Feller’s upper-lower class test in Euclidean space

Einmahl, Uwe

doi:10.1090/tran/8728

Feller’s upper-lower class test in Euclidean space
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Trans. Amer. Math. Soc. 375 (2022), 6575-6596 Request permission

Abstract:

We provide an extension of Feller’s upper-lower class test for the Hartman-Wintner LIL to the LIL in Euclidean space. We obtain this result as a corollary to a general upper-lower class test for $\Gamma _n T_n$ where $T_n=\sum _{j=1}^n Z_j$ is a sum of i.i.d. d-dimensional standard normal random vectors and $\Gamma _n$ is a convergent sequence of symmetric non-negative definite $(d,d)$-matrices. In the process we derive new bounds for the tail probabilities of $d$-dimensional normally distributed random vectors.

References

István Berkes, Siegfried Hörmann, and Michel Weber, Upper-lower class tests for weighted i.i.d. sequences and martingales, J. Theoret. Probab. 23 (2010), no. 2, 428–446. MR 2644868, DOI 10.1007/s10959-009-0219-5
Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
G. Dierickx, A general Darling-Erdös theorem and integral test for sums of independent and identically distributed random vectors in Euclidean space, Ph.D. thesis, Vrije Universiteit Brussel, Belgium, 2018.
Gauthier Dierickx and Uwe Einmahl, A general Darling-Erdős theorem in Euclidean space, J. Theoret. Probab. 31 (2018), no. 2, 1142–1165. MR 3803927, DOI 10.1007/s10959-016-0728-y
Uwe Einmahl, A note on the law of the iterated logarithm in Hilbert space, Probab. Theory Related Fields 82 (1989), no. 2, 213–223. MR 998931, DOI 10.1007/BF00354760
Uwe Einmahl, The Darling-Erdős theorem for sums of i.i.d. random variables, Probab. Theory Related Fields 82 (1989), no. 2, 241–257. MR 998933, DOI 10.1007/BF00354762
Uwe Einmahl, On the almost sure behavior of sums of i.i.d. random variables in Hilbert space, Ann. Probab. 19 (1991), no. 3, 1227–1263. MR 1112414
Uwe Einmahl, Exact convergence rates for the bounded law of the iterated logarithm in Hilbert space, Probab. Theory Related Fields 92 (1992), no. 2, 177–194. MR 1161185, DOI 10.1007/BF01194920
Uwe Einmahl and Deli Li, Characterization of LIL behavior in Banach space, Trans. Amer. Math. Soc. 360 (2008), no. 12, 6677–6693. MR 2434306, DOI 10.1090/S0002-9947-08-04522-4
Paul Erdös, On the law of the iterated logarithm, Ann. of Math. (2) 43 (1942), 419–436. MR 6630, DOI 10.2307/1968801
W. Feller, The general form of the so-called law of the iterated logarithm, Trans. Amer. Math. Soc. 54 (1943), 373–402. MR 9263, DOI 10.1090/S0002-9947-1943-0009263-7
W. Feller, The law of the iterated logarithm for identically distributed random variables, Ann. of Math. (2) 47 (1946), 631–638. MR 17894, DOI 10.2307/1969225
J. Kuelbs, Sample path behavior for Brownian motion in Banach spaces, Ann. Probability 3 (1975), 247–261. MR 372937, DOI 10.1214/aop/1176996396
V. M. Zolotarev, Concerning a certain probability problem, Theory Probab. Appl., 6 (1961), 201–204.