Marcinkiewicz laws with infinite moments

Abstract

Marcinkiewicz laws of large numbers for ϕ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Lévy-Ottaviani-Etemadi and Hoffmann-Jørgensen inequalities are revisited.

A note on Marcinkiewicz laws for strictly stationary -mixing sequences

“…If moments do not exist, our next theorem gives a similar characterization (see also Szewczak, 2010, Theorem 1, Theorem 2 and Remark 14). Define…”

“…Theorems 1 and 2 are implied respectively by Theorem 1 in Szewczak (2011), as well as Theorem 1 and 2 in Szewczak (2010) and the following two lemmas (cf. Szewczak, 2010, Remark 14).…”

A weak law of large numbers for maxima

“…For maxima, the following result may serve as a counterpart. The case where E|X 1 | r I [|X 1 |≤x] , r ∈ (0, 2), varies slowly is investigated in Szewczak (2010). Theorem 1 is also related to Theorem 4.6 in Hitczenko et al (1998) (covering a subclass of nonhypercontractive maxima) and it holds (to some extent) for strictly stationary processes (see the last section).…”

“…By the inequality (13) in Szewczak (2010) and stationarity, for s, t > 0 P(|S n | > t) ≥ (1 − ϕ 1 − max 1≤k<n P(|S k | > s))P( max 1≤k≤n |S k | > t + s).…”

On Marcinkiewicz–Zygmund laws

“…[7], [12, p. 155]) and its proof is a modification of the proof of Lemma on p. 155 in [17] with the window of the size m deleted (cf. [23]). …”