### 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 |
---|---|

Pages (from-to) | 66-86 |

Number of pages | 21 |

Journal | Journal of Mathematical Sciences |

Volume | 78 |

Issue number | 1 |

Publication status | Published - 1996 |

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### ASJC Scopus subject areas

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

### Cite this

*Journal of Mathematical Sciences*,

*78*(1), 66-86.

**Lévy processes with values in a space of continuous functions arising as limits of independent triangular arrays.** / Michaletzky, Gy; Szeidl, L.

Research output: Contribution to journal › Article

*Journal of Mathematical Sciences*, vol. 78, no. 1, pp. 66-86.

}

TY - JOUR

T1 - Lévy processes with values in a space of continuous functions arising as limits of independent triangular arrays

AU - Michaletzky, Gy

AU - Szeidl, L.

PY - 1996

Y1 - 1996

N2 - In this paper we investigate the limit distribution of the functions of independent triangular arrays Xnj, 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(Xnj) 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 Xnj 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(Xnj, 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.

AB - In this paper we investigate the limit distribution of the functions of independent triangular arrays Xnj, 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(Xnj) 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 Xnj 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(Xnj, 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.

UR - http://www.scopus.com/inward/record.url?scp=0039599318&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0039599318&partnerID=8YFLogxK

M3 - Article

VL - 78

SP - 66

EP - 86

JO - Journal of Mathematical Sciences

JF - Journal of Mathematical Sciences

SN - 1072-3374

IS - 1

ER -