论文标题
在一个随机遗忘的模型上
On a random model of forgetting
论文作者
论文摘要
Georgiou，Katkov和Tsodyk考虑了以下随机过程。令$ x_1，x_2，\ ldots $为独立，相同分布，均匀的随机点的无限顺序，$ [0,1] $。从$ s = \ {0 \} $开始，元素$ x_k $ join $ s $一个逐一按顺序进行。当输入元素大于$ s $的当前最小元素时，此最低含量将留下$ s $。令$ s（1，n）$表示第一个$ n $ elements $ x_k $ join之后的$ s $的内容。仿真表明，$ s（1，n）| $ s $ at $ n $的$ s $通常接近$ n/e $。在这里，我们首先给出了一个严格的证据，证明确实是这种情况，实际上，对称差异的差异（1，n）$和set $ \ {x_k \ ge 1-1/e：1 \ leq k \ leq n \} $最多是$ \ tilde {o}（O}（O}（\ sqrt n），我们的主要结果是对过程的更准确描述，特别是暗示$ n $倾向于无限$ n^{ - 1/2} \ big（| s（1，n）| -n/e \ big）$收敛到正常随机变量，带有方差$ 3E^{ - 2} - 2} -e^{ - 1} $。我们进一步表明，$ s（1，n）$的对称差异的动力学和集合$ \ {x_k \ ge 1-1/e：1 \ leq k \ leq n \} $以适当的缩放为三维Bessel过程收敛。
Georgiou, Katkov and Tsodyks considered the following random process. Let $x_1,x_2,\ldots $ be an infinite sequence of independent, identically distributed, uniform random points in $[0,1]$. Starting with $S=\{0\}$, the elements $x_k$ join $S$ one by one, in order. When an entering element is larger than the current minimum element of $S$, this minimum leaves $S$. Let $S(1,n)$ denote the content of $S$ after the first $n$ elements $x_k$ join. Simulations suggest that the size $|S(1,n)|$ of $S$ at time $n$ is typically close to $n/e$. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ is of size at most $\tilde{O}(\sqrt n)$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as $n$ tends to infinity $ n^{-1/2}\big( |S(1,n)|-n/e \big) $ converges to a normal random variable with variance $3e^{-2}-e^{-1}$. We further show that the dynamics of the symmetric difference of $S(1,n)$ and the set $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ converges with proper scaling to a three dimensional Bessel process.
