J.M. Marstrand, Order{two density and the strong law of large numbers, To appear in Mathematika, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....= r . Examples include Hausdor measures on deterministic and random self similar sets, mixing repellers or occupation measures of stable processes, see [BF92] PZ94] KF92] and [FX95] We remark that average densities were also used to characterize geometric regularity of sets, see [FS95] [JM96], PM97] or symmetry properties of measures, see [M98a] MP98] Our rst result shows that for the intersection local time measure on the intersection of two Brownian paths in 3 space an average density of order two may be de ned using a gauge function of purely exponential type. Theorem 1.1 ....

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika. 43 (1996) 1-22.

....objects are highly irregular. We believe that this improves the understanding of the geometry of non recti able sets and measures and we illustrate the strength of this result by means of several examples of its application to questions studied previously in [BeFi] M oPr] M o2] and [FaSp] [Ma2]. We also remark that our result is new even on the line. However, in the important case of measures on the line with positive lower and nite upper densities it is possible to deduce the Palm property of tangent measure distributions from very detailed information about the structure of these ....

....x, the right and left lower average dimensional densities coincide and each of them is half of the lower average dimensional density of at x. Of course, analogous results can be found for upper average densities. As our nal application we show in Theorem 5 how the results of [FaSp] and [Ma2] follow naturally by using the higher regularity of tangent measure distributions. Our approach removes their unnatural assumption of niteness of the upper densities for results about lower densities. Lemma 3. Suppose that a probability measure P on M(R d ) de nes an self similar random ....

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika, 43:1-22, 1996.

....and [7] have extended this result to various other classes of fractal measures with self similarity properties. Average densities have also been used for the investigation of general measures with positive lower and nite upper densities. For example, Falconer and Springer in [6] and Marstrand in [11] generalize a classical inequality of Marstrand using average densities and in [16] it is shown that the lower one sided average densities do not vanish. In [4] Bedford and Fisher ask whether the left sided and right sided average densities always agree. An answer to this question can be given in ....

J.M. Marstrand, Order{two density and the strong law of large numbers, To appear in Mathematika, 1996.

....describes regularity properties of , although this point of view seems to have so far very little rigorous justi cation. There are however interesting recent results relating the geometric regularity of measures to the relation of the average densities and the lower and upper densities, see [6] [14], 16] Let us now recall the known results about the average densities of the Brownian path fB(t) 0 t 1g. The path is equipped with a natural measure , the occupation measure de ned by (A) Z 1 0 1 A (s) ds for A IR d Borel. By classical results of Ciesielski, Taylor and Ray the ....

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika. Vol. 43, 1996, pp. 1-22.

....describes regularity properties of , although this point of view seems to have so far very little rigorous justi cation. There are however interesting recent results relating the geometric regularity of measures to the relation of the average densities and the lower and upper densities, see [6] [14], 16] Let us now recall the known results about the average densities of the Brownian path fB(t) 0 t 1g. The path is equipped with a natural measure , the occupation measure de ned by (A) Z 1 0 1 A (s) ds for A IR d Borel. By classical results of Ciesielski, Taylor and Ray the ....

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika. Vol. 43, 1996, pp. 1-22.

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J.M. Marstrand, Order{two density and the strong law of large numbers, To appear in Mathematika, 1996.

No context found.

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika. 43 (1996) 1-22.

No context found.

J.M. Marstrand. Order{two density and the strong law of large numbers. Mathematika, 43:1-22, 1996.

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