N. Patzschke and M. Z ahle, Self-similar random measures are locally scale invariant, Probab.Theory Related Fields, 97(1993), 559-574.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a typical point . More precisely, a measure distribution P on M(R d ) is an self similar random measure if P is a Palm measure invariant under the rescaling group (S ) 0 . This concept has been subject of investigations by Patzschke, U. Z ahle and M. Z ahle, see for example [PaZU] and [PaZM]. In particular, they have studied the relation of the concept to statistically self similar measures in the constructive sense. We extend the notion of self similar random measures to allow, in the natural way, for the possibility that P (f g) 0 and or that the total mass of P may be less ....

N. Patzschke and M. Zahle. Self similar random measures are locally scale invariant. Prob. Theory Rel. Fields, 97:559-574, 1993.

....point of the random measure. We can interpret Palm distributions as those distributions which have the origin as a typical point of their realizations (see [22] for details) This concept of statistical self similarity has been studied by Patzschke, U. Z ahle and M. Z ahle for example in [18] [20] where also its relation to statistically self similar measures in the constructive sense was investigated. We get the following theorem (recall that we did not require to be self similar in any sense) Theorem 2.8 At almost all points x every tangent measure distribution P 2 P( x) de nes ....

N. Patzschke and M. Z ahle, Self-similar random measures are locally scale invariant, Probab.Theory Related Fields, 97(1993), 559-574.

....of increasing interest in the last years. There is a lot of papers concerning fractal properties like multifractal spectrum, fractal derivatives, Fourier transformation, and tangential distributions of self similar random measures (see for example Falconer [F] Olsen [O] Patzschke and Z ahle [PZ1], Z1] Lau and Wang [LW] Strichartz [S] Graf [G] Arbeiter and Patzschke [AP] In all these papers the investigations were restricted to random measures with compact support. A generalization to self similar random measures with noncompact support was given in [Z3] and [A1] A First ....

Patzschke, N., and Z ahle, M., Self--similar random measures are locally scale invariant, Probab. Th. Rel. Fields 97 (1993), 559 -- 574

....a typical point . More precisely, a measure distribution P on M(R d ) is an ff self similar random measure if P is a Palm measure invariant under the rescaling group (S ff ) 0 . This concept has been subject of investigations by Patzschke, U. Zahle and M. Zahle, see for example [PaZU] and [PaZM]. In particular, they have studied the relation of the concept to statistically self similar measures in the constructive sense. Using the notion of ff self similar random measures, we may reformulate Theorem 1 as Theorem 3. Let ff 0 and 2 M(R d ) be any measure. Then, at almost all ....

N. Patzschke and M. Zahle. Self-similar random measures are locally scale invariant. Prob. Theory Rel. Fields, 97:559--574, 1993.

....theory, local characteristics of self similar sets. This idea is due to Bedford and Fisher (see [4] Closely related ideas can be found in the work of U. Zahle on self similar random measures (see [19] which is continued in a series of joint work with Patzschke and M. Zahle (see e.g. 18] [17]) Bandt joined these ideas and defined the tangent measure distributions of a measure at a point as limiting distributions of sequences of natural probability distributions on (rescaled) enlargements of the measure about this point. Roughly speaking, the weight that a tangent measure distribution ....

N. Patzschke and M. Z ahle. Self-similar random measures are locally scale invariant. Prob. Th. Rel.Fields, 97:559--574, 1993.

....dimensions are calculated. The last section deals with the tangential distribution of the random self similar measure Phi. 2. Random self similar measures In this section we recall the definition and some properties of random self similar measures. We use the same notions as in [PZU] and in [PZM]. For further details see these papers. 8 Math. Nachr. 181 (1996) Let K ae IR d be a fixed compact set with K = int K. Without loss of generality we assume diamK = 1. Our primary object is a probability measure on Sim N Theta[0; 1] N , where Sim is the set of all similarities in IR d ....

....the tangential distribution not only for Phi almost all x (where dim( Phi; x) ff(1) but also for Phi q almost all x (where dim( Phi; x) ff(q) The techniques of the proof are the same as in [G, 6.3] In the special case, where p i = r D i with probability one, Patzschke and M. Z ahle [PZM] showed that the tangential distribution of Phi (w. r. t. the normalizing function i(x; r) r D ) is a self similar measure in the sense of U. Z ahle [Z] Acknowledgements The frist author was supported by the Deutsche Forschungsgemeinschaft (DFG) Bonn. 42 Math. Nachr. 181 (1996) ....

Patzschke, N., and Z ahle, M.: Self--Similar Random Measures are Locally Scale Invariant. Probab. Th. Rel. Fields 97 (1993), 559 -- 574

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N. Patzschke and M. Z ahle, Self-similar random measures are locally scale invariant, Probab.Theory Related Fields, 97(1993), 559-574.

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N. Patzschke and M. Z ahle. Self-similar random measures are locally scale invariant. Prob. Th. Rel.Fields, 97:559--574, 1993.

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