## Random PermanentsThe determination of permanent random measures and the representation of symmetric statistics as functionals of symmetrization random measures with some deterministic kernels, make it possible to clarify the influence of properties of a random measure on the limiting results for symmetric statistics and also to study the influence of the characteristic structure of these kernels. This approach in the theory of symmetric statistics has inspired the authors to investigate random permanents and their generating functions in detail. New limiting results for random permanents are basically obtained by employing the algebraic and analytical properties of the permanents of sampling matrices and their generating functions. This notion allows clarification of different schemes in the asymptotic analysis of symmetric statistics as the size of a sample n tends to infinity. |

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### Contents

Preface | 3 |

Permanents in the series scheme | 25 |

Permanents with increasing degrees | 45 |

Normal approximation | 61 |

Symmetric statistics | 97 |

Mixed sampling permanents | 135 |

Rate of convergence | 155 |

Dependent variables | 167 |

Bibliographical comments | 179 |

191 | |

### Common terms and phrases

analogous approximation condition 3.15 asymptotic analysis asymptotic representation Borovskikh and Korolyuk central limit theorem Chapter Charlier-Poisson complete the proof conditions 6.1 Consider Corollary counting measure decomposition formula 1.11 defined degenerate kernel dimension distributed random variables elementary symmetric polynomials fixed following estimate following form following representation Gaussian Hermite polynomials identically distributed random independent and identically inequality inner sum Ito-Poisson integral Jx Jx Kaneva Korolyuk 1993a large numbers law of large Lemma limiting result ln[l measurable space multiple sampling matrix order statistics parameter permanent functional permanent measure permanent multiple functionals permanent random measure permanent symmetric statistics permanents of matrices permanents of sampling Poisson approximation condition Poisson approximation scheme Poisson distributed Poisson random measure proof of theorem random elements random permanent real-valued right-hand side sampling function satisfies the estimate shift decomposition formula simple sample statistical image stochastic integral symmetric polynomials takes place taking into account theorem 4.1 weak convergence weak limit

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