The Asymptotic Distributions of the Absolute Maximum of the Generalized Wiener Random Field

Transcription

1 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. The Asmptotc Dstrbutons of the Absolute Maxmum of the Generalze Wener anom el Anre P. Trfonov, Alexanra V. Salnkova,3, Alexaner V. Zakharov, Oleg V. Chernoarov 3,4, # an Dara S. ozhkova 3 Department of ao Phscs, Voronezh State Unverst, Unverstetskaa sq.,, Voronezh 3948, ussa. Department of Informaton Technologes an Computer-ae Desgn, Voronezh State Techncal Unverst, Moscow avenue, 4, Voronezh 3946, ussa. 3 Internatonal Laborator of Statstcs of Stochastc Processes an Quanttatve nance, Natonal esearch Tomsk State Unverst, Lenn avenue, 36, Tomsk 6345, ussa. 4 Dept. of Electroncs an Nanoelectroncs, Natonal esearch Insttute Moscow Power Engneerng Insttute, Krasnokazarmennaa st., 4, Moscow 5, ussa. Corresponng author # Orc: Abstract The technque s presente for obtanng the asmptotc approxmatons of the strbuton functons of the absolute mum of the generalze Wener ranom fels. Base on the propose approaches, the close expressons are foun for the strbuton functons of the absolute mum of Wener ranom fels wth the constants an pecewse constant rft an ffuson. B means of statstcal smulaton, t s establshe that the ntrouce analtcal formulas successfull agree wth the corresponng expermental ata referrng to a we range of the ranom fel parameters values. Kewors: Wener ranom fel, Dstrbuton functon of the absolute mum, Local Markov approxmaton, Statstcal smulaton. INTODUCTION The task of fnng the probablstc strbutons for the absolute (greatest) ma of Gaussan ranom processes an fels arses n varous applcatons of phscs, nclung rao phscs an rao engneerng [, ]. Despte a great number of theoretcal an expermental papers evote to the stu of the ma of stochastc functons, the general expressons for the strbutons for the absolute mum of Gaussan ranom fel reman unetermne et. In [3, 4], the asmptotcall exact (wth h ncreasng) expressons are presente for the strbuton functon h of the absolute mum of the homogeneous Gaussan ranom fel. In practce, the essentall heterogeneous Wener processes an fels are commonl foun. As Wener ranom process L t, t, we conser the purel ffuson an centere Gaussan Markov ranom process wth the zero rft coeffcent, the constant ffuson coeffcent an the t, t mn t t [5]. Such process covarance functon, escrbes Brownan moton of partcles n varous envronments, fluctuatons of the generator oscllaton phase n the presence of thermal an shot nose, as well as the behavor of ecson statstcs of the sequental etector of sgnals aganst Gaussan nterferences, etc. As the generalzaton of Wener process n case of the two- L,, mensonal efntonal oman, Wener fel, s consere [3, 7, 8]. Ths fel s Gaussan centere ranom fel wth the covarance functon,,, mn, mn. The sectons of, Wener ranom fel, Wener ranom processes. L b the η an κ varables are In [3] the asmptotcall exact (wth h ncreasng) expresson h h of the threshol s obtane for the probablt h crossng b the absolute mum of Wener ranom fel L, wthn the efntonal oman set b the contons,,,. The analss of the asmptotc expresson [3] shows that ts accurac for the fnte values of h ma be unsatsfactor. Thus, the values of the probablt h calculate b the formula from [3] ma be greater than one, whle the corresponng values of the strbuton functon h ma be less than zero. Beses, h an h, f h, an ths contracts the unerstanng of the functons h an h as the probablt measures lng wthn the range of values from to. It shoul also be note that the researcher often nee to know the strbuton of the mum of Wener fel wthn the ntervals 79

