Markov Properties of $k$-Record Processes via Order Statistics

Abstract

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.