布朗运动首冲时及Mills率的研究
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摘要
在概率理论中,大偏差理论(或者说是尾部概率)及小偏差理论(或者说是小球概率)在一定背景下是两个互补的方向.大偏差理论是一个更经典的方向,它是用来研究随机变量X与它均值偏差的概率,即寻找Prob(|X-M|＞t)的上界.小偏差理论是用来研究随机变量X非常小的概率,即寻找Prob(|X|＜t)的上界.近年来,大偏差理论,也可以称为指数型小值概率的渐近计算理论,已经有了许多新的研究成果.已证实,大偏差理论是处理统计、工程、统计力学及应用概率领域中许多问题的重要工具.小偏差理论也被发现与Strassen迭代算法的收敛率及经验过程有密切联系.在大偏差方向上有许多优秀的研究成果,见文献.而在小偏差理论中许多很完美的成果可以在文献中找到.本篇文章所研究的概率问题是将上面提到过的概率中左面的随机变量替换为布朗运动,再将概率右面的简单区域替换为非随机函数及随机函数等各种复杂区域.但无论情况多么复杂,对于布朗运动在各种区域上的出逃概率的研究都是基于大偏差和小球概率理论.
本文第二部分内容是关于Mills率的研究.Mills率定义为一个超过某点的标准正态概率除以在这个点上的标准正态密度.对于Mills率的上、下界估计是由概率及统计等许多领域的研究所引起的.在各种背景下研究它也已经有很长的历史了.
Li研究了布朗运动在单一随机区域中的出逃概率.利用Slepian不等式给出了关于布朗运动在单一随机区域中的出逃概率的上、下界的渐近估计.我们改进了此方向问题的研究.在第2章中,我们考虑了布朗运动在极大极小随机区域中的出逃概率.并给出了这两种概率的上、下界的渐近估计.
Lifshits和Shi研究了布朗运动在单一随机椭球区域中出逃概率的上、下界的渐近估计,并证明了上、下界估计的渐近等价性.我们对此方向问题进行了更深入地研究.在第3章中,我们考虑了布朗运动在极大极小随机椭球区域中的出逃概率.最后得到了这两类概率的渐近估计,并进一步地证明了上、下界估计的渐近等价性.
Li,Lifshits和Shi分别研究了布朗运动在随机区域中出逃概率的渐近估计.我们改进了他们的研究.在第4章中,我们考虑了带有漂移项的布朗运动在随机椭球区域中的出逃概率问题.并给出了这类概率的渐近估计.
Li研究了布朗运动在无界凸区域中出逃概率的渐近估计,但是其中的上界估计在部分参数区间上不是精确的.基于他的研究,在第5章中,我们研究了布朗运动在无界凸区域中出逃概率的上界渐近估计.我们创造性地构造了一个高斯随机过程,将此过程应用到Slepian不等式中.最后结合高斯方法给出了一个一般性的概率上界,并说明了此上界在部分参数区间上优于已有上界估计.
第6章研究了一维Mills的上、下界估计和多维Mills率的上界估计.对于一维Mills率,非常精细的估计已经利用连分数表示出来,但没有一个简单明了的表达式.我们利用多项式的方法给出了精确的多项式表达式,改进了现有的估计结果.在多维Mills率问题上,利用在给定高斯密度下控制点的几何属性,我们给出了一个在一般凸区域中的多维高斯概率的上界估计.利用已得一维Mills率的估计验证了在部分参数区间上我们的上界估计要优于已有的估计.
In probability theory,the large deviation theory(or the tail probabilities) and the small deviation theory(or the small ball probabilities) are in a sense two complementary directions.The large deviation theory,which is a more classical direction,seeks to control the probability of deviation of a random variable X from its mean M,i.e.one looks for upper bounds on Prob(|X - M|＞t).The small deviation theory seeks to control the probability of X being very small,i.e.it looks for upper bounds on Prob(|X|＜t).In recent years,there has been renewed interest in the topic of large deviations,namely, the asymptotic computation of small probability on an exponential scale.The large deviations estimates have proved to be the crucial tool required to handle many questions in statistics,engineering,statistical mechanics,and applied probability.It has also been found that the small deviation estimates has close connections with the rate of convergence in Strassen's law of the iterated logarithm,and empirical processes.There are a number of excellent texts on large deviation,see e.g.recent books.A recent exposition of the state of the art in small deviation theory can be found in Li.The problem we consider is that replacing random variable X on the left side of the above probability by the Brownian motion,and replacing simple domain on the right side of the above probability by random domains.No matter how difficult,the research on the exit probability of a Brownian motion from various domains is based on the large deviation theory and the small deviation theory.
