Estimating tails of independently stopped random walks using concave approximations of hazard functions

This paper considers logarithmic asymptotics of tails of randomly stopped sums. The stopping is assumed to be independent of the underlying random walk. First, finiteness of ordinary moments is revisited. Then the study is expanded to more general asymptotic analysis. Results are applicable to a large class of heavy-tailed random variables. The main result enables one to identify if the asymptotic behaviour of a stopped sum is dominated by its increments or the stopping variable. As a consequence, new sufficient conditions for the moment determinacy of compounded sums are obtained.

Logarithmic Asymptotics for Probability of Component-Wise Ruin in a Two-Dimensional Brownian Model

We consider a two-dimensional ruin problem where the surplus process of business lines is modelled by a two-dimensional correlated Brownian motion with drift. We study the ruin function P ( u ) for the component-wise ruin (that is both business lines are ruined in an infinite-time horizon), where u is the same initial capital for each line. We measure the goodness of the business by analysing the adjustment coefficient, that is the limit of - ln P ( u ) / u as u tends to infinity, which depends essentially on the correlation ρ of the two surplus processes. In order to work out the adjustment coefficient we solve a two-layer optimization problem.

Large deviations of means of heavy-tailed random variables with finite moments of all orders

Logarithmic asymptotics of the mean process {Sn∕n} are investigated in the presence of heavy-tailed increments. As a consequence, a full large deviations principle for means is obtained when the hazard function of an increment is regularly varying with index α∈(0,1). This class includes all stretched exponential distributions. Thus, the previous research of Gantert et al. (2014) is extended. Furthermore, the presented proofs are more transparent than the techniques used by Nagaev (1979). In addition, the novel approach is compatible with other common classes of distributions, e.g. those of lognormal type.

Laplace transform asymptotics and large deviation principles for longest success runs in Bernoulli trials

The longest stretch L(n) of consecutive heads in n independent and identically distributed coin tosses is seen from the prism of large deviations. We first establish precise asymptotics for the moment generating function of L(n) and then show that there are precisely two large deviation principles, one concerning the behavior of the distribution of L(n) near its nominal value log1∕pn and one away from it. We discuss applications to inference and to logarithmic asymptotics of functionals of L(n).

Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

Let W = {Wn: n ∈ N} be a sequence of random vectors in Rd, d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W, that is, for any vector q > 0 in Rd, we find that logP(there exists n ∈ N: Wnuq) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {an}, {vn}, (1 / vn)logP(Wn / an ≥ uq) has a, continuous in q, limit JW(q). We allow the scalings {an} and {vn} to be regularly varying with a positive index. This approach is general enough to incorporate sequences W, such that the probability law of Wn / an satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.

Logarithmic Asymptotics for Multidimensional Extremes Under Nonlinear Scalings

Let W = { W n : n ∈ N} be a sequence of random vectors in R d , d ≥ 1. In this paper we consider the logarithmic asymptotics of the extremes of W , that is, for any vector q > 0 in R d , we find that logP(there exists n ∈ N: W n u q ) as u → ∞. We follow the approach of the restricted large deviation principle introduced in Duffy (2003). That is, we assume that, for every q ≥ 0, and some scalings {a n }, {v n }, (1 / v n )logP( W n / a n ≥ u q ) has a, continuous in q , limit J W ( q ). We allow the scalings {a n } and {v n } to be regularly varying with a positive index. This approach is general enough to incorporate sequences W , such that the probability law of W n / a n satisfies the large deviation principle with continuous, not necessarily convex, rate functions. The equations for these asymptotics are in agreement with the literature.