For n + 1 particles moving independently on a straight line, we study the question of how long the leading position of one of them can last. Our focus is the asymptotics of the probability p(T,n) that the leader time will exceed T when n and T are large. It is assumed that the dynamics of particles are described by independent, either stationary or self-similar, Gaussian processes, not necessarily identically distributed. Roughly, the result for particles with stationary dynamics of unit variance is as follows: L= -log p(T,n) /(Tlog n)=1/d+o(1), where d/(2pi) is the power of the zero frequency in the spectrum of the leading particle, and this value is the largest in the spectrum. Previously, in some particular models, the asymptotics of L was understood as a sequential limit first over T and then over n. For processes that do not necessarily have non-negative correlations, the limit over T may not exist. To overcome this difficulty, the growing parameters T and n are considered in the domain clog T 1 . The Lamperti transform allows us to transfer the described result to self-similar processes with the normalizer of log p(T,n) becoming log T log n.

中文翻译：

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一维随机粒子追踪问题中的领导指数

对于在直线上独立运动的 n+1 个粒子，我们研究其中一个粒子的领先地位能持续多久的问题。我们的重点是当 n 和 T 很大时领导时间将超过 T 的概率 p(T,n) 的渐近性。假设粒子的动力学由独立的、静止的或自相似的高斯过程描述，不一定相同分布。粗略地说，单位方差平稳动力学的粒子的结果如下： L= -log p(T,n) /(Tlog n)=1/d+o(1)，其中 d/(2pi) 是幂前导粒子谱中的零频率，该值在谱中最大。以前，在某些特定模型中，L 的渐近性被理解为首先在 T 上然后在 n 上的序列极限。对于不一定具有非负相关性的过程，T 的限制可能不存在。为了克服这个困难，在域阻塞T 1 中考虑了增长参数T 和n。Lamperti 变换允许我们将描述的结果转移到自相似过程，其中 log p(T,n) 的归一化器变为 log T log n。