Non-central limit theorems for non-linear functional of Gaussian fields (original) (raw)
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Let X=(X,, t CR) be a stationary Gaussian process on (R, .F, P) with time-shift operators ( CJ,, s E R) and let H(X) = L'(f), u(X), P) denote the space of square-integrable functionals of X. Say that Y t H(X) with EY =O satisfies the Central Limit Theorem (CLT) if A family of martingales (Z,(t), t 2 0) is exhibited for which 2, ("c) = Z,, and martingale techniques and results are used to provide suficient conditions on X and Y for the CLT. These conditions are then shown to be necessary for slightly more restrictive central limit behavior of Y.
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Limit theorems for statistical functionals with applications to dimension estimation
dim haus (E) ≤ dim − box (E) and dim pack (E) ≤ dim + box (E) for any bounded set E ∈ R d and dim haus (F) ≤ dim pack (F) Hamann [24] to dependent observations. Keller [35] extended the method of Grasberger and Procaccia [22] to The first part of this thesis solves a problem originating from the work of Keller [35]. Note that µ(B(x, ε)) can always be estimated byμ(B(x, ε)), whereμ is the empirical probability measure of independent identically distributed random variables. It is evident thatμ(B(x, ε)) = 0 if no observation falls into B(x, ε) hence logμ(B(x, ε)) does not make sense at all. This was pointed by Keller [35]. One of purposes of this work is to show how such procedure is meaningful if enough data is available to fall into B(x, ε). The second part of the dissertation deals with a different but related problem. In the late 1980's Brosamler [8] and Schatte [46] independently proved a new type of limit theorems. This type of statements extends the classical central limit theorem to a pathwise version and is therefore called the almost sure central limit Philipp [38] gave a general condition for the validity of (1) so that a large class of dependent sequences satisfies the ASCLT. Later Peligrad and Shao [44] proved (1) directly for associated, strongly mixing and ρ-mixing sequences under the same conditions that assure the usual central limit theorem. Statements of type (1) with some non-normal limiting distribution function G are usually called almost sure (or pointwise) limit theorems (ASLT). The first result in this field belongs to Peligrad and Révész [43]. They showed that a weak convergence of properly normalized and centered partial sum of i.i.d. random variables to a limiting α-stable distribution G α (0 < α < 2) implies the corresponding ASLT. Analogous result was proved by Berkes and Dehling [5] for the normal limiting distribution. Thus for i.i.d. random variables, almost sure limit theorems are weaker results than corresponding classical limit theorems. Moreover, Berkes, Dehling and Móri [6] provided counterexamples which show that the reverse is not valid. An excellent survey on this topic can be found in Berkes [3] as well as in Atlagh and Weber [1]. Recently Berkes and Csáki [4] obtained a general result in the almost sure limit theory. They used it to prove almost sure versions of several classical limit theorems. In particular they proved the ASLT for U-statistics under finite second moment of the kernel.