Limit theorems for subsequences of arbitrarily-dependent sequences of random variables.

*(English)*Zbl 0571.60027The purpose of the present article is to formulate and to prove rigorously and in considerable generality a heuristic subsequence principle put forward by the reviewer: Given a limit theorem for independent identically distributed random variables under certain moment conditions, there exists an analogous theorem such that an arbitrary sequence of random variables satisfying the same moment conditions contains a subsequence all of whose further subsequences satisfy the analogous theorem. P. Révész, J. Komlós, V. F. Gapoškin, I. Berkes and the reviewer [cf. the bibligraphy of the article under review] have verified the principle in numerous important cases (e.g., the strong law of large numbers, the central limit theorem, the law of the iterated logarithm) by ad hoc methods based on obtaining a suitable martingale difference sequence very close to a certain subsequence of the arbitrarily given sequence of random variables and then proving the analogous theorem for such martingale difference sequences.

The approach of the present article is quite different. First the principle is formulated rigourously as follows: A statute A is defined to be a measurable subset of \(P(R)\times R^{\infty}\) (where P(R) is the Polish space of all probability measures on R) such that \(\lambda^*\{x\in R^{\infty}:(\lambda,x)\in A\}=1\) (where \(\lambda^*=\lambda \otimes \lambda \otimes...)\); A is called a limit statute, if (\(\lambda\),x)\(\in A\) and \(\sum | y_ i-x_ i| <\infty\), then (\(\lambda\),y)\(\in A\) (where \(x=(x_ i)\), \(y=(y_ i))\). One of the main theorems of the paper is the following: If \(X=\{X_ i\}\) is an arbitrary sequence of random variables whose laws form a tight set in P(R) and A is any limit statute, then there exists a subsequence Y of X and a random measure \(\omega\) \(\to \mu (\omega)\in P(R)\) such that (\(\mu\) (\(\omega)\), Z(\(\omega)\))\(\in A\) a.s. for any subsequence Z of Y. Taking \(A=\{(\lambda,x):\) either \(\int^{\infty}_{-\infty}\) \(| t| \lambda (dt)=\infty\) or \(\int^{\infty}_{-\infty}\) \(| t| \lambda (dt)<\infty\) and \(\sum^{N}_{i=1}x_ i/N\to \int^{\infty}_{-\infty}\) \(t\lambda\) (dt)\(\}\) in this theorem gives a result due to Komlós. Other limit statutes give theorems due to Révész, Gapoškin, Berkes and the reviewer.

The technique of proof of the theorem is as follows. One extracts a subsequence Y which can be associated with an exchangeable sequence \(\xi\) in such a way that theorems for \(\xi\) are true for Y. The random measure \(\mu\) appearing in the theorem is the canonical measure associated with the exchangeable process \(\xi\) (i.e., given \(\mu\), \(\xi\) is an independent identically distributed sequence). Another important theorem gives the functional forms of weak limit theorems. An assertion of Révész concerning unconditionally a.s. convergent subsequences is also proved. In conclusion, the author gives a simple and elegant technique for proving wide classes of theorems illustrating the subsequence principle.

The approach of the present article is quite different. First the principle is formulated rigourously as follows: A statute A is defined to be a measurable subset of \(P(R)\times R^{\infty}\) (where P(R) is the Polish space of all probability measures on R) such that \(\lambda^*\{x\in R^{\infty}:(\lambda,x)\in A\}=1\) (where \(\lambda^*=\lambda \otimes \lambda \otimes...)\); A is called a limit statute, if (\(\lambda\),x)\(\in A\) and \(\sum | y_ i-x_ i| <\infty\), then (\(\lambda\),y)\(\in A\) (where \(x=(x_ i)\), \(y=(y_ i))\). One of the main theorems of the paper is the following: If \(X=\{X_ i\}\) is an arbitrary sequence of random variables whose laws form a tight set in P(R) and A is any limit statute, then there exists a subsequence Y of X and a random measure \(\omega\) \(\to \mu (\omega)\in P(R)\) such that (\(\mu\) (\(\omega)\), Z(\(\omega)\))\(\in A\) a.s. for any subsequence Z of Y. Taking \(A=\{(\lambda,x):\) either \(\int^{\infty}_{-\infty}\) \(| t| \lambda (dt)=\infty\) or \(\int^{\infty}_{-\infty}\) \(| t| \lambda (dt)<\infty\) and \(\sum^{N}_{i=1}x_ i/N\to \int^{\infty}_{-\infty}\) \(t\lambda\) (dt)\(\}\) in this theorem gives a result due to Komlós. Other limit statutes give theorems due to Révész, Gapoškin, Berkes and the reviewer.

The technique of proof of the theorem is as follows. One extracts a subsequence Y which can be associated with an exchangeable sequence \(\xi\) in such a way that theorems for \(\xi\) are true for Y. The random measure \(\mu\) appearing in the theorem is the canonical measure associated with the exchangeable process \(\xi\) (i.e., given \(\mu\), \(\xi\) is an independent identically distributed sequence). Another important theorem gives the functional forms of weak limit theorems. An assertion of Révész concerning unconditionally a.s. convergent subsequences is also proved. In conclusion, the author gives a simple and elegant technique for proving wide classes of theorems illustrating the subsequence principle.

##### MSC:

60Fxx | Limit theorems in probability theory |

60G42 | Martingales with discrete parameter |

60B05 | Probability measures on topological spaces |

60G07 | General theory of stochastic processes |

##### Keywords:

subsequence principle; strong law of large numbers; central limit theorem; law of the iterated logarithm; martingale difference sequences; exchangeable sequence; exchangeable process; functional forms of weak limit theorems
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\textit{D. J. Aldous}, Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 59--82 (1977; Zbl 0571.60027)

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##### References:

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[11] | Komlós, J.: A generalisation of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18, 217-229 (1967) · Zbl 0228.60012 · doi:10.1007/BF02020976 |

[12] | Meyer, P.A.: Probability and potentials. Waltham, Mass.: Blaisdell 1966 · Zbl 0138.10401 |

[13] | Parthasarathy, K.R.: Probability measures on metric spaces. New York-London: Academic Press 1967 · Zbl 0153.19101 |

[14] | Révész, P.: The laws of large numbers. New York-London: Academic Press 1967 |

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