Asymptotic behavior of market shares for a stochastic growth model

Consider a market with finitely many firms, each enjoying an uncertain growth. We assume the (random) growth rates of these firms are independent and identically distributed. We show that the asymptotic probability distribution of the market shares gives each firm an equal probability of dominating the market (while all other firms are almost extinguished). In particular, these conclusions are independent of the initial market shares of the firms. We conclude that sample paths fluctuate between almost domination and almost extinction. We also demonstrate that these fluctuations may be very slow. Extensions to situations where the total demand is restricted are discussed.