In the article authors develop an approach to calculating the statistic development probability for composite functions of square values in Gaussian casual process trajectories. Calculating distribution density for additive composite functions is based on standard Wiener process trajectories. Authors have developed a density formula for uniformly convergent decomposition, with x = 0. The convergence is exponentially fas

Bugayevskaya, A. N.

Esin, V. A.

Motkina, N. N.

Virchenko, Yu. P.

Vitokhina, N. N.

Zinchenko, N. A.

English

DSpace at Belgorod State University

COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI) 3192    Cite This Paper: Natalia N. Vitokhina, Yuriy P. Virchenko, Natalya A. Zinchenko, Natalya N. Mot’kina, Vladimir A. Esin, Anna N. Bugayevskaya. (2019). The distribution density of square value probabilities functionality from trajectories of wiener process, 8(6), COMPUSOFT, An International Journal of Advanced Computer Technology. PP. 3192-3196.   This work is licensed under Creative Commons Attribution 4.0 International License.    THE DISTRIBUTION DENSITY OF SQUARE VALUE PROBABILITIES FUNCTIONALITY FROM TRAJECTORIES OF WIENER PROCESS  Natalia N. Vitokhina, Yuriy P. Virchenko, Natalya A. Zinchenko, Natalya N. Mot’kina, Vladimir A. Esin, Anna N. Bugayevskaya Belgorod State University, 85, Pobedy St., Belgorod, 308015, Russia  Corresponding author:  Natalia N. Vitokhina Department of Theoretical and Mathematical Physics, Belgorod State University, 85, Pobedy St., Belgorod, 308015, Russia E-mail: vitohina@bsu.edu.ru  Abstract: In the article authors develop an approach to calculating the statistic development probability for composite functions of square values in Gaussian casual process trajectories. Calculating distribution density for additive composite functions is based on standard Wiener process trajectories. Authors have developed a density formula for uniformly convergent decomposition, with x = 0. The convergence is exponentially fast. The calculation of the approximated probability is presented:   cwJT Pr .  Keywords: stationary processes; casual Gaussian processes; Wiener process; the density of distribution of probabilities; additive functionality; return transformation of Laplace; calculating the successive approximations; accuracy assessment.  I. INTRODUCTION In work the task about the calculating the density of distribution of probabilities of a random variable  wTJwhere is considered  wTJ      TdttuuJ02  (1) - functionality in space of L2 [6]  T,0 .  Let  - trajectories of standard Wiener process on [0, T] with a population mean of E  ttw 2 . This task, as well as other tasks about the calculating the density of distributions of probabilities of casual values of additive square functionalities from trajectories of Gaussian casual processes, is classical.   From the analytical point of view, Gaussian casual processes are, apparently, the simplest casual processes. It is connected with the fact that the Gaussian form of private multipoint distributions of Gaussian processes allows to calculate obviously characteristic functionality of each such process and to apply to their research well-developed analysis methods in Hilbert space [1,2]. Now the class of Gaussian casual processes, at least stationary, is well studied. For Gaussian processes, the problem about the calculating the characteristic functions for random variables such is essentially solved. On the basis of this method a large number of solutions of the specific objectives having    ; 0, , 0w t t T T  Available online at: https://ijact.in Date of  Submission Date of  Acceptance 03/06/2019 17/06/2019 Date of  Publication 03/07/2019 Page numbers 3192-3196 (5 Pages) ISSN:2320-0790 COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI) 3193  various appendices [3,4] is so far received). For example, the first such result for the simplest stationary Gaussian process (Ornstein-Ulenbek's process) was received still in [5]. However, the problem of restoration of the density of distribution on the basis of the turning-out formulas for characteristic functions remains the guaranteed accuracy which is still poorly investigated in sense of receiving the approximations with suitable for use in all range of change of a random variable. The usual approach to the solution of this task (see, for example, [6]) leads to approximate formulas for the density of distribution of f (x),   ,0x  suitable for the assessment of probabilities of big evasion, i.e. in asymptotic area x   changes of values of a random variable.  In this work we investigate a problem of calculating the successive approximations of the density of distribution of probabilities of f (x) for casual values of functionality (1) in any piece [0, M], > M> 0.  