Another part in this paper is the study on the Mills' ratio.Mills' ratio is the standard normal probability beyond a certain point divided by the standard normal density at that point.Bounds on Mills' ratio arise naturally in many areas of probability and statistics. There is also a long history of studying them in various settings.
Li have studied the first exit time of a Brownian motion from a single random domain. We develop this problem.In Chapter 2,we study the first exit time of a Brownian motion from the minimum and maximum random domains and obtain the upper and lower estimates of this two probabilities.
Lifshits and Shi have studied the first exit time of a Brownian motion from a single parabolic domain.We develop this problem.In Chapter 3,we study the first exit time of a Brownian motion from the minimum and maximum parabolic random domains.Finally, we obtain the upper and lower estimates and also prove that they are still asymptotically equivalent.
Lifshits and Shi have studied the first exit time of a Brownian motion from a parabolic domain.We develop this problem.In Chapter 4,we study the first exit time of a Brownian motion with a drift from a random parabolic domain.Finally,the upper and lower estimates of the exit probability are obtained.We also prove that they are still asymptotically.
In Chapter 5,we study the first exit time of a Brownian motion from an unbounded convex domain.Li have studied this problem and obtained the upper and lower estimates. However,the upper estimates is not very sharp for some parameter.Based on his research,we construct a Gaussian process.Applying this process to Slepian's inequality and using Gaussian technique,a new upper estimates is obtained.For some parameter, the new upper estimates is more sharper than the old one.
In Chapter 6,we study the univariate Mills' ratio and the multivariate Mills' ratio.It is well-known that the univariate Mills' ratio can be estimated by the continued fraction, but the estimate is not simple.We give a formula by polynomial,and improving previous estimates.In addition,an upper bound of multivariate Gaussian probability for a general convex domain is given based on a geometric observation.By using the estimates of the univariate Mills' ratio,we checked that the bound is sharper than known ones on multivariate Mills' ratio in many case.
本文第二部分内容是关于Mills率的研究.Mills率定义为一个超过某点的标准正态概率除以在这个点上的标准正态密度.对于Mills率的上、下界估计是由概率及统计等许多领域的研究所引起的.在各种背景下研究它也已经有很长的历史了.
Li研究了布朗运动在单一随机区域中的出逃概率.利用Slepian不等式给出了关于布朗运动在单一随机区域中的出逃概率的上、下界的渐近估计.我们改进了此方向问题的研究.在第2章中,我们考虑了布朗运动在极大极小随机区域中的出逃概率.并给出了这两种概率的上、下界的渐近估计.
Lifshits和Shi研究了布朗运动在单一随机椭球区域中出逃概率的上、下界的渐近估计,并证明了上、下界估计的渐近等价性.我们对此方向问题进行了更深入地研究.在第3章中,我们考虑了布朗运动在极大极小随机椭球区域中的出逃概率.最后得到了这两类概率的渐近估计,并进一步地证明了上、下界估计的渐近等价性.
Li,Lifshits和Shi分别研究了布朗运动在随机区域中出逃概率的渐近估计.我们改进了他们的研究.在第4章中,我们考虑了带有漂移项的布朗运动在随机椭球区域中的出逃概率问题.并给出了这类概率的渐近估计.
Li研究了布朗运动在无界凸区域中出逃概率的渐近估计,但是其中的上界估计在部分参数区间上不是精确的.基于他的研究,在第5章中,我们研究了布朗运动在无界凸区域中出逃概率的上界渐近估计.我们创造性地构造了一个高斯随机过程,将此过程应用到Slepian不等式中.最后结合高斯方法给出了一个一般性的概率上界,并说明了此上界在部分参数区间上优于已有上界估计.