2 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. mn, mn,, where mn an mn. The asmptotc formula [3] obtane for the case of mn, mn oes not account for the nfluence of nonzero values of mn an mn on the strbuton of the absolute mum of the fel. In a number of practcal applcatons of statstcal rao phscs an rao engneerng, t s necessar to calculate the probablstc strbutons for the absolute ma of heterogeneous Gaussan ranom fels whch are the atve superposton of both Wener ranom fel an a certan etermne fel. Such ranom fels (whch further we wll refer to as the generalze Wener fels) are nvolve when we process of the quas-etermnstc an stochastc sgnals wth unknown scontnuous power parameters [9-] an the submages wth a pror unknown szes, etc. Unlke Wener ranom fel, the generalze Wener fel has the mathematcal expectaton changng wthn the efntonal oman. It oes not allow usng the technque [3, 4] for obtanng the asmptotc strbutons of the absolute mum of the generalze Wener fels. In [9-], b means of the methos of Markov processes theor [5, 6], the exact expresson s foun for the strbuton functon of the absolute mum of the generalze Wener ranom process wth the constant an pecewse constant rft an ffuson coeffcents. However, we o not have methos for the exact soluton of a smlar task n case of the generalze Wener ranom fel et. Below, the asmptotcall exact expressons are ntrouce for the strbuton functon of the absolute mum of the generalze Wener ranom fle wth both constant an pecewse constant rft an ffuson, wthout the prevousl specfe flaws [3]. The applcablt lmts of the obtane asmptotc expressons are expermentall establshe b means of computer statstcal smulaton. THE DISTIBUTION O THE ABSOLUTE MAXIMUM O THE WIENE ANDOM IELD Accorng to [3, 7, 8], uner Wener fel we unerstan the L,, centere heterogeneous Gaussan ranom fel, wth the covarance functon,,, L, L, mn, mn,. Let us fn the asmptotcall exact (wth h ncreasng) expresson for the strbuton functon h of the absolute mum of Wener ranom fel wthn the efntonal mn, mn,, oman Λ set b the contons, mn, mn. It shoul be note that the strbuton functon of the absolute mum of Wener fel wthn the arbtrar efntonal oman mn, (), mn, can be eas calculate b the formula h h, where h s the strbuton functon of the absolute mum of the fel uner, mn,, mn. It s known from [3] that the strbuton h of the absolute mum of Wener ranom fel L, uner the large values of h s etermne b the behavor of the fel n the neghborhoo of the values m, m belongng to the efntonal oman Λ an settng the fel sperson as the mum one. In the present case, t s obvous that m, m. Therefore, n orer to fn the asmptotcall exact (wth h ncreasng) expresson for the strbuton functon h, t s suffcent to stu the behavor of the covarance functon of the fel, L n the small neghborhoo of pont (,). Uner,,,, the covarance functon () can be asmptotcall presente as follows where,,,,,, t, t,mn t t, () (3) an enotes the hgher-orer nfntesmal terms compare wth. We esgnate t L,, as the statstcall nepenent an jontl Gaussan centere ranom processes wth the covarance functons t,t (3). Such processes are entcall equal to zero uner t, an the are Wener ranom processes uner t [5]. rom Eqs. (), (), t follows that the covarance functons of Gaussan ranom fels L, an L L L, conce asmptotcall, whle. Therefore, Wener ranom fel, L converges n strbuton to the sum of the statstcall nepenent Gaussan ranom processes L an L uner,. Thus, the strbuton functon h of the absolute mum of the ranom fel L, can be presente n the form of h P sup L sup L h xw x x, h (4) 793

