马尔可夫Andrei Markov

马尔可夫的母亲是Nadezhda Petrovna，她是一位国家工作人员的女儿，他的父亲是马尔可夫 Grigorievich 马尔可夫，一位乡村执事的儿子。马尔可夫 Grigorievich 马尔可夫曾在教会神学院学习，之后找到了一份文书的工作。全家搬到了圣彼得堡，在那里马尔可夫 Grigorievich在林业部门任职，后来成为多个家庭和地产的管理人。马尔可夫 Grigorievich结过两次婚；与第一任妻子Nadezhda育有两个儿子和几个女儿。马尔可夫 Andreyevich是两个男孩中较大的一个，较小的是Vladimir。尽管Vladimir在25岁时死于肺结核，但他已经作为数学家赢得了国际声誉。

早年，马尔可夫身体不好，直到十岁都只能靠拐杖行走。他在圣彼得堡文理中学第五中学接受中等教育，在那里他展现出卓越的数学才能，但在其他科目上表现相当差。他在中学期间写了他的第一篇数学论文，但论文中关于线性微分方程积分的结果并不新颖。然而，写这篇论文确实使他结识了亚历山大·科尔金和叶戈尔·佐洛塔廖夫，这两位是大学里的顶尖教授。显然，数学是马尔可夫在大学学习的正确科目，1874年，他进入了圣彼得堡大学的物理数学系。在那里，他参加了由亚历山大·科尔金和叶戈尔·佐洛塔廖夫主持的研讨班，但也听了数学系主任巴夫尼提·列波维奇·切比雪夫的讲座。这些对马尔可夫特别有启发，因为巴夫尼提·列波维奇·切比雪夫经常通过提出新问题和难题供学生研究来鼓励研究氛围。

马尔可夫于1878年毕业，因提交了当年院系设定的奖项主题——On the integration of differential equations by means of 连分数的最佳论文而获得金牌。他现在希望接受培训成为一名大学教授，并在接下来的两年里攻读硕士学位（这相当于博士学位的水平）。他于1880年因其学位论文On the binary quadratic forms with positive determinant而获得该学位。这篇学位论文非常出色[4]：-

这项工作受到巴夫尼提·列波维奇·切比雪夫的高度评价，代表了数论的圣彼得堡学派最杰出的成就之一，甚至可能是整个俄罗斯数学最杰出的成就之一。只需回想一下当时法国和德国最杰出的数论学家所关注的有理逼近领域中的那些问题，就能体会到马尔可夫在该领域中深入了多少。因此，尽管这篇学位论文立即发表（以法语发表在Mathematische Annalen上），但它直到1910年至1920年代柏林数学家费迪南德·格奥尔格·弗罗贝尼乌斯和罗伯特·雷马克试图掌握马尔可夫工作中所包含的那套思想时，才被西欧数学家普遍吸收，这或许并不令人惊讶。

提交硕士学位论文后，马尔可夫 开始在圣彼得堡大学担任 privatdozent 并攻读博士学位（相当于 教授资格论文（Habilitation））。1884年，他凭借学位论文 On certain applications of continued fractions 获得博士学位。

马尔可夫从小就认识Maria Ivanova Valvatyeva，因为她是他父亲所管理的地产主人的女儿。马尔可夫曾辅导Maria Ivanova数学，后来向她求婚。然而，Maria Ivanova的母亲不允许女儿嫁给她的地产管理人的儿子，直到马尔可夫获得了足够的社会地位。1883年，Maria Ivanova的母亲同意了这桩婚事，婚礼于当年举行。

马尔可夫于1886年成为圣彼得堡大学的编外教授，1893年成为正式教授。巴夫尼提·列波维奇·切比雪夫于1886年提议马尔可夫为俄罗斯科学院的通讯院士。他于1890年当选为编外院士，1896年当选为正式院士。他于1905年正式退休，但一生中大部分时间继续授课。

马尔可夫 的早期工作主要在数论与分析、代数连分数、积分的极限、逼近论和级数的收敛性方面。1900年后，马尔可夫 将他老师 巴夫尼提·列波维奇·切比雪夫 开创的连分数方法应用于 概率论 [4]：-

马尔可夫 是 巴夫尼提·列波维奇·切比雪夫 在概率论中的思想和研究方向的最优雅的代言人。尤其引人注目的是他关于 雅各布·伯努利 定理（称为大数定律）的研究、关于 巴夫尼提·列波维奇·切比雪夫 提出的概率论两个基本定理的研究，以及关于最小二乘法的研究。

他还研究了相互依赖变量的序列，希望以最一般的形式建立概率的极限定律。他在相当一般的假设下证明了中心极限定理。马尔可夫 尤其因对 马尔可夫 链的研究而被铭记，这种随机变量序列中，未来的变量由当前的变量决定，但与当前状态如何从其前驱状态产生的方式无关。这项工作开创了概率论的一个全新分支，并启动了随机过程理论。1923年，诺伯特·维纳 成为第一个严格处理连续 马尔可夫 过程的人。一般理论的基础在1930年代由 安德雷·柯尔莫哥洛夫 提供。

