通过顺序统计量研究k记录过程的马尔可夫性质
Markov Properties of $k$-Record Processes via Order Statistics
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中文总结 AI 辅助
研究k记录过程的马尔可夫性质，基于运行顺序统计量序列直接推导其概率结构，证明相关条件独立性性质，在统一框架中恢复经典分布结果，还处理指数情况并给出指数分布特征。
中文摘要 AI 辅助
k记录值理论（类型2 k记录）在部分极值研究和基于记录数据的统计推断中起着重要作用。常见方法是将与分布函数F相关的k记录分析简化为从变换后的分布F₁:k(x)=1-(1-F(x))^k的普通记录分析。本文基于运行顺序统计量序列Un=Xn-k+1:n直接推导k记录过程的概率结构，表明(Un)关于其自然滤子形成马尔可夫链并给出明确转移核。关键步骤是高阶统计量的条件独立性性质。在F连续时，通常的类型2 k记录时间几乎必然与(Un)的记录时间一致，从而在统一框架中恢复经典分布结果。还处理了指数情况，其中k记录值形成具有独立指数增量和伽马分布边缘的随机游走，并记录了指数分布的相应特征。
英文摘要
The theory of $k$-record values (Type 2 $k$-records) plays an important role in the study of partial extremes and in statistical inference based on record data. A common approach reduces the analysis of $k$-records associated with a distribution function $F$ to that of ordinary records from the transformed distribution $F_{1:k}(x)=1-(1-F(x))^k$. This representation is widely used to derive distributional and inferential results, often without an explicit construction of the underlying stochastic mechanism, and relies on a structural property of order statistics that, although classical, is typically invoked without proof. We give a direct derivation of the probabilistic structure of $k$-record processes based on the sequence of running order statistics $U_n=X_{n-k+1:n}$, the $k$-th largest among the first $n$ observations. We show that $(U_n)$ forms a Markov chain with respect to its natural filtration, with an explicit transition kernel. The key step is a conditional-independence property of upper order statistics, which we isolate and prove: conditionally on $U_n$, the $k-1$ observations exceeding $U_n$ are distributed as order statistics from the distribution truncated at $U_n$, independently of the whole past trajectory. Under continuity of $F$, the usual Type 2 $k$-record times coincide almost surely with the record times of $(U_n)$. This yields a transparent construction of the $k$-record process as the record process of a Markov chain, and classical distributional results -- including the representation through $F_{1:k}$ and the joint density of the first $m$ $k$-record values -- are recovered in a unified framework. We also treat the exponential case, in which the $k$-record values form a random walk with independent exponential increments and Gamma-distributed marginals, and record a corresponding characterisation of the exponential distribution.
