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تسجيل مستخدم جديدGenerating Pareto records
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We present, (partially) analyze, and apply an efficient algorithm for the simulation of multivariate Pareto records. A key role is played by minima of the record-setting region (we call these generators) each time a new record is generated, and two highlights of our work are (i) efficient dynamic maintenance of the set of generators and (ii) asymptotic analysis of the expected number of generators at each time.
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For iid $d$-dimensional observations $X^{(1)}, X^{(2)}, ldots$ with independent Exponential$(1)$ coordinates, consider the boundary (relative to the closed positive orthant), or frontier, $F_n$ of the closed Pareto record-setting (RS) region [ mbox{R
S}_n := {0 leq x in {mathbb R}^d: x otprec X^{(i)} mbox{for all $1 leq i leq n$}} ] at time $n$, where $0 leq x$ means that $0 leq x_j$ for $1 leq j leq d$ and $x prec y$ means that $x_j < y_j$ for $1 leq j leq d$. With $x_+ := sum_{j = 1}^d x_j$, let [ F_n^- := min{x_+: x in F_n} quad mbox{and} quad F_n^+ := max{x_+: x in F_n}, ] and define the width of $F_n$ as [ W_n := F_n^+ - F_n^-. ] We describe typical and almost sure behavior of the processes $F^+$, $F^-$, and $W$. In particular, we show that $F^+_n sim ln n sim F^-_n$ almost surely and that $W_n / ln ln n$ converges in probability to $d - 1$; and for $d geq 2$ we show that, almost surely, the set of limit points of the sequence $W_n / ln ln n$ is the interval $[d - 1, d]$. We also obtain modifications of our results that are important in connection with efficient simulation of Pareto records. Let $T_m$ denote the time that the $m$th record is set. We show that $F^+_{T_m} sim (d! m)^{1/d} sim F^-_{T_m}$ almost surely and that $W_{T_m} / ln m$ converges in probability to $1 - d^{-1}$; and for $d geq 2$ we show that, almost surely, the sequence $W_{T_m} / ln m$ has $liminf$ equal to $1 - d^{-1}$ and $limsup$ equal to $1$.
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