On the other law of the iterated logarithm for self-normalized sums

Cai, Guorong

doi:10.1590/s0001-37652008000300002

An. Acad. Bras. Ciênc.

2008

DOI: 10.1590/s0001-37652008000300002

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On the other law of the iterated logarithm for self-normalized sums

Guorong Cai

Abstract: In this note, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.

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2022

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On the other LIL for variables without finite variance

Pakshirajan¹,

M.²

2022

Theor. Probability and Math. Statist.

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In this paper we give a simpler proof of Jain’s [Z. Wahrsch. Verw. Gebiete 59 (1982), no. 1, 117–138] result concerning the Other Law of the Iterated Logarithm for partial sums of a class of independent and identically distributed random variables with infinite variance but in the domain of attraction of a normal law. Jain’s result is less restrictive than ours but depends heavily on the techniques of Donsker and Varadhan in the theory of Large deviations. Our proof involves elementary properties of slowly varying functions. We assume that the distribution of random variables X n X_n satisfies the condition that lim x → ∞ log ⁡ H ( x ) ( log ⁡ x ) δ = 0 \lim _{ x\rightarrow \infty } \frac {\log H(x)}{(\log x)^\delta } = 0 for some 0 > δ > 1 / 2 0 >\delta > 1/2 , where H ( x ) = E ( X 1 2 I ( | X 1 | ≤ x ) ) H(x)=\mathsf E\left (X_1^2 I(|X_1|\le x)\right ) is a slowly varying function. The condition above is not very restrictive.

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On the other LIL for variables without finite variance

Pakshirajan¹,

M.²

2022

Theor. Probability and Math. Statist.

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show abstract

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On the other law of the iterated logarithm for self-normalized sums

Abstract: In this note, we obtain a Chung's integral test for self-normalized sums of i.i.d. random variables. Furthermore, we obtain a convergence rate of Chung law of the iterated logarithm for self-normalized sums.

Cited by 1 publication

References 6 publications

On the other LIL for variables without finite variance

On the other LIL for variables without finite variance

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