3 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. where x P sup L x, x P sup L x are the strbuton functons of the absolute ma of the L, corresponngl, ranom processes L, w x x x mum of the ranom process s the probablt enst of the absolute L uner, an, are the ntervals of the possble values of the parameters η an κ set b the contons mn,, mn,, corresponngl. B applng the results presente n [9, ], we obtan exp u x x u u,,. Here,,, mn x exp t mn (5), an x t s the probablt ntegral []. Then, from Eq. (4), takng nto account Eq. (5), we get h exp x x exp u h x u u x. The accurac of Eq. (6) ncreases wth h. In [3], uner ncreasng h, the asmptotcall exact expresson h h that the s obtane for the probablt threshol h s exceee b the realzaton of Wener ranom L, uner,, whence we fn that fel mn mn h exph h (6) Let us compare the asmptotcall exact expressons (6) an (7). Uner h, from the formula (6) we get h exph h h (7). Therefore, the expressons (6) an (7) conce asmptotcall n the case of h. or the fnte values of h, the formula (7) can prove the essentall conservatve values for the probablt h, whch ma be greater than, whle the approprate values of the functon h ma appear less than zero. Then, b applng the formula (7), we obtan h 5. 7, h 4. 7, f h. 3, an h 5. 88, h 4. 88, h, whle h an h f. uner h. At the same tme, the values of α calculate b the formula (6) o not excee an the corresponng values of h are nonnegatve that full comples wth the meanng of the functons h h as the probablt measures lng an wthn the range of values from to. In orer to establsh the borers of applcablt of the asmptotcall exact formulas (6), (7) for the fnte values of h, the statstcal computer smulaton s carre out of the value of L, wth the absolute mum of Wener ranom fel the covarance functon (). Durng the smulaton, the samples X j L,,, j,, of Wener j n lk l k j ranom fel are forme wth the scretzaton step. 5 b the η an κ varables. Here n lk are nepenent Gaussan ranom numbers wth the zero mathematcal expectaton an the unt sperson. The value X of the absolute mum of the fel, m L wthn the efntonal oman Λ s foun as the greatest value, j mn X for all, j mn,, where s the nteger part. The expermental values for the probabltes h an h for the varous threshols h are etermne as the relatve frequences of the varable X exceeng, or not exceeng the threshols, respectvel. m gure : The probablt of the absolute mum of Wener ranom fel exceeng the threshol In g. the expermental values of the probablt h h are presente obtane urng the processng 4 of no less than realzatons of Wener ranom fel. Here, b contnuous lnes the corresponng theoretcal epenences h are also rawn calculate b the formula (6). The confence lmts evate from the expermental values for h wth the probablt of.95 an b no more than % uner. an b no more than 4 % uner

4 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. Curve an crosses correspon to. 4, curve an trangles to. 7, curve 3 an squares to mn mn mn mn mn mn.9. B crcles, the expermental values for α are shown uner, an the corresponng mn mn theoretcal curve calculate b the formula (6) conces wth the curve. or the comparson, b ashe lne the theoretcal epenence h s plotte calculate b the formula (7) for the case of. rom g. an other smulaton mn mn results, t follows that the theoretcal formula (6) ntrouce above successfull approxmates the expermental ata for all the values of h, mn an mn. The formula (7) obtane n [3] has a low accurac uner small values of h, an that s wh t can be use for calculatng h uner h 3, when.. THE DISTIBUTION O THE ABSOLUTE MAXIMUM O THE GENEALIZED WIENE ANDOM IELD WITH CONSTANT DIT AND DIUSION Let us conser the generalze Wener ranom fel,, wth the mathematcal expectaton L z L, S,,, z (8) an the covarance functon,,, L, L, L, L, mn, mn,. We see that the sectons L, v,, const L (9) v, const an of the fel, L b the η an κ varables are Gaussan Markov ffuson ranom processes wth the constant rft K, z K, K z an ffuson K coeffcents, corresponngl [5]. Let us fn the asmptotcall exact (wth z ncreasng) expresson for the strbuton functon h of the absolute mum of the ranom fel, oman Λ set b the contons,,, L wthn the efntonal. Usng ths expresson makes t eas to wrte own the strbuton functon h for arbtrar efntonal oman, mn, mn,. We