谢尔盖·伯恩斯坦 继续发展 马尔可夫 链的理论，他写道（例如见 [4]）：-

马尔可夫 关于概率计算的经典课程，以及他的原创论文，作为准确性和阐述清晰性的典范，在很大程度上促进了概率论转变为数学中最完善的领域之一，并促进了 巴夫尼提·列波维奇·切比雪夫 的方法和研究方向的广泛传播。他以 巴夫尼提·列波维奇·切比雪夫 的精神对观察到的随机现象之间的依赖性进行的深刻分析，使 马尔可夫 能够通过引入和研究相依随机量，以本质性的方式扩展概率论。

马尔可夫也对诗歌感兴趣，并对诗歌风格进行了研究——也许令人惊讶的是，安德雷·柯尔莫哥洛夫有着相似的兴趣。然而，值得指出的是，尽管马尔可夫发展了他的马尔可夫链理论作为纯数学工作，没有考虑物理应用，但他确实将这些思想应用于文学文本中的两种状态链，即元音和辅音。因此，他对诗歌的兴趣并非完全独立于他的数学工作。

作为一名讲师，马尔可夫对他的学生要求很高[1]：-

他的讲座以无可挑剔的论证严谨性而著称，他在学生中培养了那种不把任何事情视为理所当然的数学思维方式。他在课程中纳入了许多最近的研究成果，同时常常省略传统问题。讲座很难，只有认真的学生才能理解。……在讲座期间，他不关心黑板上方程的顺序，也不关心自己的外表。

马尔可夫经历了俄罗斯政治活动激烈的时期，由于持有坚定的观点，他深深卷入其中。马克西姆·高尔基，俄罗斯短篇小说作家、小说家和左翼活动家，于1902年当选为俄罗斯科学院成员，但他的当选很快因政治原因被沙皇下令撤销。马尔可夫强烈抗议，并拒绝接受次年授予他的荣誉。1907年6月，沙皇尼古拉解散了以左翼多数当选的第二杜马。马尔可夫拒绝了他的成员资格，可能预料会遭受严重后果，但当局选择不以一位年迈而杰出的院士为例。1913年，自1613年以来在俄罗斯掌权的罗曼诺夫王朝庆祝了他们300年的权力。这不太可能改善他们已经虚弱的地位。马尔可夫通过举行自己的庆祝活动来表示他对庆祝活动的不赞成——他庆祝了大数定律200周年！俄罗斯革命于1917年初开始，因为食品供应短缺。那年9月，马尔可夫请求科学院将他派往俄罗斯内陆一个贫困城镇。他被派往扎赖斯克，一个小乡村城镇，在那里他在中学教数学，没有收到任何报酬。他回到圣彼得堡，但他的健康现在正在恶化，他做了一次眼部手术。尽管到1921年他的状况如此糟糕，几乎无法站立，但他继续在大学讲授概率论。他于1922年7月去世，之前经历了数月最严重的痛苦。

马尔可夫有一个儿子（同名），生于1903年9月9日，追随父亲也成为了一位著名数学家。

Andrei Andreyevich Markov's mother was Nadezhda Petrovna, who was the daughter of a state worker, and his father was Andrei Grigorievich Markov, the son of a country deacon. Andrei Grigorievich Markov studied at a church seminary, then got a job as a clerk. The family moved to St Petersburg where Andrei Grigorievich served in the Forestry Department and then became a manager of various households and estates. Andrei Grigorievich married twice; with his first wife Nadezhda he had two sons and several daughters. Andrei Andreyevich was the oldest of the two boys while the younger was Vladimir. Although Vladimir died from tuberculosis at the age of 25, he had already gained an international reputation as a mathematician.

In his early years Markov was in poor health and up to the age of ten he could only walk with the assistance of crutches. His secondary schooling was at St Petersburg Gymnasium No 5 where he showed outstanding talents for mathematics but performed rather poorly in other subjects. He wrote his first mathematics paper while at the Gymnasium but his results on integration of linear differential equations which were presented in the paper were not new. However, writing the paper did result in him meeting Korkin and Zolotarev, two of the leading professors at the university. It was clear that mathematics was the right subject for Markov to study at university and, in 1874, he entered the Physics and Mathematics Faculty of St Petersburg University. There he enrolled in the seminar run by Korkin and Zolotarev but also attended lectures by Chebyshev, the head of the mathematics department. These were particularly stimulating to Markov, since Chebyshev often encouraged an atmosphere of research by posing new questions and problems for his students to investigate.

Markov graduated in 1878 having won the gold medal for submitting the best essay for the prize topic set by the faculty in that year - On the integration of differential equations by means of continued fractions. He now wished to train to become a university professor and worked for his Master's degree over the next two years (this was at a level equivalent to a doctorate). He was awarded the degree in 1880 for his thesis On the binary quadratic forms with positive determinant. This thesis was outstanding [4]:-

This work, very highly esteemed by Chebyshev, represents one of the finest achievements of the St Petersburg school of number theory, and perhaps even of all Russian mathematics. It is enough to recall the sorts of questions in the field of rational approximation which at that time preoccupied the most prominent number theorists of France and Germany, to appreciate how much deeper into the field Markov had penetrated. it is therefore perhaps not surprising that, although the dissertation was published immediately (in French in Mathematische Annalen), it did not become generally absorbed by west European mathematicians, until from 1910 to the 1920s the Berlin mathematicians Frobenius and Remak attempted to master the set of ideas contained in Markov's work.