第6章研究了一维Mills的上、下界估计和多维Mills率的上界估计.对于一维Mills率,非常精细的估计已经利用连分数表示出来,但没有一个简单明了的表达式.我们利用多项式的方法给出了精确的多项式表达式,改进了现有的估计结果.在多维Mills率问题上,利用在给定高斯密度下控制点的几何属性,我们给出了一个在一般凸区域中的多维高斯概率的上界估计.利用已得一维Mills率的估计验证了在部分参数区间上我们的上界估计要优于已有的估计.
In probability theory,the large deviation theory(or the tail probabilities) and the small deviation theory(or the small ball probabilities) are in a sense two complementary directions.The large deviation theory,which is a more classical direction,seeks to control the probability of deviation of a random variable X from its mean M,i.e.one looks for upper bounds on Prob(|X - M|＞t).The small deviation theory seeks to control the probability of X being very small,i.e.it looks for upper bounds on Prob(|X|＜t).In recent years,there has been renewed interest in the topic of large deviations,namely, the asymptotic computation of small probability on an exponential scale.The large deviations estimates have proved to be the crucial tool required to handle many questions in statistics,engineering,statistical mechanics,and applied probability.It has also been found that the small deviation estimates has close connections with the rate of convergence in Strassen's law of the iterated logarithm,and empirical processes.There are a number of excellent texts on large deviation,see e.g.recent books.A recent exposition of the state of the art in small deviation theory can be found in Li.The problem we consider is that replacing random variable X on the left side of the above probability by the Brownian motion,and replacing simple domain on the right side of the above probability by random domains.No matter how difficult,the research on the exit probability of a Brownian motion from various domains is based on the large deviation theory and the small deviation theory.
Another part in this paper is the study on the Mills' ratio.Mills' ratio is the standard normal probability beyond a certain point divided by the standard normal density at that point.Bounds on Mills' ratio arise naturally in many areas of probability and statistics. There is also a long history of studying them in various settings.
Li have studied the first exit time of a Brownian motion from a single random domain. We develop this problem.In Chapter 2,we study the first exit time of a Brownian motion from the minimum and maximum random domains and obtain the upper and lower estimates of this two probabilities.
Lifshits and Shi have studied the first exit time of a Brownian motion from a single parabolic domain.We develop this problem.In Chapter 3,we study the first exit time of a Brownian motion from the minimum and maximum parabolic random domains.Finally, we obtain the upper and lower estimates and also prove that they are still asymptotically equivalent.
Lifshits and Shi have studied the first exit time of a Brownian motion from a parabolic domain.We develop this problem.In Chapter 4,we study the first exit time of a Brownian motion with a drift from a random parabolic domain.Finally,the upper and lower estimates of the exit probability are obtained.We also prove that they are still asymptotically.
In Chapter 5,we study the first exit time of a Brownian motion from an unbounded convex domain.Li have studied this problem and obtained the upper and lower estimates. However,the upper estimates is not very sharp for some parameter.Based on his research,we construct a Gaussian process.Applying this process to Slepian's inequality and using Gaussian technique,a new upper estimates is obtained.For some parameter, the new upper estimates is more sharper than the old one.
In Chapter 6,we study the univariate Mills' ratio and the multivariate Mills' ratio.It is well-known that the univariate Mills' ratio can be estimated by the continued fraction, but the estimate is not simple.We give a formula by polynomial,and improving previous estimates.In addition,an upper bound of multivariate Gaussian probability for a general convex domain is given based on a geometric observation.By using the estimates of the univariate Mills' ratio,we checked that the bound is sharper than known ones on multivariate Mills' ratio in many case.
引文
[1] Dembo A, Zeitouni O. Large Deviation Techniques and Applications. New York: Springer-Verlag, 1998.
[2] Hollander F D. Large Deviation. Providence: American Mathematical Society, 2000.
[3] Li W V, Shao Q M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic Processes: Theory and Methods. Handbook of Statistics, 2001, 19:533-598.
