Summary
Let S n =ξ 1+...+ξ n , n≧1, be the partial sums of stationary, dependent random variables in ℝm. The probability space can be partitioned into I t ∪I r , where I t = {∥S n∥→∞} and I r ={each S n is limit point of (S n)n≧1}. This result follows from the inclusion{∥S n ∥>ɛ for n>0}⊂I t a.s., which is obtained by using Kac's inequality.
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Berbee, H. Recurrence and transience for random walks with stationary increments. Z. Wahrscheinlichkeitstheorie verw Gebiete 56, 531–536 (1981). https://doi.org/10.1007/BF00531431
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DOI: https://doi.org/10.1007/BF00531431