Strong and conditional invariance principles for samples attracted to stable laws

Abstract

We prove almost sure convergence of a representation of normalized partial sum processes of a sequence of i.i.d. random variables from the domain of attraction of an -stable law, ¡2. We obtain an explicit form of the limit in terms of the LePage series representation of stable laws. One consequence of these results is a conditional invariance principle having applications to option pricing as well as to resampling by signs and permutations.

Unit Root Inference for Non-Stationary Linear Processes Driven by Infinite Variance Innovations

“…By Corollary 1 of LePage et al (1997), we can consider a probability space where ; U and are de…ned together with f" t g t2N distributed like f" t g t2N , and such that a , and argue that on such a probability space it holds that .13) for bounded and continuous real f , where E j"j denotes expectation under P j"j . Then it follows that on a general probability space convergence in (A.13) holds in the weak sense instead of in probability.…”

“…binomial sequence with P ( i = 1) = p = 1 P ( i = 1), fU i g is an i.i.d. sequence of uniform [0; 1] random variables, and the sequences f i g; f i g and fU i g are jointly independent of one another; see LePage et al (1997). The sequence of jump magnitudes f 1= i g is the weak limit in R 1 of the order statistics of fj" t j T t=1 g, f i g is the weak limit of their respective signs, and fU i g, of their relative location in the sample.…”

Option pricing for infinite variance data

“…This result was later generalized by LePage et al [8] who showed that Equation (3) remains valid for i.i.d normalized innovations {X i } in the domain of attraction of a symmetric α-stable law with index 0 < α < 2.…”

“…Reliable and consistent estimation for the option pricing formula is highly relevant. This is because C appears impossible to compute exactly in closed form and must be approximated numerically in practice (see [8,Remark 4] on the practical difficulties). In the special case of the Pareto distribution, a Monte Carlo estimate of C n defined in Equation (2) can be used to approximate C. Such an estimate is easily derived by simulating δ 1 , .…”

“…The following is a restatement of the option pricing formula from [8] for the case of symmetric data. THEOREM 1 (Option pricing formula for stock price differences) Let {X i } be a sequence of i.i.d.…”