In this chapter, we study a random walk whose increments have a (right) heavy-tailed distribution with a negative mean. We also consider applications to queueing and risk processes.The maximum of such a random walk is almost surely finite, and our interest is in the tail asymptotics of the distribution of this maximum, for both infinite and finite time horizons; we are further interested in the local asymptotics for the maximum in the case of an infinite time horizon. We use direct probabilistic techniques and show that, under the appropriate subexponentiality conditions, the main reason for the maximum to be far away from zero is again that a single increment of the walk is similarly large.We present here two approaches for deriving such results, the first using a first renewal time at which the random walk exceeds a "tilted" level and the second using classical ladder epochs and heights. It turns out that the former approach is more direct since it is based on more elementary arguments, and we start with it in Sects. 5.1 and 5.2. In Sect. 5.1 we deal with the infinite time horizon and first obtain a general lower bound, and then the correct asymptotics, for the distribution of the maximum. Similar results for finite time horizons (with uniformity in time) are given in Sect. 5.2.We then turn to the classical ladder heights approach. This allows us to obtain both tail and local asymptotics for the maximum of the random walk in the case of an infinite time horizon. In Sects. 5.3 and 5.4 we recall known basic results on the ladder structure and on taboo renewal measures. In Sect. 5.5 we give results on bounds and asymptotics for both the tail and the local probabilities of the first ascending ladder height; this will lead to another proof of the tail asymptotics for the infinite-time maximum of the walk (Sect. 5.6) and also to the asymptotics for the local probabilities of the distribution of the maximum (Sect. 5.7). In Sect. 5.8 we present the asymptotics for the density of the maximum in the case where the density exists; this is also based on the ladder-heights representation. In each of three Sects. 5.6-5.8, we show that the corresponding condition on the distribution to belong to the appropriate class is not only sufficient but also necessary, for the desired asymptotics to hold. In Sect. 5.9, we consider the three particular cases where the S.

No abstract

We study conditions under whichwhere S τ is a sum ξ 1 + . . . + ξ τ of random size τ and M τ is a maximum of partial sums M τ = max n≤τ S n . Here ξ n , n = 1, 2, . . . , are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where τ is independent of the summands; also, in a particular situation, we deal with a stopping time.Also we consider the case where Eξ > 0 and where the tail of τ is comparable with or heavier than that of ξ, and obtain the asymptoticsThis case is of a primary interest in the branching processes.In addition, we obtain new uniform (in all x and n) upper bounds for the ratio P{S n > x}/P{ξ 1 > x} which substantially improve Kesten's bound in the subclass S * of subexponential distributions.

Abstract. In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability P{W > x}. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load.

No abstract

Abstract. We consider the sums Sn = ξ 1 + · · · + ξn of independent identically distributed random variables with negative mean value. In the case of subexponential distribution of the summands, the asymptotic behavior is found for the probability of the event that the maximum of sums max(S 1 , . . . , Sn) exceeds high level x. The asymptotics obtained describe this tail probability uniformly with respect to all values of n.

We present upper and lower bounds for the tail distribution of the stationary waiting time D in the stable GI/GI/s FCFS queue. These bounds depend on the value of the traffic load ρ which is the ratio of mean service and mean interarrival times. For service times with intermediate regularly varying tail distribution the bounds are exact up to a constant, and we are able to establish a "principle of s − k big jumps" in this case (here k is the integer part of ρ), which gives the most probable way for the stationary waiting time to be large .Another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time.

Abstract. We consider the sums S n = ξ 1 + · · · + ξ n of independent identically distributed random variables. We do not assume that the ξ's have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M > x} as x → ∞, provided that M = sup{S n , n ≥ 1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity.We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite-and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution F I .Keywords: supremum of sums of random variables, large deviation probabilities, subexponential distribution, integrated weighted tail distribution IntroductionLet ξ, ξ 1 , ξ 2 , . . . be independent random variables with common non-degenerate distribution F on the real line R. We let F (x) = F ((−∞, x]) and F (x) = 1−F (x). In general, for any distribution G, we denote its tail by G(x) = G((x, ∞)). In this paper, an important role is played by the negative truncated mean functionwhere ξ − = max{−ξ, 0}; the function m(x) is continuous, m(0) = 0 and m(x) > 0 for any x > 0. Put S 0 = 0, S n = ξ 1 + · · · + ξ n , and M = sup {S n , n ≥ 0}.