### Abstract

In this paper we investigate the limit distribution of the functions of independent triangular arrays X_{nj}, 1 ≤ j ≤ k(n), n ≥ 1. According to LeCam's theorem, if f belongs to the class of functions PD[0,2] (which is slightly weaker than the assumptions that f(0) = 0, and f has the second derivative at zero), then the distribution of ∑_{j=1} ^{k(n)} f(X_{nj}) is shift convergent. He also gives the explicit form of the characteristic function of the limit infinitely divisible distribution. We consider the class of functions PD[0,1] and prove a similar statement. Since in the definition of the sequence of centering constants the truncation points depend only on the value of X_{nj} and not on the function f, this makes the analysis of the joint distribution of random variables in the above form considerably easier. Also we analyze the process of partial sums ∑_{j=1}^{[uk(n)]} f(X_{nj}, t), 0 ≤ u ≤ 1. where f(x,t) is a parametric family of functions depending continuously on the parameter t. In the case of power functions we give an explicit representation of the limit process in term of Poissonian integrals.

Original language | English |
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Pages (from-to) | 66-86 |

Number of pages | 21 |

Journal | Journal of Mathematical Sciences |

Volume | 78 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 1996 |

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### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Applied Mathematics