[4] Li W V. The first exit time of a brownian motion from an unbounded convex domain. The Annals of Probability, 2003, 31 (2): 1078-1096.
[5] Lifshits M A, Shi Z. The first exit time of brownian motion from a parabolic domain. Bernoulli, 2002, 8(6):745-765.
[6] Ledoux M, Talagrand M. Probability on Banach Space. New York: Springer-Verlag Berlin Heidelberg, 1991.
[7] Ledoux M. Isoperimetry and gaussian analysis. Lectures on Probability Theory and Statistics, 1996, 1648:165-294.
[8] Bogachev V I. Gaussian measure. New York: American Mathematical Society, 1998.
[9] Varadhan S R S. Large Deviations and Applications. Philadelphia: Society for Industrial and Applied Mathematics, 1984.
[10] Rockafellar R T. Convex Analysis. Princeton: Princeton University Press, 1970.
[11] Chung K L. On the maximum partial sums of sequences of independent random variables. Transactions of the American Mathematical Society, 1948, 64(2):205-233.
[12] Baldi P, Roynette B. Some exact equivalents for the brownian motion in hoder norm. Probability Theory and Related Fields, 1992, 93(4):457-484.
[13] Kuelbs J, Li W V. Small ball estimates for brownian motion and the brownian sheet. Journal of Theoretcal Probability, 1993, 6(3):547-577.
[14] Mogulskii A A. Small deviations in a space of tragectories. Theory of Probability and Its Applications, 1974, 19(4):726-736.
[15] Shi Z. Small ball probabilities for a wiener process under weighted sup-norm, with an application to the supremum of bessel local times. Journal of Theoretical Probability, 1996, 9(4):915-929.
[16] Berthet P, Shi Z. Small ball estimates for brownian motion under a weighted sup-norm. Studia Scientiarum Mathematicarum Hungarica, 2000, 36(1/2) :275-289.
[17] Novikov A A. On estimates and asymptotic behavior for non-exit probabilities of a wiener process to a moving boundary. Mathematics of the USSR-Sbornik, 1981, 38(4):495-505.
[18] Li W V. Small deviations for gaussian markov processes under the sup-norm. Journal of Theoretical Probability, 1999, 12(4):971-984.
[19] Li WV. A gaussian correlation inequality and its applications to small ball probabilities. Electronic Communications in Probability, 1999, 4:111-118.
[20] Biane P, Yor M. Valeurs principales associees aux temps locaux browniens. Bulletin Mathematique, 1987, 111(1):23-101.
[21] Revuz D, Yor M. Continuous Martingales and Brownian Motion. New York: Springer-Verlag Berlin Heidelberg, 1999.
[22] Gordon R D. Values of mill's ratio of area to bounding ordinate and of the normal probability intergal for large values of the argument. Annals of Mathematical Statistics, 1941, 12(3):364-366.
[23] Sampford M R. Some inequalities on mill's ratio and related functions. Annals of Mathematical Statistics, 1953, 24(1):130-132.
[24] Shenton L R. Inequalities for the normal integral including a new continued fraction. Biometrika, 1954, 41(1/2):177-189.
[25] Steck G P. Lower bounds for the multivariate normal mill's ratio. Annals of Probability, 1979, 7(3):547-551.
[26] Hashorva E, Husler J. On multivariate gaussian tails. Annals of the Institute of Statistical Mathematics, 2003, 55(3):507-522.
[27] Mills J P. Table of the ratio: area to bounding ordinate, for any portion of normal curve. Biometrika, 1926, 18(3/4):395-400.
[28] Birnbaum Z W. An inequality for mill's ratio. The Annals of Mathematical Statistics, 1942, 13(2):245-246.
[29] Hardy G H, Littlewood J E, Polya G. Inequalities. London: Cambridge University Press, 1934.
[30] Murty V N. On a result of birnbaum regarding the skewness of x in a bivariate normal population. Journal of the Indian Society of Agricultural Statistics, 1952, 4:85-87.
[31] Laplace M, Simon P. Traite de Mecanique Celeste. Paris: Bachelier, 1805.
