When studying for a degree at University College London, Xu was familiar with Clem's random variables and probability distribution (published in 1937) and mastered the tools of characteristic function, so he was very interested in limit theory. 1947, he and Robbins co-authored the paper Total Convergence and Law of Large Numbers, which introduced the concept of total convergence for the first time. At that time, the main international interest in probability was the limit distribution of the sum of independent random variables, which was transforming from classical to modern results. Some famous probability experts, such as Colmo Gorov, Qin Xin, Gnedenko, Levi and Ferrer, are attacking this problem. In 1947, the main result is obtained: the necessary and sufficient conditions for the row sum of triangular array of infinitesimal random variables with independent rows to converge to the given infinite separable law in distribution. Because of the lack of information at that time, he didn't understand other people's work. At that time, he wrote to Zhong Kailai and said ... I am worried that the ongoing work will be more important than others ... "Later, when he learned about the work of Gnedenko and Colmo Gorov, he stopped publishing his own research. In fact, Xu's method is different from that of the Russians, and Xu's method is more direct. 1968, when the English version of Limit Distribution of the Sum of Independent Random Variables written by Gnedenko and Colmo Gorov was reprinted, Xu's work was printed as an appendix for the first time. But Xu didn't see this book before his death, let alone the publication of this part of his works.

In the mid-1950s, Xu was interested in Markov processes. He studied the differentiability of transition probability function in jump process by pure analysis method. He did some limit theorems of Markov chains, but they were not published, and the discussion class was interrupted by the Great Leap Forward. After 1959, his interest turned to combinatorial design. It should also be mentioned that he finished a paper in 1945. In this paper, the distribution of two highly correlated random variables is approximated by the characteristic function method for the first time, and the asymptotic expansion of sample variance and the estimation of residual term are given. The difficulty here is to deal with two-dimensional distribution, which is a problem of mathematical statistics, but methods and tools are commonly used characteristic functions in probability theory. Since 1970s, this work has aroused many in-depth international studies.

Xu's interest in combinatorial mathematics in his later years was caused by his triangular scheme. He thinks that the matrix method can be systematically introduced into combinatorial mathematics. From 196 1, he presided over a seminar on experimental design, reported this work, carried out research, and published an article in advances in mathematics under his pseudonym, which is the result of this seminar. In this paper, a matrix lemma is used to deal with the uniqueness and non-uniqueness of various formats (including Zhang Liqian's results) with υ=2. Later, at the beginning of 1966, he systematically reported the work of BIB in the discussion class. During the Cultural Revolution, he did not stop studying. At that time, he couldn't see any magazines. It was not until 1970 that he was paralyzed and was allowed to read magazines. In two months, he read all the chronicles of mathematical statistics after 1966 "Cultural Revolution", understood the international academic trends, wrote the last paper on bibs and coding, and entrusted the manuscript of this article to Professor Duan Xuefu.

Xu is very talented and has a strong ability to master foreign languages. He studied French in his spare time in middle school and was able to write short articles and dialogues two years later. Besides English in class, he also taught himself German and Russian. In the early days of liberation, in order to translate a large number of Soviet textbooks, he studied hard by himself, and in a short time he was able to translate some important textbooks, such as Fichkin Goldz's Three-volume Course of Calculus and Gnedenko's Course of Probability. He also mastered many teaching contents by himself, such as Lebesgue integral, measure theory, functional analysis and so on. Apart from his talent, his hard work and perseverance are important reasons for his achievements. When I was teaching in The National SouthWest Associated University, Kunming, I lived in poverty and lacked materials. Sometimes it is difficult to find reference books. On his bookshelf, he copied the whole book "Function Theory" by hand.

In the 1950s, he was already a famous professor. Once he saw a good book, he read it carefully and did a lot of exercises. He tried to answer the exercises of Tang Song's Theory of Real Variable Functions and Anderson's Introduction to Multivariate Statistical Analysis chapter by chapter. He can deepen some exercises and turn them into small research exercises, and some can become papers. He is very strict about the publication of papers. He once said: "I don't want my article to be famous in famous magazines." I want the magazine to be famous for my articles. " Although he is a member of the department and can recommend papers for publication in Science Journal as soon as possible, most of his own papers are published in Journal of Peking University. Some of his papers are as long as dozens of pages, while others are as short as a little more than one page. It's all about solving problems, being unpretentious and concise. There are only more than 30 papers published in his life, but most of them are very weighty works. He often writes down some small results in the margins of books and thinks they are not worth publishing.

In 1962, when he taught in the discussion class that the quadratic distribution of normal variables is a necessary and sufficient condition for x2, he had already dealt with the degenerate situation. This result was published abroad as a separate paper in 1966. He is very serious about the quoted results, and he must be able to prove them completely. In his research work, two points are very obvious. One is the pursuit of elementary proof. He thinks that elementary methods are more meaningful than difficult methods, so his lectures can attract many people to listen. His analysis of the problem is very clear, and the solution of the problem seems natural and easy. Another feature is that the proof requirement is calculus, without any geometric intuition. In order to fully illustrate his point of view, in the winter of 1964, when he discussed the systematic teaching of point set topology in his class, every proof was derived from set operation, and was later interrupted by the socialist education movement. In teaching, he advocates that "good works show conciseness" and should explain original and true ideas to students, while in form, he tries to be concise in the method of proof without using redundant words. His lecture is a perfect combination of profound thoughts and perfect forms. His Chinese and foreign students praised him: "His lecture was perfect." As a teacher and scientist, he has great influence on students and colleagues. Someone recalled: "Xu insisted on explaining things in simple terms and never avoided difficulties." In particular, the highest goal and academic level of calm, clear and silent dedication attracted us. "

He stubbornly engaged in scientific research and teaching for a long time despite his illness, and worked for the scientific cause of the motherland to the last breath. 1970 12 18 in the morning, he died in Peking University.

After springer Publishing House published The Complete Works of Xu, there was a sentence in the book review:

Xu is recognized as the first mathematician in China who enjoys an international reputation in mathematical statistics and probability theory.