The Harald Cramér Volume

Abstract

Harald Cramér was nominated Professor in Actuarial Mathematics and Mathematical Statistics at the University of Stockholm in the late twenties. 1959 a book was “dedicated to Harald Cramér on his sixty-fifth birthday, in recognition of his pathbreaking work in probability and statistics”, with the title Probability and Statistics, The Harald Cramér Volume (Almqvist & Wiksell, Uppsala 1959; John Wiley & Sons, New York). The development during these three decades can be described in Cramér's own words in a recently published „Éloge de M. Paul Lévy.” “Pendant ces années” (the late twenties) “la théorie moderne des probabilités a commencé ce développement puissant qui, depuis cette époque, a continué sans cesse, et qui a fait de la théorie des probabilités une des branches les plus fécondes de l'analyse moderne, ayant les rapports intimes avec tous les domaines des mathématiques pures, et en même temps des applications pratiques dont l'étendue et l'importance vont toujours en croissant.”A few historical notes will be given as a background for the review of The. Harald Cramér Volume, 1903-1934 Filip Lundberg formed the collective theory of risk. This theory is a particular part of the general theory of stochastic processes which latter was drafted, in its main lines, by Kolmogoroff, 1931 and 1933. (A stochastic process is constituted by a variable depending on chance which is simultaneously a function of one or more variates. For readers not used to this terminology such a variable may be exemplified with the total risk costs for a certain period paid for a group of policies issued by a fire insurance company. For simplicity, it is assumed that the group involves only agricultural risks, similar to each other and situated in the open country so that no conflagration can occur. Further we assume that during the period of observation no change occurs in the composition of the group neither through underwriting, nor through cancellation, nor through alteration.