A General Darling–Erdős Theorem in Euclidean Space

Abstract

We provide an improved version of the Darling-Erdős theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance principle in this setting which has other applications as well such as an integral test refinement of the multidimensional Hartman-Wintner LIL. We also identify a borderline situation where one has weak convergence to a shifted version of the standard limiting distribution in the classical Darling-Erdős theorem.

Feller's upper-lower class test in Euclidean space

“…< ∞ By a standard argument (see, for instance, Sect. 5 in [6]), we then can infer from Theorem 2.3 in [4] that…”

“…Feller also stated an interesting conjecture in his paper [12], namely that the above upper-lower class test still holds if one replaces B n by nσ 2 n . This conjecture has finally been proven in [4] leading to the following result in dimension 1, P{|S n | ≤ σ n √ nφ n eventually} = 1 or 0 (1.3) according as…”

“…Moreover, it was shown in [4] that (1.3) We return to the d-dimensional case: Feller's upper-lower class test (1.3) was actually obtained in [4] as a corollary to a d-dimensional result. To formulate this result we need some extra notation: we are now assuming that X n = (X…”

Darling–Erdős theorem for Lévy processes at zero

“…mean zero and variance one case that for (1.1) to hold it is necessary and sufficient that LLt E ξ 2 1 1{|ξ 1 | ≥ t} → 0, as t → ∞. Einmahl and Mason (1989) have obtained martingale Darling-Erdős theorems, and recently Dierickx and Einmahl (2018) have established multivariate versions. Corresponding results for Brownian motion were established by Khoshnevisan et al (2005).…”

Darling--Erdős theorem for Lévy processes at zero

“…mean zero and variance one case that for (1) to hold it is necessary and sufficient that LLt E ξ 2 1 1{|ξ 1 | ≥ t} → 0, as t → ∞. Einmahl and Mason [8] have obtained martingale Darling-Erdős theorems, and recently Dierickx and Einmahl [6] have established multivariate versions. Corresponding results for Brownian motion were established by Khoshnevisan et al [9].…”