2022/08/14 by Li, Anshui
#62B10 #62H10 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2208.06791
Let X1,~X2,⋯ be a sequence of i.i.d random variables which are supposed to be observed in sequence. The nth value in the sequence is a k-record~value if exactly k of the first n values (including Xn) are at least as large as it. Let \bf Rk denote the ordered set of k-record values. The famous Ignatov's Theorem states that the random sets \bf Rk(k=1,2,⋯) are independent with common distribution. We introduce one new record named recent-k-record (RkR in short) in this paper: Xn is a j-RkR if there are exactly j values at least as large as Xn in Xn-k,~Xn-k+1,⋯,~Xn-1. It turns out that RkR brings many interesting problems and some novel properties such as prediction rule and Poisson approximation which are proved in this paper. One application named "No Good Record" via the Lovász Local Lemma is also provided. We conclude this paper with some possible connection with scan statistics.