Probabilities of Large Deviations of Sums of Independent Random Variables with Common Distribution Function in the Domain of Attraction of the Normal Law

Large deviations perspective on ordinal optimization of heavy-tailed systems

“…For extensions to other types of heavy-tailed random variables, see for instance, Nagaev (1979) and Rozovskii (1993). A review is given by Mikosch & Nagaev (1998), and a recent study is Denisov et al (2008).…”

Transition phenomena for ladder epochs of random walks with small negative drift

“…In particular, if (17) holds with v = 0, then P(τ (a) > n) ∼ P(τ (0) > n). Roughly speaking, (3) give a rather good approximation in the case when n is much smaller l * 1 a and n are comparable, then one has to use a correction factor, given by the right hand side of (18).…”

“…[11] and by Rozovskii [17] that if P(X > x) is regularly varying at infinity with index p < −2, then, under some additional restrictions on the left tail,…”

On the Distribution of the Nearly Unstable AR(1) Process with Heavy Tails

“…The analysis of the finite sum Y K = K n=0 ρ n X n can be done following the same lines as the analysis of the random walk with heavy-tailed increments (see [5] for an extensive study of such results, including the partial sums with nonidentically distributed increments). In particular, the style of the proofs we give for the region where both the normal approximation and the heavy-tailed asymptotic play a role, resembles in spirit the work done by Rozovskiǐ [23] in the random walk setting. Theorems 2.2 and 2.3, below, give the asymptotic behavior of the partial sums Y K for moderate and large values of x, respectively.…”