Foreword ..................................................... VII
Preface ....................................................... IX
1. Basic Concepts .............................................. 1
1.1. Bipartite Graphs in Complex Stochastic Systems ......... 1
1.2. Perfect Matchings ...................................... 2
1.3. Permanent Function ..................................... 4
1.4. P-statistics ........................................... 6
1.5. The H-decomposition .................................... 8
1.6. P-statistics .......................................... 12
1.7. Examples .............................................. 14
1.8. Bibliographic Details ................................. 16
2. Properties of P-statistics ................................. 19
2.1. Preliminaries: Martingales ............................ 19
2.2. P-decomposition of a P-statistic ...................... 21
2.3. Variance Formula for a P-statistic .................... 27
2.4. Bibliographic Details ................................. 32
3. Asymptotics for Random Permanents .......................... 35
3.1. Introduction .......................................... 35
3.2. Preliminaries ......................................... 37
3.2.1. Limit Theorems for Exchangeable
Random Variables ............................... 37
3.2.2. Law of Large Numbers for Triangular Arrays ..... 40
3.2.3. More on Elementary Symmetric Polynomials ....... 41
3.3. Limit Theorem for Elementary Symmetric Polynomials .... 43
3.4. Limit Theorems for Random Permanents .................. 45
3.5. Additional Central Limit Theorems ..................... 55
3.6. Strong Laws of Large Numbers .......................... 59
3.7. Bibliographic Details ................................. 65
4. Weak Convergence of Permanent Processes .................... 67
4.1. Introduction .......................................... 67
4.2. Weak Convergence in Metric Spaces ..................... 68
4.3. The Skorohod Space .................................... 71
4.4. Permanent Stochastic Process .......................... 74
4.5. Weak Convergence of Stochastic Integrals
and Symmetric Polynomials Processes ................... 75
4.6. Convergence of the Component Processes ................ 78
4.7. Functional Limit Theorems ............................. 81
4.8. Bibliographic Details ................................. 86
5. Weak Convergence of P-statistics ........................... 87
5.1. Multiple Wiener-Ito Integral. Limit Law
for U-statistics ...................................... 88
5.1.1. Multiple Wiener-Ito Integral
of a Symmetric Function ........................ 88
5.1.2. Classical Limit Theorems for U-statistics ...... 92
5.1.3. Dynkin-Mandelbaum Theorem ...................... 94
5.1.4. Limit Theorem for U-statistics of
Increasing Order ............................... 96
5.2. Asymptotics for P-statistics ......................... 100
5.3. Examples ............................................. 107
5.4. Bibliographic Details ................................ 120
6. Permanent Designs and Related Topics ...................... 121
6.1. Incomplete U-statistics .............................. 121
6.2. Permanent Design ..................................... 125
6.3. Asymptotic properties of USPD ........................ 129
6.4. Minimal Rectangular Schemes .......................... 134
6.5. Existence and Construction of MRS .................... 140
6.5.1. Strongly Regular Graphs ....................... 140
6.5.2. MRS and Orthogonal Latin Squares .............. 142
6.6. Examples ............................................. 144
6.7. Bibliographic Details ................................ 147
7. Products of Partial Sums and Wishart Determinants ......... 149
7.1. Introduction ......................................... 149
7.2. Products of Partial Sums for Sequences ............... 151
7.2.1 Extension to Classical U-statistics ............ 157
7.3. Products of Independent Partial Sums ................. 160
7.3.1 Extensions ..................................... 165
7.4. Asymptotics for Wishart Determinants ................. 167
7.5. Bibliographic Details ................................ 169
References ................................................... 171
Index ........................................................ 177
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