Strong approximation of the St. Petersburg game

Let (Formula presented.) be i.i.d. random variables with (Formula presented.) (Formula presented.) and let (Formula presented.). The properties of the sequence (Formula presented.) have received considerable attention in the literature in connection with the St. Petersburg paradox (Bernoulli 1738). Let (Formula presented.) be a semistable Lévy process with underlying Lévy measure (Formula presented.). For a suitable version of (Formula presented.) and (Formula presented.), we prove the strong approximation (Formula presented.) a.s. This provides the first example for a strong approximation theorem for partial sums of i.i.d. sequences not belonging to the domain of attraction of the normal or stable laws. © 2016 Informa UK Limited, trading as Taylor & Francis Group

Selection from a stable box

Let $\{X_j\}$ be independent, identically distributed random variables. It is well known that the functional CUSUM statistic and its randomly permuted version both converge weakly to a Brownian bridge if second moments exist. Surprisingly, an infinite-variance counterpart does not hold true. In the present paper, we let $\{X_j\}$ be in the domain of attraction of a strictly $\alpha$-stable law, $\alpha\in(0,2)$. While the functional CUSUM statistics itself converges to an $\alpha$-stable bridge and so does the permuted version, provided both the $\{X_j\}$ and the permutation are random, the situation turns out to be more delicate if a realization of the $\{X_j\}$ is fixed and randomness is restricted to the permutation. Here, the conditional distribution function of the permuted CUSUM statistics converges in probability to a random and nondegenerate limit.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6014 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotics of trimmed CUSUM statistics

There is a wide literature on change point tests, but the case of variables with infinite variances is essentially unexplored. In this paper we address this problem by studying the asymptotic behavior of trimmed CUSUM statistics. We show that in a location model with i.i.d. errors in the domain of attraction of a stable law of parameter $0<\alpha <2$, the appropriately trimmed CUSUM process converges weakly to a Brownian bridge. Thus, after moderate trimming, the classical method for detecting change points remains valid also for populations with infinite variance. We note that according to the classical theory, the partial sums of trimmed variables are generally not asymptotically normal and using random centering in the test statistics is crucial in the infinite variance case. We also show that the partial sums of truncated and trimmed random variables have different asymptotic behavior. Finally, we discuss resampling procedures which enable one to determine critical values in the case of small and moderate sample sizes.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ318 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Split invariance principles for stationary processes

The results of Koml\'{o}s, Major and Tusn\'{a}dy give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294--2320] obtained Wiener approximation of a class of dependent stationary processes with finite $p$th moments, $2<p\le4$, with error term $o(n^{1/p}(\log n)^{\gamma})$, $\gamma>0$, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249--280] removed the logarithmic factor, reaching the Koml\'{o}s--Major--Tusn\'{a}dy bound $o(n^{1/p})$. No similar results exist for $p>4$, and in fact, no existing method for dependent approximation yields an a.s. rate better than $o(n^{1/4})$. In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to $o(n^{1/p})$ for arbitrary $p>2$. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems.Comment: Published in at http://dx.doi.org/10.1214/10-AOP603 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Kadec-Pelczynski theorem in LpL^p , 1≤p<21 ≤ p < 2

By a classical result of Kadec and Pełczyński (1962), every nor- malized weakly null sequence in Lp, p > 2, contains a subsequence equivalent to the unit vector basis of ℓ2 or to the unit vector basis of ℓp. In this paper we investigate the case 1 ≤ p < 2 and show that a necessary and sufficient condition for the first alternative in the Kadec-Pełczyński theorem is that the limit random measure μ of the sequence satisfies ∫R x2 dμ(x) ∈ Lp/2. © 2015 American Mathematical Society