After submitting his master's thesis, Markov began to teach at St Petersburg University as a privatdozent while working for his doctorate (equivalent to the habilitation). He was awarded his doctorate in 1884 for his dissertation On certain applications of continued fractions.

Markov had known Maria Ivanova Valvatyeva since they were children for she was the daughter of the owner of the estate which his father was managing. Markov had tutored Maria Ivanova in mathematics and later he proposed marriage to her. However Maria Ivanova's mother would not allow her daughter to marry the son of her estate manager until Markov had gained sufficient social status. In 1883 Maria Ivanova's mother agreed to the marriage which took place in that year.

Markov became an extraordinary professor at St Petersburg University in 1886 and an ordinary professor in 1893. Chebyshev proposed Markov as an adjunct of the Russian Academy of Sciences in 1886. He was elected as an extraordinary member in 1890 and an ordinary academician in 1896. He formally retired in 1905 but continued to teach for most of his life.

Markov's early work was mainly in number theory and analysis, algebraic continued fractions, limits of integrals, approximation theory and the convergence of series. After 1900 Markov applied the method of continued fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory [4]:-

Markov was the most elegant spokesman for Chebyshev's ideas and directions of research in probability theory. Especially remarkable is his research relating to the theorem of Jacob Bernoulli known as the Law of Large Numbers, to two fundamental theorems of probability theory due to Chebyshev, and to the method of least squares.

He also studied sequences of mutually dependent variables, hoping to establish the limiting laws of probability in their most general form. He proved the central limit theorem under fairly general assumptions. Markov is particularly remembered for his study of Markov chains, sequences of random variables in which the future variable is determined by the present variable but is independent of the way in which the present state arose from its predecessors. This work founded a completely new branch of probability theory and launched the theory of stochastic processes. In 1923 Norbert Wiener became the first to treat rigorously a continuous Markov process. The foundation of a general theory was provided during the 1930s by Andrei Kolmogorov.

Sergei Bernstein, who continued to develop the theory of Markov chains, wrote (see for example [4]):-

A A Markov's classic course on the computation of probabilities, and his original memoirs, models of accuracy and clarity of exposition, contributed to a very large extent to the transformation of the theory of probability into one of the most perfected areas of mathematics, and to the wide dissemination of Chebyshev's methods and directions of research. His profound analysis in the spirit of Chebyshev of the dependencies among observed random phenomena allowed Markov to extend probability theory in an essential way through the introduction and investigation of dependent random quantities.

Markov was also interested in poetry and he made studies of poetic style - perhaps surprisingly Kolmogorov had similar interests. It is worth pointing out, however, that although Markov developed his theory of Markov chains as a purely mathematical work without considering physical applications, he did apply the ideas to chains of two states, namely vowels and consonants, in literary texts. His interest in poetry was not, therefore, an entirely separate interest from his mathematical work.

As a lecturer, Markov demanded much of his students [1]:-

His lectures were distinguished by an irreproachable strictness of argument, and he developed in his students that mathematical cast of mind that takes nothing for granted. He included in his courses many recent results of investigations, while often omitting traditional questions. The lectures were difficult, and only serious students could understand them. ... During his lectures he did not bother about the order of equations on the blackboard, nor about his personal appearance.

Markov lived through a period of great political activity in Russia and, having firm opinions, he became heavily involved. Maksim Gorky, the Russian short-story writer, novelist and left wing activist, was elected a member of the Russian Academy of Sciences in 1902, but his election was soon withdrawn for political reasons on the Tsar's orders. Markov protested strongly and refused to accept honours awarded him on the following year. In June 1907 Tsar Nicholas dissolved the Second Duma which had been elected with majority on the left. Markov repudiated his membership and might have expected to suffer severe consequences but the authorities chose not to make an example of an elderly and distinguished academician. In 1913 the Romanov dynasty, which had been in power in Russia since 1613, celebrated their 300 years of power. This was not likely to improve their already weak position. Markov showed his disapproval of the celebration but holding celebrations of his own - he celebrated 200 years of the Law of Large Numbers! The Russian Revolution began early in 1917 as food supplies ran low. In September of that year Markov requested the Academy to send him to a disadvantaged town in the Russian interior. He was sent to Zaraisk, a small country town, where he taught mathematics in the secondary school without receiving any remuneration. He returned to St Petersburg but his health was now deteriorating and he had an eye operation. Although by 1921 he was in such a bad way that he was hardly able to stand, yet he continued to lecture on probability at the university. His death in July 1922 came after months of the most severe suffering.

Markov had a son (of the same name) who was born on 9 September 1903 and followed his father in also becoming a renowned mathematician.