take nto account that wthn the oman Λ the L, mathematcal expectaton (8) of the ranom fel reaches the absolute mum at the pont (,), whle the fel sperson at ths pont s mnmum. The fel mathematcal expectaton ecreases wth η an κ ncreasng proportonall to zηκ, whle the fel sperson ncreases proportonall to the ηκ. Therefore, uner z, the poston of the absolute mum of the fel, L s locate n the small neghborhoo of the pont (,). Beses, f z, then the coornates, of the fel absolute mum m m poston converge to the values, n mean square [3]. Then, n orer to fn the asmptotcall exact (wth z ncreasng) expresson for the strbuton functon h of the absolute mum of the fel L,, t s suffcent to stu the behavor of ts mathematcal expectaton an covarance functon n the small neghborhoo of the pont (,). Uner,, the mathematcal expectaton (8) s represente n the form of S, S S, S t zt. () In turn, f,,, the correlaton functon (9) takes the form,,,,, then,, () We esgnate t t, t mnt t. (), L,, as statstcall nepenent an jontl Gaussan ranom processes wth the mathematcal expectatons S t () an the covarance functons t,t (). rom Eqs. (8)-(), t follows that the mathematcal expectatons an the correlaton functons of Gaussan ranom fels L, an, L L L conce asmptotcall, f,. Therefore, uner, Gaussan ranom fel, L converges n strbuton to the sum of the statstcall nepenent Gaussan ranom processes L an L. Thus, smlarl Eq. (4), the strbuton functon ranom fel, h of the absolute mum of the L can be presente as follows h P sup L sup L h x w x x, h (3) where x P sup L x, x P sup L x are the strbuton functons of the absolute ma of the ranom processes L, L ; w x x x s the corresponng probablt enst of the absolute mum of the ranom process L ; an, are the ntervals for the possble values of the η an κ parameters set b the,,, respectvel. The contons, ranom processes L an L are the contnuous Gaussan Markov ffuson ones [5] wth the z rft coeffcents an the unt ffuson coeffcents. The 795

5 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. strbutons of the absolute ma of such processes are alrea known from [9, ] an etermne as x, z x zxz x z x exp exp, x, x, exp x z z z, (4) where,. Then, from Eq. (3) takng nto account Eq. (4), we have z x z h exp exp zx z x exp exp z h x z z z x. (5) gure : The probablt of the absolute mum of the generalze Wener ranom fel exceeng the threshol uner. The accurac of the expresson (5) ncreases wth z. In orer to establsh the borers of applcablt of the asmptotcall exact formula (5) for the fnte values of z, we carr out the statstcal computer smulaton etermnng the value of the absolute mum of the generalze Wener L, wth the mathematcal expectaton (8) ranom fel an the covarance functon (9). Durng the smulaton, the j j lk,, j,, l k samples X L, j n zj of the ranom fel are forme wth the scretzaton step.5 b the η an κ varables. Here, as above, n lk are nepenent Gaussan ranom numbers wth the zero mathematcal expectaton an the unt sperson. The value L, s foun as X m of the absolute mum of the fel X for all, j, h an h the greatest value j probabltes,. The expermental values for the for the varous threshols h are etermne as the relatve frequences of the varable X exceeng an not exceeng the threshols, respectvel. m gure 3: The probablt of the absolute mum of the generalze Wener ranom fel exceeng the threshol uner 4. In g. the expermental values for the probablt h h are presente obtane urng the processng 4 of no less than realzatons of the ranom fel uner an n g. 3 uner 4. In that gures, b contnuous lnes the corresponng theoretcal epenences of h are rawn calculate b the formula (5). The confence lmts conce wth the corresponng ones n g.. Curves an squares correspon to z, curves an trangles to z. 5, curves 3 an crosses to z, curves 4 an crcles to z. 5. rom gs., 3 an other smulaton results, t follows that the theoretcal formula (5) s alrea a successful approxmaton of the expermental ata uner z

