2≠ζ2ノ]。 we have easily 噺・…]−r・T)2・・T)−3/2・・/・・卿[−1・a/・・2−1・u/・ασ〃2] ヰ鴎・脚[−lru/r・σ」)2], E[・…明一(・・T)・2[…ダ3/2・・/・・M〆?u・・パ・・/・ノ ・〆万伽戸・UP[一・r・/・ノ2/・』[−Sru/r・σ〃2] =E[・r・、の]・mp[−lru/rασノ)2]. E[α瑚一r・・(・・T)・2・r・・ノー’・・rp[−lrr・/・ノ2+r・/r・・σ])2月』[−r・/・ノ2] …ff’r・/ω・r・/r・/E・〃岬[−lr・/r・σ刀2挿/r・σ〃rヱー〆21.Φr−u/rασ川 +reUP [一 ra/σノ2/4]−rα/r〆Eσ)rヱー/2iilΦ ra/r〆2σ)川 (・ap[−lru/r・σ〃2]一,/2ii’ru/r・・))Φr−−u/rασ川】・
‘ t 50 K.1覗ASE [1] [2] (3) REFERENCES R.J.. Adler and A:M・Hasofer:. LeVel crossings for random fields, Ann・ of Prob., 4 (1976), 1−12. K〔Ito: The expected nuinber of zeros of continuous stationaTy Gaussian proceSses, J. Math. Kyoto Univ.,3 (1964〕,207−216. Yu. K. BelyaevクV. P. Nosko and A・A. Sveshnikov:Probability chara¢teristics of upwards excursions of a two−dimensional random field・ Th. of Prob. and Math. Stat., 6 (1972), 24−31. 1 .. 〆 、