[32] Ruben H. A new asymptotic expansion for the normal probability integral and mill's ratio. Journal of the Royal Statistical Society. Series B, 1962, 24(1):177-179.
[33] Franklin J, Friedman B. A convergent asymptotic representation for integrals. Mathematical Proceedings of the Cambridge Philosophical Society, 1957, 53(3):612—619.
[34] Ruben H. A convergent asymptotic expansion for mills' ratio and the normal probability integral in terms of rational functions. Mathematische Annalen, 1963, 151(4):355-364.
[35] Ruben H. Irrational fraction approximations to mills' ratio. Biometrika, 1964, 51(3/4):339-345.
[36] Kerridge D F, Cook G W. Yet another series for the normal integral. Biometrika, 1976, 63(2):401-403.
[37] Dudley R M. Some inequalities for continued fraction. Mathematics of Computation, 1987, 49(180) :585-593.
[38] Savage I R. Mills' ratio for multivariate normal distributions. Journal of research, National Bureau of Standards. Section B, 1962, 66(3):93-96.
[39] Gjacjauskas E. Estimation of the multidimensional normal probability law for the vanishing hyperangle. Lithuanian Mathematical Journal, 1973, 13(3):406-410.
[40] Satish I. On a lower bound for the multivariate normal mills' ratio. Annals of Probability, 1986, 14(4): 1399-1403.
[41] Gordon Y. Some inequalities for gaussian processes and applications. Israel Journal of Mathematics, 1985, 50(4):265-289.
[42]Ciesielski Z,Taylor S J.First passage time and sojourn times for brownian motion in space and the exact hausdorff measure of the sample path.Transactions of the American Mathematical Society,1962,103(3):434-450.
[43]Lu D,Li W V.A note on multivariate gaussian estimates.Journal of Mathematical Analysis and Applications,2009,354(2):704-707.
[44]Klugman S A,Panjer H H,Willmot G E.Loss Models:From Data to Decisions.New Jersey:John Wiley and Sons,Inc,1998.
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[52]Karatzas I,Shreve S E.Brownian Motion and Stochastic Calculus.New York:Springer-Verlag New York Inc,1988.
[2] Hollander F D. Large Deviation. Providence: American Mathematical Society, 2000.
[3] Li W V, Shao Q M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic Processes: Theory and Methods. Handbook of Statistics, 2001, 19:533-598.
[4] Li W V. The first exit time of a brownian motion from an unbounded convex domain. The Annals of Probability, 2003, 31 (2): 1078-1096.
[5] Lifshits M A, Shi Z. The first exit time of brownian motion from a parabolic domain. Bernoulli, 2002, 8(6):745-765.
[6] Ledoux M, Talagrand M. Probability on Banach Space. New York: Springer-Verlag Berlin Heidelberg, 1991.
[7] Ledoux M. Isoperimetry and gaussian analysis. Lectures on Probability Theory and Statistics, 1996, 1648:165-294.
[8] Bogachev V I. Gaussian measure. New York: American Mathematical Society, 1998.
[9] Varadhan S R S. Large Deviations and Applications. Philadelphia: Society for Industrial and Applied Mathematics, 1984.
[10] Rockafellar R T. Convex Analysis. Princeton: Princeton University Press, 1970.
[11] Chung K L. On the maximum partial sums of sequences of independent random variables. Transactions of the American Mathematical Society, 1948, 64(2):205-233.
[12] Baldi P, Roynette B. Some exact equivalents for the brownian motion in hoder norm. Probability Theory and Related Fields, 1992, 93(4):457-484.
[13] Kuelbs J, Li W V. Small ball estimates for brownian motion and the brownian sheet. Journal of Theoretcal Probability, 1993, 6(3):547-577.
[14] Mogulskii A A. Small deviations in a space of tragectories. Theory of Probability and Its Applications, 1974, 19(4):726-736.
[15] Shi Z. Small ball probabilities for a wiener process under weighted sup-norm, with an application to the supremum of bessel local times. Journal of Theoretical Probability, 1996, 9(4):915-929.
