On the rate of approximation in the central limit theorem

Pricing Discrete European Barrier Options Using Lattice Random Walks

“…In the literature there are results similar to the one in the derived from Taylor expansions of the initial value f or U t f . See for instance the work by Butzer et al (1975) or Heston and Zhou (2000). However, these results require more local regularity of the initial value than in .…”

“…Kreiss, Thomée, and Widlund (1970), Heston and Zhou (2000), and Walsh and Walsh (2002) examined, among other things, the dependence between the smoothness of f and the convergence rate in . Butzer, Hahn, and Westphal (1975) investigated the convergence rate when the initial values f are differentiable functions. Finally, Löfström (1968) presented sharp estimates on the difference in uniformly for all if f belongs to a so‐called Besov space.…”

On probabilities of large deviations in Banach spaces

“…The proofs are based on the application of the Taylor's formula, properties of the limiting Gaussian distribution, induction in n and iterations of the estimates. Methods of such kind were developed earlier for the estimation of the convergence speed in the Central Limit Theorem (Bergstrom 1951;Kuelbs and Kurtz 1974;Butzer et al 1975;Paulauskas 1976;Zolotarev 1976;Sazonov 1981;Bolthausen 1982;Bentkus and Ra6kauskas 1982a, b;Haeusler 1984;G6tze 1986;etc.). Our proofs differ from that usually used to obtain limit theorems for probabilities of large deviations (Ibragimov and Linnik 1965;Petrov 1975;Rudzkis et al 1979; etc.)…”

Pricing Discrete European Barrier Options Using Lattice Random Walks

“…Kreiss, Thomée, and Widlund (1970), Heston and Zhou (2000), and Walsh and Walsh (2002) examined, among other things, the dependence between the smoothness of f and the convergence rate in equation (3.1). Butzer, Hahn, and Westphal (1975) investigated the convergence rate when the initial values f are differentiable functions. Finally, Löfström (1968) presented sharp estimates on the difference in equation (3.1) uniformly for all x ∈ R if f belongs to a so-called Besov space.…”

“…In the literature there are results similar to the one in the equation (3.3) derived from Taylor expansions of the initial value f or U t f . See for instance the work by Butzer et al (1975) or Heston and Zhou (2000). However, these results require more local regularity of the initial value than in equation (3.3).…”