TY - JOUR

T1 - On almost sure limit theorems

AU - Ibragimov, I. A.

AU - Lifshits, M. A.

PY - 1999/1/1

Y1 - 1999/1/1

N2 - By a sequence of random vectors {ζk}, we can construct empirical distributions of the type Qn = (log n)-1 Σnk=1 δζk/k. Statements on the convergence of these or similar distributions with probability 1 to a limit distribution are called almost sure theorems. We propose several methods which permit us to easily deduce the almost sure limit theorems from the classical limit theorems, prove the invariance principle of the type "almost sure," and investigate the convergence of generalized moments. Unlike the majority of the preceding papers, where only convergence to the normal law is considered, our results may be applied in the case of limit distributions of the general type.

AB - By a sequence of random vectors {ζk}, we can construct empirical distributions of the type Qn = (log n)-1 Σnk=1 δζk/k. Statements on the convergence of these or similar distributions with probability 1 to a limit distribution are called almost sure theorems. We propose several methods which permit us to easily deduce the almost sure limit theorems from the classical limit theorems, prove the invariance principle of the type "almost sure," and investigate the convergence of generalized moments. Unlike the majority of the preceding papers, where only convergence to the normal law is considered, our results may be applied in the case of limit distributions of the general type.

KW - Convergence almost sure

KW - Invariance principle

KW - Limit theorems

KW - Sums of independent variables

KW - Weak dependence

UR - http://www.scopus.com/inward/record.url?scp=0033622215&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0033622215

VL - 44

SP - 254

EP - 272

JO - Theory of Probability and its Applications

JF - Theory of Probability and its Applications

SN - 0040-585X

IS - 2

ER -