II. MATERIAL AND METHODS As trajectories of standard Wiener process { w(t); t ≥0},   ttw 2 1 are continuous E   ttw 2  with probability, honor probably for each of them the random variable is defined  wTJ . We will proceed [2] of the following formula for the making function of a random variable (1)           2/12/1expE  TchwTJTQ  ,    21 TQTQ   . The density of distribution of f (x) random variable wTJ  is defined by the return transformation of Laplace      cicidTchxeixf 2/12/121  (2) where c>0, and a section in the plane λ is made by a negative part of the valid axis. Let's enter density    xTfTxg 22  at   ,0xfor which replacement of a variable of integration in (2) we receive         cicidchxeixg 2/12/121   .2/12/12exp12/12exp21 cicidxi (3)  III. RESULTS AND DISCUSSION Let's prove the following theorem [7, 8].THEOREM. The density of g (x) is represented as the following absolutely meeting row       02411exp412!4!2132llxlllllxxg (4)  for which N-y rest       NllxlllllxxNg2411exp412!4!21321  it is estimated by the size       2411exp412!4!2321 NxNNNNxxNg  (5)  PROOF: Let's put in (3) c=0 as the available features lie on a negative half shaft. We deform an integration contour in the contour of C consisting of consecutive passing of ways { s-iε; s   0; }, { eis; s  2;2}, {-s+iε; s  ;0 }. Such deformation is possible since.                 2/1Im22/1Re2212/12/122/1 ichchchchch      2/cos2/1212/cos2/1221 RshRch  , whereieR  and on an arch of a circle  ;2;  assessment for mo the module of sub integral expression in is carried out (3),         ,2/12/cos2/1cosexp2/12/1expRshxRchx  guaranteeing at x> 0,    2/cos2/1R  the performance of a condition of Jordan on this arch at as much as small > 0, since 0cos  . The same takes place for an arch2,;   In (3) we will make replacement of a variable of integrationq2/1 , then qdqdq 2,2  . At the same time the contour of C in the plane λ after the transition to a limit0  to turn into a straight line { q=is, s  R } into the planes q. After these transformations we have: COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI) 3194      .2/12exp122exp2  iidqqqxqqixg Let's pass to integration on a variable s, q = is, dq = with ids. We receive       dsisisxssixg2/12exp122exp2  (6) In the last integral we will makeshift  sxis  14  of a variable of integration, therefore, we will receive:             dsxxisixxisxsixg2/112142exp1182142exp2            .2/12exp2/1exp122exp14116exp2 dsisxxsxisx                      (7)  In (7) we will spread out a denominator of sub integral expression in meeting at any x> 0 and any s  R row       ilslxlllllisx2exp02exp!2!!1212/12exp21exp1   Its convergence is uniform in any strip [0, M] ×R the planes (x, s), M> 0.  Substituting the last expression in (7), we will receive          2exp14116exp2xsxisxxg        dsxlilsl llll2exp2exp0 2!4!21            2exp140 2!4!21116exp2xsxisl llllx  .2/12exp dsxllxilsx     The shift of operators of summation and integration is based on uniform convergence of a row on s at any fixed x.  In everyone composed the sums we will make a shiftsxils   on an integration variable, we will receive               (8)  Let's present integral in the form of the sum of two integrals according to the expression standing in a pre-exponential bracket. The integral corresponding to the composed is addressing in zero,          02/1exp2exp0 2!4!21dsxllxsl llllsi  in a type of oddness of sub integral function. The integral corresponding to composed xl 4/1will be transformed as follows        .2/1exp410 2!4!212exp1 xllll lllldsxsx Substitution of this expression in (8) taking into account the value of integral of Poisson leads to formula (4).  As a row (4) sign-variable Ng  does not surpass the rest of a row Ng first composed from among rejected. Therefore, the assessment takes place (5).  INVESTIGATION. The assessment takes place [9]    2/52231 NexNg . (9)  PROOF: Let's find assessment of the rest of a row (4). For this purpose, we will write down on the basis of (4) expression for g(x) in shape:      ,012lxlhlalxg Where     ,412!4!2 llllla    ,24/1exp2/3 xlxxlh  also, we will find a maximum on x functions  xNh . Equating zero a derivative on x this function:      ,02324121 xNxNhxxNh  We find the solution x  of this equation - a point of the only maximum of a function  xNh .          1201 2 !2 1/ 4exp 164 !llll lg x x isxl         2 1/ 2exp .l lxs dsx     COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI) 3195  ,24132 Nx    .3412/323 NexNh  Therefore, rest N assessment      .2412!4!232321 NNNNexNhNaxNg Let's estimate from the above coefficient Na , having made the received assessment of more transparent,     Nl lNl lNl llNNNNa1 211lnexp1 2111 212!2!!12   ,2/1ln121exp1121expNeNNl l  in view of the justice of inequalities   xx 1ln  at x> 0 and  .ln11111NNxdxNl l Then, as 13, takes place        (9) Let's estimate the accuracy of the approximations of the probability of PR   cwTJ  received on the basis of functions Ng , N = 1 now   cwTJ  Ng . 