Limit Theorems in Probability, Statistics and Number Theory by Peter Eichelsbacher Guido Elsner Holger Kösters Matthias Löwe Franz Merkl & Silke Rolles

Author:Peter Eichelsbacher, Guido Elsner, Holger Kösters, Matthias Löwe, Franz Merkl & Silke Rolles
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

which together with (38) gives (15). □

3.2 Proof of Proposition 1.2

The proof of Proposition 1.2 will be a consequence of the following two propositions and theorem, which are the large time analogs of their small time versions given in Propositions 5.1 and 5.5 and Theorem 3.1 of [12].

Recall the definition of U(x) given in (12). Note that, after integrating by parts,

(39)

The function U(x) is continuous, in fact, differentiable, at each x > 0, with

Further,

The right-hand side here could be 0 only if is constant on (0, x], and since , as long as for some x > 0, we see that x  − 2 U(x) is strictly decreasing for large enough x, and as x  0, while x  − 2 U(x)  0 as .

In view of the monotonicity of x  − 2 U(x) just established, for each λ > 0, once t is large enough, depending on λ, for , the function

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