[16] Berthet P, Shi Z. Small ball estimates for brownian motion under a weighted sup-norm. Studia Scientiarum Mathematicarum Hungarica, 2000, 36(1/2) :275-289.
[17] Novikov A A. On estimates and asymptotic behavior for non-exit probabilities of a wiener process to a moving boundary. Mathematics of the USSR-Sbornik, 1981, 38(4):495-505.
[18] Li W V. Small deviations for gaussian markov processes under the sup-norm. Journal of Theoretical Probability, 1999, 12(4):971-984.
[19] Li WV. A gaussian correlation inequality and its applications to small ball probabilities. Electronic Communications in Probability, 1999, 4:111-118.
[20] Biane P, Yor M. Valeurs principales associees aux temps locaux browniens. Bulletin Mathematique, 1987, 111(1):23-101.
[21] Revuz D, Yor M. Continuous Martingales and Brownian Motion. New York: Springer-Verlag Berlin Heidelberg, 1999.
[22] Gordon R D. Values of mill's ratio of area to bounding ordinate and of the normal probability intergal for large values of the argument. Annals of Mathematical Statistics, 1941, 12(3):364-366.
[23] Sampford M R. Some inequalities on mill's ratio and related functions. Annals of Mathematical Statistics, 1953, 24(1):130-132.
[24] Shenton L R. Inequalities for the normal integral including a new continued fraction. Biometrika, 1954, 41(1/2):177-189.
[25] Steck G P. Lower bounds for the multivariate normal mill's ratio. Annals of Probability, 1979, 7(3):547-551.
[26] Hashorva E, Husler J. On multivariate gaussian tails. Annals of the Institute of Statistical Mathematics, 2003, 55(3):507-522.
[27] Mills J P. Table of the ratio: area to bounding ordinate, for any portion of normal curve. Biometrika, 1926, 18(3/4):395-400.
[28] Birnbaum Z W. An inequality for mill's ratio. The Annals of Mathematical Statistics, 1942, 13(2):245-246.
[29] Hardy G H, Littlewood J E, Polya G. Inequalities. London: Cambridge University Press, 1934.
[30] Murty V N. On a result of birnbaum regarding the skewness of x in a bivariate normal population. Journal of the Indian Society of Agricultural Statistics, 1952, 4:85-87.
[31] Laplace M, Simon P. Traite de Mecanique Celeste. Paris: Bachelier, 1805.
[32] Ruben H. A new asymptotic expansion for the normal probability integral and mill's ratio. Journal of the Royal Statistical Society. Series B, 1962, 24(1):177-179.
[33] Franklin J, Friedman B. A convergent asymptotic representation for integrals. Mathematical Proceedings of the Cambridge Philosophical Society, 1957, 53(3):612—619.
[34] Ruben H. A convergent asymptotic expansion for mills' ratio and the normal probability integral in terms of rational functions. Mathematische Annalen, 1963, 151(4):355-364.
[35] Ruben H. Irrational fraction approximations to mills' ratio. Biometrika, 1964, 51(3/4):339-345.
[36] Kerridge D F, Cook G W. Yet another series for the normal integral. Biometrika, 1976, 63(2):401-403.
[37] Dudley R M. Some inequalities for continued fraction. Mathematics of Computation, 1987, 49(180) :585-593.
[38] Savage I R. Mills' ratio for multivariate normal distributions. Journal of research, National Bureau of Standards. Section B, 1962, 66(3):93-96.
[39] Gjacjauskas E. Estimation of the multidimensional normal probability law for the vanishing hyperangle. Lithuanian Mathematical Journal, 1973, 13(3):406-410.
[40] Satish I. On a lower bound for the multivariate normal mills' ratio. Annals of Probability, 1986, 14(4): 1399-1403.
[41] Gordon Y. Some inequalities for gaussian processes and applications. Israel Journal of Mathematics, 1985, 50(4):265-289.
[42]Ciesielski Z,Taylor S J.First passage time and sojourn times for brownian motion in space and the exact hausdorff measure of the sample path.Transactions of the American Mathematical Society,1962,103(3):434-450.
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