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23 Balan, R. M. and Kulik, R. , Self-normalized weak invariance principle for mixing sequences. Lab. Reas. Probab. Stat., Univ. Ottawa-Carleton Univ., Tech. Rep. Ser. , 417

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Abstract  

Consider the set Θn of all a n-sized increment processes of the uniform empirical process α n on [0, 1]. We assume that a n ↓ 0, na n ↑ ∞, d n = na n(log n)−1 → ∞ and na n(log n)−7/3 = O(1). In Berthet (1996, 2005) the fourth assumption was shown to be critical with respect to the pointwise rates of convergence in the functional law of Deheuvels and Mason (1992) for Θn because strong approximation methods become ineffective at such a small scale a n. We are now able to study directly these small empirical increments and compute the exact rate of clustering of Θn to any Strassen function having Lebesgue derivative of bounded variation by making use of a sharp small deviation estimate for a Poisson process of high intensity due to Shmileva (2003a). It turns out that the best rates are of order d n 1/4(log n)−1 and are faster than in the Brownian case whereas the slowest rates are of order d n −1/2 and correspond to the apparently crude ones obtained in Berthet (2005) by means of Gaussian small ball probabilities. These different sharp properties of the empirical and Brownian paths imply an almost sure lower bound in the strong invariance principle and provide a new insight into the famous KMT approximation of α n.

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LetS n be the partial sums of ?-mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S n/sn) converges almost surely to ?-8 8 f(x)dF(x). We also obtain strong approximation forH(n)=?k=1 n k -1 f(S k/sk)=logn ?-8 8 f(x)dF(x) which will imply the asymptotic normality ofH(n)/log1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ?-mixing random variables.

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Abstract  

Aggregated processes appear in many areas of statistics, natural sciences and economics and studying their behavior has a considerable importance from a purely probabilistic point of view as well. Granger (1980) showed that aggregating processes of simple structure can lead to processes with much more complex dynamics, in particular, aggregating random coefficient AR(1) processes can result in long memory processes. This opens a new way to analyze complex processes by constructing such processes from simple ‘building blocks’ via aggregation. The basic statistical problem of aggregation theory is, given a sample {Y 1 (N), …, Y n (N)} of size n of the N-fold aggregated process, to draw conclusions for the structure of the constituting processes (“disaggregation”) and use this for describing the asymptotic behavior of the aggregated process. Probabilistically, this requires determining the limit distribution of nonlinear functionals of {Y 1 (N), …, Y n (N)}, which depends sensitively on the relative order of n and N. In this survey paper, we give a detailed asymptotic study of aggregated linear processes with an arbitrary (possibly infinite) number of parameters and apply the results to the disaggregation problem of AR(1) and AR(2) processes. We also discuss the problem of long memory of aggregated processes.

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