Probability background

Mathematical background of the Actuarial Science is Theory of Probability, Mathematical Statistics and Theory of Random Processes.

From the mathematical point of view, the Risk Theory, being highly appreciated by the insurance theory and practice, is no more than a subsection of Theory of Probability called "boundary problem of the random processes". The latter problem appears in diverse applications outside Economics.

The balance between the fundamental and applied knowledge was discussed by many generations. The Russian-speaking persons bear in mind the famous fable of I.A. Krylov written back in 19th century; the balance is illustrated in an inerasable manner:

The Russian (and the Soviet, as its part) probability school was created by P.L. Tchebyschev, A.A. Markov, A.M. Liapounov, S.N. Bernstein, A.N. Kolmogorov, B.V. Gnedenko, A.Ya. Khinchine, Yu.V. Linnik, L.N. Bolshev, A.V. Skorokhod, V.S. Koroljuk, and - in general - by the Moscow, Leningrad, Kiev, Novosibirsk, Vilnius, Tbilisi, Tashkent schools which collaborated a lot. It contributed considerably to the understanding by humankind of the Event of Randomness.

Telling about his own career, Harald Cramér, one of the founders of the Risk Theory and modern Actuarial Science, wrote a paper "Half a century with Probability theory: some personal recollections" (The Annals of Probability, 1976, Vol. 4, No. 4, 509-546;  the Russian translation of this paper was published in 1979), where the name of Kolmogorov was mentioned 42 times, of Khinchine - 20 times, and of Gnedenko - 6 times.

In particular, Cramér (on p. 517, in Section 3.4. "The new Russian school") wrote:

"All through the latter part of the 1920's, it was clear that a strong new activity in mathematical probability was taking place in the Soviet Union. In a remarkable paper of 1927, S.N. Bernstein discussed the extension of the central limit theorem to sums of random variables which are not required to be independent. He introduced an important method of dealing with such cases, to which I shall return later.
But the main Russian works of this period were written by two quite young mathematicians, A.Ya. Khintchine and A.N. Kolmogorov, who were to be leaders in the coming development of the field. In a joint paper of 1925 they proved the famous "three series theorem", giving necessary and sufficient conditions for the convergence of a series, the terms of which are independent random variables. The probability that such a series is convergent can only be equal to zero or one, which is a particular case of the so called "zero-one law," discovered at the same time...",

an further (on p. 535, in Section 6.4. "Moscow, 1955"):

"In May 1955 the University of Moscow celebrated its bicentennial, and I was invited to attend, representing the Stockholm University. It was a great event, and it gave me an opportunity to make the personal acquaintance of the Soviet mathematicians, whose work had meant so much for the advancement of probability theory. Unfortunately, Khintchine was ill - he died shortly afterwards - but I met Kolmogorov, who gave the impression of being a great scientific personality, and I had some interesting conversations with him. I was also happy to meet other members of their probabilistic group. There was Dynkin, who was beginning his great work on Markov processes, Gnedenko who in collaboration with Kolmogorov had written the book on limit problems referred to above, Linnik who was beginning his work on large deviations so closely connected with my own work of 1937, Yaglom and Rozanov who were to do outstanding work on stationary processes, and many others. They formed a group of wonderful scientific activity, and were preparing to start their new Journal of Probability Theory and its Applications, which soon acquired an internationally leading position in the field.

In connection with this Bicentennial there was also a mathematical conference, and I gave a lecture on my recent work on the ruin problem. Later in the same year, Kolmogorov spent some time in Stockholm as the guest of our university, and gave a series of lectures on limit theorems in probability to our group".

Though many of the above mentioned colleagues have passed away, their works are available for those who wish to study the Probability Theory.

The journal "Theory of Probability and its Applications" founded by Kolmogorov is published quarterly since 1956 and until now. Many prominent works have appeared in this journal; highly professional papers are published there nowadays.

The highly professional Encyclopedia "Probability and Mathematical Statistics" has appeared in 1999. It fixes the current state of arts in this field.

The Russian library of professional monographs covering many aspects of the Probability Science is very large. It comprises both original monographs and translations. In 1996 the First World Congress of the Bernoulli Society was held in Tashkent (Uzbekistan). Useful may be the catalogue which makes a rather complete list of the probability publications, mainly from the publishers "Mir", "Nauka", "Finansy i Statistika".