6 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. The Dstrbuton Of The Absolute Maxmum Of The Generalze Wener anom el Wth Pecewse Constant Drft An Dffuson Let us conser the generalze Wener ranom fel,, wth the mathematcal expectaton S, z z mn, mn, z, an the covarance functon z z,,,, gmn, mn, g mn,, mn,,, g. We see that the sectons L, v,, const L L, (6) (7) v, const an of the fel, L b the η an κ varables are Gaussan Markov ffuson ranom processes wth the pecewse constant rft an ffuson coeffcents [5]. Let us fn the asmptotcall exact (wth z an z ncreasng) expresson for the strbuton functon h of the absolute mum of the ranom fel L, wthn the efntonal oman Λ set b the contons mn,,. We take nto account that wthn the oman mn, Λ the mathematcal expectaton (6) of the ranom fel L, reaches the absolute mum at the pont (,). Inee,, z z z S, f,, whle S, f,. The fel sperson, ncreases proportonall to the bηκ wth η an κ, where b. Therefore, smlarl to [9,, ], we can show that uner z, z the poston of the absolute mum of the L, s locate n the small neghborhoo of the pont fel (,). Beses, f z, z, then the coornates m m, of the fel absolute mum poston converge to the values, n mean square [3]. Then, n orer to fn the asmptotcall exact (wth ncreasng expresson for the strbuton functon mum of the fel L, h z,, ) of the absolute, t s suffcent to stu the behavor of ts mathematcal expectaton an covarance functon n the small neghborhoo of the pont (,). Uner,, the mathematcal expectaton (6) s wrtten own n the form of S S S, (8), mn, t zt, t, S t (9), t. In turn, f,,, the correlaton functon (7) permts the presentaton, then,,,,,, t, t, gmnt, t g mn, t t, We esgnate (). () t L,, as statstcall nepenent an jontl Gaussan ranom processes wth the mathematcal expectatons S t (9) an the covarance functons t,t (). Such processes are entcall equal to zero uner t, whle uner t the are contnuous Gaussan Markov ffuson ranom processes wth the pecewse constant rft an ffuson coeffcents [5]. rom Eqs. (6)-(8), (), t follows that the mathematcal expectatons an the correlaton L, an functons of Gaussan ranom fels L L L conce asmptotcall, when,. Therefore, uner Gaussan ranom fel, L converges n strbuton to the sum of the statstcall nepenent Gaussan ranom processes L an L. Thus, smlarl Eq. (4), the strbuton functon ranom fel, h of the absolute mum of the L can be presente as follows h P sup L sup L h x w x x, h () where x P sup L x, x P sup L x are the strbuton functons of the absolute ma of the ranom processes L, L ; w x x x s the corresponng probablt enst of the absolute mum of the ranom process L ; an, are the ntervals for the possble values of the η an κ parameters set b the,, contons mn, mn results obtane n [9, ], we come to, respectvel. B applng the z x exp x V x,, (3) exp z g z g z g g g, 797

7 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. exp V x, x x x,,, where,,, mn,. Dfferentatng the expresson (3) prouces, mn w x exp z x x z V x, x,, x exp x x (4),. B substtutng Eqs. (3), (4) n Eq. (), we fn the asmptotcall exact (wth ncreasng z an z ) expresson for the strbuton functon the ranom fel. h of the absolute mum of In orer to establsh the borers of applcablt of the asmptotcall exact formulas ()-(4) for the fnte values of z,,, we carr out the statstcal computer smulaton of the value of the absolute mum of the generalze Wener L, wth the mathematcal expectaton (6) ranom fel an the covarance functon (7) uner z z z, g. Durng the smulaton, base on the nepenent Gaussan ranom numbers n lk wth the zero mathematcal expectatons an the unt spersons, the samples z X j L, j l k mn, mn j, j,, j,, of the ranom fel are forme wth the scretzaton step.5 b the η an κ varables. The value X m of the L, s foun as the