Limit theorems for increments of sums of independent random variables

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Limit theorems for increments of sums of independent random variables. / Frolov, A. N.

In: Theory of Probability and its Applications, Vol. 48, No. 1, 2003, p. 93-107.

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Frolov, A. N. / Limit theorems for increments of sums of independent random variables. In: Theory of Probability and its Applications. 2003 ; Vol. 48, No. 1. pp. 93-107.

BibTeX

@article{1cf604821894472ab25d3f0201edf2a2,

title = "Limit theorems for increments of sums of independent random variables",

abstract = "We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cram{\'e}r condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erdos-R{\'e}nyi law and Mason's extension of this law, the Shepp law, the Cs{\"o}rgo- R{\'e}v{\'e}sz theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index α ∈ (1,2] and the parameter of symmetry β = -1.",

keywords = "Erdos-R{\'e}nyi law, Increments of sums of independent random variables, Large deviations, Law of the iterated logarithm, Shepp law, Strong approximations laws, Strong law of large numbers",

author = "Frolov, {A. N.}",

year = "2003",

doi = "10.1137/S0040585X980245",

language = "English",

volume = "48",

pages = "93--107",

journal = "Theory of Probability and its Applications",

issn = "0040-585X",

publisher = "Society for Industrial and Applied Mathematics",

number = "1",

}

RIS

TY - JOUR

T1 - Limit theorems for increments of sums of independent random variables

AU - Frolov, A. N.

PY - 2003

Y1 - 2003

N2 - We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cramér condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erdos-Rényi law and Mason's extension of this law, the Shepp law, the Csörgo- Révész theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index α ∈ (1,2] and the parameter of symmetry β = -1.

AB - We investigate the almost surely asymptotic behavior of increments of sums of independent identically distributed random variables satisfying the one-sided Cramér condition. We establish that, irrespective of the length of the increments, the norming sequence in strong limit theorems for increments of sums is determined by a behavior of the inverse function to the function of deviations. This allows for unifying the following well-known results for increments of sums: the strong law of large numbers, the Erdos-Rényi law and Mason's extension of this law, the Shepp law, the Csörgo- Révész theorems, and the law of the iterated logarithm. In the case of large increments, we derive new results for random variables from the domain of attraction of a stable law with index α ∈ (1,2] and the parameter of symmetry β = -1.

KW - Erdos-Rényi law

KW - Increments of sums of independent random variables

KW - Large deviations

KW - Law of the iterated logarithm

KW - Shepp law

KW - Strong approximations laws

KW - Strong law of large numbers

UR - http://www.scopus.com/inward/record.url?scp=2142754400&partnerID=8YFLogxK

U2 - 10.1137/S0040585X980245

DO - 10.1137/S0040585X980245

M3 - Article

AN - SCOPUS:2142754400

VL - 48

SP - 93

EP - 107

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 1

ER -

ID: 75021622