Convergence rate in integral limit theorem with stable limit law

On the rate of convergence of sums of extremes to a stable law

“…Clearly, the rate of convergence in (1.4) is related to that of (1.1). For reader's convenience we will clarify and augment certain Berry Esseen type results for the convergence of the total sum in (1.1), which are to be found in the work of Bergstr6m (1952b), Cram4r (1963, Satyabaldina (1972Satyabaldina ( , 1973, Paulauskas (1973), Christoph (1979Christoph ( , 1980, Hall (1981aHall ( , 1981b, Dubinskaite (1983) and Mijnheer (1987). Roughly speaking, the rates of convergence depend on how close the tails of F and G are to each other.…”

“…For the proofs of (T.3), (T.4) and (T.5) we need the following fact which can be derived by successive integration by parts (see Christoph (1979), page 97, or Dubinskaite (1983), page 50, formula (17)). Whenever (1.14) holds for some k > 1, then…”

“…Following the ideas of Christoph (1979) and Dubinskaite (1983) we introduce the integral differences H I "= H and for k > 2…”

Fourier Analysis of Semistable Distributions

“…If the description in (1)(2)(3)(4) holds for some constants α ∈ (0, 2), c > 1 and θ ∈ R and functions M L (·) and M R (·), then it can be shown that cy α (t) = y α (c 1/α t) + ibt, t ∈ R, for a suitable constant b ∈ R. Conversely, considering a nonvanishing characteristic function φ(t), let y(t) be the uniquely defined continuous complex-valued function such that y(0) = 0 and φ(t) = e y(t) , t ∈ R; see [2,Theorem 7.6.2]. If the functional equation cy(t) = y(qt) + ibt, t ∈ R, is satisfied for some constants c > 1, q > 1 and b ∈ R, which is a slight modification of the property that Lévy [11,Sect.…”

“…Motivated by the author's earlier results on St. Petersburg games, described in [3], Megyesi and the author [6] proved that if the underlying distribution function F of X is in the domain of geometric partial attraction of a semistable law of index α ∈ (0, 2), with a distribution function G α (·) given by (1)(2)(3)(4), then in fact a centering sequence c n and a norming sequence a n = n 1/α (1/n) can be explicitly specified for all n ∈ N, where (s), 0 < s < 1, is a right-continuous positive function that is slowly varying at zero, such that the corresponding full sequence V n is stochastically compact in distribution, and the class G α = {G α,κ (·) : 1 ≤ κ ≤ c} of its subsequential asymptotic distributions is parameterized by the values of some finite interval [1, c]. Here, if the original attracting G α (·) is determined by M L (·) and M R (·) in (2) and (3), then the semistable G α,κ (·) is determined by M L,κ (x) = M L (x/κ 1/α ), x < 0, and M R,κ (x) = M R,κ (x/κ 1/α ), x > 0, that satisfy (2) and (3) with the same multiplicative period c 1/α .…”