英文标题：
《Record statistics for random walk bridges》
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作者：
Claude Godreche, Satya N. Majumdar, Gregory Schehr
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最新提交年份：
2016
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英文摘要：
We investigate the statistics of records in a random sequence $\\{x_B(0)=0,x_B(1),\\cdots, x_B(n)=x_B(0)=0\\}$ of $n$ time steps. The sequence $x_B(k)$\'s represents the position at step $k$ of a random walk `bridge\' of $n$ steps that starts and ends at the origin. At each step, the increment of the position is a random jump drawn from a specified symmetric distribution. We study the statistics of records and record ages for such a bridge sequence, for different jump distributions. In absence of the bridge condition, i.e., for a free random walk sequence, the statistics of the number and ages of records exhibits a `strong\' universality for all $n$, i.e., they are completely independent of the jump distribution as long as the distribution is continuous. We show that the presence of the bridge constraint destroys this strong `all $n$\' universality. Nevertheless a `weaker\' universality still remains for large $n$, where we show that the record statistics depends on the jump distributions only through a single parameter $0<\\mu\\le 2$, known as the L\\\'evy index of the walk, but are insensitive to the other details of the jump distribution. We derive the most general results (for arbitrary jump distributions) wherever possible and also present two exactly solvable cases. We present numerical simulations that verify our analytical results.
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中文摘要：
我们研究了$n$时间步长的随机序列$\\{x_B（0）=0，x_B（1），\\cdots，x_B（n）=x_B（0）=0\\}中记录的统计。序列$x_B（k）$表示从原点开始和结束的$n$步数的随机游走“桥”在步骤$k$处的位置。在每一步中，位置的增量是从指定的对称分布中提取的随机跳跃。我们研究了这种桥序列在不同跳跃分布下的记录统计和记录年龄。在没有桥条件的情况下，即对于自由随机游走序列，记录的数量和年龄的统计数据对所有$n$都表现出“强”的普遍性，也就是说，只要分布是连续的，它们就完全独立于跳跃分布。我们证明了桥约束的存在破坏了这种强大的“全部n$”普适性。然而，对于较大的$n$来说，仍然存在一个“较弱”的普遍性，我们表明，记录统计数据仅通过一个参数$0<\\mu\\le 2$依赖于跳跃分布，该参数被称为行走的列维指数，但对跳跃分布的其他细节不敏感。我们在可能的情况下得出了最一般的结果（对于任意跳跃分布），并给出了两种完全可解的情况。我们通过数值模拟验证了我们的分析结果。
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分类信息：

一级分类：Physics 物理学
二级分类：Statistical Mechanics 统计力学
分类描述：Phase transitions, thermodynamics, field theory, non-equilibrium phenomena, renormalization group and scaling, integrable models, turbulence
相变，热力学，场论，非平衡现象，重整化群和标度，可积模型，湍流
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一级分类：Quantitative Finance 数量金融学
二级分类：Statistical Finance 统计金融
分类描述：Statistical, econometric and econophysics analyses with applications to financial markets and economic data
统计、计量经济学和经济物理学分析及其在金融市场和经济数据中的应用
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