greatest absolute mum of the fel value X j for all mn, j mn, probabltes h an h j n lk,. The expermental values for the for the varous threshols h are etermne as the relatve frequences of the varable X exceeng an not exceeng the threshols, respectvel. m gure 4: The probablt of the absolute mum of the generalze Wener ranom fel exceeng the threshol uner,. mn mn gure 5: The probablt of the absolute mum of the generalze Wener ranom fel exceeng the threshol uner, 3. mn mn In g. 4 the expermental values for the probablt h h are presente obtane urng the processng 4 of no less than realzatons of the ranom fel uner, an n g. 5 uner mn mn, 3. Here, b contnuous mn mn lnes the corresponng theoretcal epenences h are also rawn calculate b the formulas ()-(4). The confence lmts conce wth the corresponng ones n g.. Curves an squares correspon to z, curves an trangles to z.5, curves 3 an crosses to z, curves 4 an crcles to z. 5. rom gs., 3 an other smulaton results, t follows that the theoretcal formulas ()-(4) start successfull approxmatng the expermental ata uner z.5,,. 798

8 Internatonal Journal of Apple Engneerng esearch ISSN Volume, Number 8 (7) pp esearch Ina Publcatons. CONCLUSION To calculate the characterstcs of the generalze Wener ranom fels absolute mum values, there can be use the lmtng laws of the absolute mum strbuton obtane for the case of the unlmte ncrease n the threshol an when Wener fel s statstcall equvalent to the sum of the two nepenent Wener ranom processes. The asmptotc formulas for the characterstcs thus etermne turn out to be more accurate than the common ones, f the threshol values are fnte. Whle comparng the ntrouce expressons wth the expermental ata prouce urng the smulaton n a number of partcular cases we come to the concluson that that the successfull escrbe the true strbutons n terms of a we range of the ranom fels parameters values. ACKNOWLEDGEMENT Ths stu was fnancall supporte b the ussan Scence ounaton (research project No ). 99, Detecton of Stochastc Sgnals wth Unknown Parameters (n ussan), Voronezh State Unverst, Voronezh. [] Trfonov, A.P., 984, Detecton of Sgnals wth Unknown Parameters (n ussan), n: P.A. Bakut (E.), Sgnal Detecton Theor, ao Svaz', Moscow, pp [] Salnkova, A.V., Chernoarov, O.V., an Golpaegan, L.A., 7, On probablt of the Gaussan anom Processes Crossng the Barrers, Proc. 7 3r Internatonal Conference on ronters of Sgnal Processng (ICSP 7), Pars, rance, pp. -7. [] Trfonov, A.P., an Shnakov, Yu.S., 986, Jont Dscrmnaton of Sgnals an Estmaton of ther Parameters aganst Backgroun (n ussan), ao Svaz', Moscow. [3] Ibragmov, I.A, an Has mnsk,.z, 98, Statstcal Estmaton Asmptotc Theor, Sprnger, New York. EEENCES [] Tkhonov, V.I., 97, Outlers of anom Processes (n ussan), Nauka, Moscow. [] Tkhonov, V.I., an Khmenko, V.I., 998, Level- Crossng Problems for Stochastc Processes n Phscs an ao Engneerng: A Surve, Journal of Communcatons Technolog an Electroncs, 43(5), pp [3] Pterbarg, V.I., 996, Asmptotc Methos n the Theor of Gaussan Processes an els. Amercan Mathematcal Soc., Provence. [4] Pterbarg, V.I., an atalov, V.., 995, The Laplace Metho for Probablt Measures n Banach Spaces, ussan Mathematcal Surves, 5(6), pp [5] Dnkn, E.B., 6, Theor of Markov Processes, Dover Publcatons Inc., New York. [6] Chernoarov, O.V., Sa S Thu Mn, Salnkova, A.V., Shakhtarn, B.I. an Artemenko, A.A., 4, Applcaton of the Local Markov Approxmaton Metho for the Analss of Informaton Processes Processng Algorthms wth Unknown Dscontnuous Parameters, Apple Mathematcal Scences, 8(9), pp [7] Chentsov, N.N., 956, Wener anom els Depenng on Several Parameters (n ussan), Dokl. Aka. Nauk SSS, 6(4), pp [8] Gooman, V., 976, Dstrbuton Estmates for unctonal of the Two-Parameter Wener Process, Ann. of Probablt, 4(6), pp [9] Trfonov, A.P., Nechaev, E.P., an Parfenov, V.I., 799