2, … [6] As distribution density    xTgTxf 22  ,        ,10221Pr cRcdxxTgTcwTJ   Where,       .2/0TcdxxgcR   Having designated the right part of inequality (5) by means of  xNQ , we have:     xNQxNg 1 . Let's define function now             .2/01,1  TcdxxNgxgcNRcNRcNP Our task is receiving the top assessment for evasion     cNPcwTJ Pr . Follows from inequality (5) that      xNQxNgxNQ  1 . Integrating ranging from 0 to with / T 2, we receive       2/02/012/0TcdxxNQTcdxxNgTcdxxNQ. Therefore,          ,2/02/01  TcdxxNQdxTcxNgcNRcR  it gives a required assessment           2/0PrTcdxxNQcNRcRcNPcwTJ      (10) At last, to make an assessment (10) obvious, we will calculate the integral standing in the right part     2/02/324/1exp22/0TcxdxxNNaTcdxxNQ. Replacement of a variable of integration2/32,2/1xdxdyxy   leads to a formula       2/14/14/124/12exp4/182/0 cNTErfcNNacNTdyyNNaTcdxxNQ. From here, using Erfc(x) <1, we find uniform in parameters with and T assessment  322/0 eNTcdxxNQ  . Thinner assessment considering the size of parameters with and T, turns out the use of standard inequality of [10] Erfc(x) xx2exp  ,    cNTNTNacTcdxxNQ24/12exp24/122/0      .2exp12/52cTNTNec   IV. CONCLUSION The task of calculating the characteristic functions for random variables of type (1) in Gaussian casual processes are essentially solved [5]. The task of the density distribution of f on the basis of characteristic functions formulas is studied insufficiently. Such a task involves acquiring approximations with guaranteed accuracy, applicable for the whole random variable change diapason. The task of calculating successive approximations for density distribution of f(x) probabilities for all random composite functions (1) on every [0, M],  >M > 0 interval has been considered.  Accuracy assessment for probability approximations of PR   cwTJ   received on the basis ofNg , N = 1, 2… functions is presented.    COMPUSOFT, An international journal of advanced computer technology, 8(6), June-2019 (Volume-VIII, Issue-VI) 3196  V. SUMMARY An impression in the form of absolutely meeting row for a density of distribution of probabilities is gained:       cicidchxeixg 2/12/121       . The top assessment for evasion is also received     cNPcwTJ Pr  . More exact assessment which considers the size of parameters with and T has an appearance      cNPcwTJ Pr   .2exp12/52cTNTNec  VI. REFERENCES [1]. Gikhman I.I., A.V. Skorokhod A.V., 1973. Theory of Random processes, V.II: Science. Publ. Nauka, Moscow. 641 pages. [2]. Ibragimov I.A., Rozanov Yu.A., 1970. Gaussian Random processes. Publ. Nauka Moscow: Science. 384 pages. (In Russian) [3]. Mazmanishvili A. S., 1987. Continual integration as a method of the solution of physical tasks. Kiev: Scientific thought. 224 pages. (In Russian) [4]. Arato M., 1982. Linear Stochastic Systems with Constant Coefficients. A Statistical Approach: Springer-Verlag. Berlin. 304 p. [5]. Ziegert A.J.F., 1957. A systematic approach to class problems in the theory of noise and other random phenomena. part II, examples. Trans. IRE. IT-3: 38-44. [6]. Zolotarev V.M., 1961. Concerning a Certain Probability Problem. Moscow: Theory Probability and its applications. 6, No. 2: 201-204. (In Russian). Accessed from:  http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tvp&paperid=4769&option_lang=rus [7]. Vitokhina N.N., 2006. Distributions of probabilities in a problem of registration of stochastic radiation in quantum optics [Text]: Math PhD Thesis: 01.01.03 / N.N. Vitokhina. Belgorod. 155 pages. (In Russian) [8]. Virchenko Yu.P., Vitokhina N.N., 2003. About the local limit theorem for distribution of probabilities of additive square functionality from trajectories of complex-valued Wiener process. St. Petersburg: Mathematical models in education, science, and the industry. International Academy of Sciences of the higher school, St. Petersburg office. 48-50. (In Russian) [9]. Virchenko Yu.P., Vitokhina N. N, 2005. The distribution density of square values probabilities functionality from trajectories of the Wiener process. Belgorod: Scientific sheets of BELGU. Physical-Math sciences. No. 2(22). release 11. 16-22. (In Russian) available from: Virchenko Yu.P., Vitokhina N.N. The probability distribution density of random values of squared functional on Wiener process trajectories. ArXiv: math-ph/0510028 vl. [10]. Kuo H.-H, 1979. Gaussian measures in Banachian spaces: World. Moscow. Publ. Springer-Verlag Berlin Heidelberg, Vol III, page. 176 pages. ISBN: 978-3-540-37508-1.                            22301 2 !2 1 1 1exp4 44 !lllll lx xl                  2/302c TNQ x dxeN 

The distribution density of square value probabilities functionality from trajectories of wiener process

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The distribution density of square value probabilities functionality from trajectories of wiener process

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