Summary
Let Y 1 , Y 2 , ... be a sequence of i.i.d. random variables with distribution P(Y 1 = k) = p k (k = ±1, ±2,...), E(Y 1) = 0, E(Y 21 ) = σ2<∞. Put T n = Y 1+...+Y n and N(x,n) = # {k:0<k≦n, T k = x}. Extending the result of Révész (1981) it is shown that for appropriate Skorohod construction we have
provided all moments E(¦Y 1¦m), m≧0 exists where L is the local time of a Wiener process. Certain rate of convergence is given also under weaker conditions and for ¦L(x,nσ 2)-σ2 N(x, n)¦ too, when x is fixed.
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Dedicated to Prof. I. Vincze on the occasion of his 70-th birthday
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Csáki, E., Révész, P. Strong invariance for local times. Z. Wahrscheinlichkeitstheorie verw Gebiete 62, 263–278 (1983). https://doi.org/10.1007/BF00538801
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DOI: https://doi.org/10.1007/BF00538801