We introduce a recursive algorithm which enables the computation of the distribution of epidemic size in a stochastic SIR model for very large population sizes. In the important parameter region where the model is just slightly supercritical, the distribution of epidemic size is decidedly bimodal. We find close agreement between the distribution for large populations and the limiting case where the distribution is that of the time a Brownian motion hits a quadratic curve. The model includes the possibility of vaccination during the epidemic. The effects of the parameters, including vaccination level, on the form of the epidemic size distribution are explored.

Stockholm University, Faculty of Science, Department of Mathematics.

A Maxtrimmed St. Petersburg Game2016In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 29, no 1, p. 277-291Article in journal (Refereed)

Abstract [en]

Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that s(n)/n log(2)n ->(p) 1 as n -> infinity It is also known that almost sure convergence fails. However, Csorgo and Simons (Stat Probab Lett 26: 65-73, 1996) have shown that almost sure convergence holds for trimmed sums, that is, for S-n - max(1 <= k <= n) X-k. Since our actual distribution is discrete there may be ties. Our main focus in this paper is on the maxtrimmed sum, that is, the sum trimmed by the random number of observations that are equal to the largest one. We prove an analog of Martin-Lof's (J Appl Probab 22: 634-643, 1985) distributional limit theorem for maxtrimmed sums, but also for the simply trimmed ones, as well as for the total maximum. In a final section, we interpret these findings in terms of sums of (truncated) Poisson random variables.

Let S-n, n >= 1, be the successive sums of the payoffs in the classical St. Petersburg game. The celebrated Feller weak law states that S-n/(n log(2) n) ->(P) 1 as n ->infinity. In this paper we review some earlier results of ours and extend some of them as we consider an asymmetric St. Petersburg game, in which the distribution of the payoff X is given by P(X = sr(k-1)) = pq(k-1), k = 1, 2,..., where p + q = 1 and s, r > 0. Two main results are extensions of the Feller weak law and the convergence in distribution theorem of Martin-Lof (1985). Moreover, it is well known that almost-sure convergence fails, though Csorgo and Simons (1996) showed that almost-sure convergence holds for trimmed sums and also for sums trimmed by an arbitrary fixed number of maxima. In view of the discreteness of the distribution we focus on 'max-trimmed sums', that is, on the sums trimmed by the random number of observations that are equal to the largest one, and prove limit theorems for simply trimmed sums, for max-trimmed sums, as well as for the 'total maximum'. Analogues with respect to the random number of summands equal to the minimum are also obtained and, finally, for joint trimming.

Stockholm University, Faculty of Science, Department of Mathematics.

Extreme-trimmed St. Petersburg games2015In: Statistics and Probability Letters, ISSN 0167-7152, E-ISSN 1879-2103, Vol. 96, p. 341-345Article in journal (Refereed)

Abstract [en]

Let S-n, n >= 1, describe the successive sums of the payoffs in the classical St. Petersburg game. Feller's famous weak law, Feller (1945), states that s(n)/n log(2) n (sic) 1 as n -> infinity. However, almost sure convergence fails, more precisely, lim supn ->infinity S-n/n log(2) n = +infinity a.s. and lim inf(n ->infinity) S-n/n log(2) n = 1 a.s. as n -> infinity. Csorgo and Simons (1996) have shown that almost sure convergence holds for trimmed sums, that is, for S-n - max(1 <= k <= n) X-k and, moreover, that this remains true if the sums are trimmed by an arbitrary fixed number of maximal sums. A predecessor of the present paper was devoted to sums trimmed by the random number of maximal summands. The present paper concerns analogs for the random number of summands equal to the minimum, as well as analogs for joint trimmings.

We study the maximum of a Brownian motion with a parabolic drift; this is a random variable that often occurs as a limit of the maximum of discrete processes whose expectations have a maximum at an interior point. We give new series expansions and integral formulas for the distribution and the first two moments, together with numerical values to high precision.

7.

Martin-Löf, Anders

Stockholm University, Faculty of Science, Department of Mathematics.

Two modifications of the Petersburg game are considered: (1) truncation, so that the player has a finite capital at his disposal and (2) a cost of borrowing capital, so that the player has to pay interest on the capital needed. In both cases, limit theorems for the total net gain are derived, so that it is easy to judge if the game is favourable or not.

The boook is a compilation of 21 of the papers presented at the International Cramér Symposium on Insurance Mathematics (ICSIM) held at Stockholm University 0n 11-14 June, 2013.