Relations for product moments and covariances of kth records from discrete distributions

The aim of this paper is to establish recurrence relations satisfied by product moments and covariances of kth records arising from discrete distributions. They will be evaluated for geometric underlying distribution. Then we use these results to obtain formulas for correlation coefficients of geometric kth records. We consider all three known types of kth records: strong, ordinary, and weak.


Introduction
Let X 1 , X 2 , . . . be a sequence of independent and identically distributed (iid) random variables (rv's) with cumulative distribution function (cdf) F. Let X 1:n ≤ · · · ≤ X n:n stand for the respective order statistics of a random sample (X 1 , X 2 , . . . , X n ). For positive integer k, we define occurrence times (T (k) n , n ≥ 0) and values of kth records (R (k) n , n ≥ 0) as follows: Note that for k = 1 the sequences of 1st record times (T (1) n , n ≥ 0) and 1st records (R (1) n , n ≥ 0) reduce to the sequences of record times (T n , n ≥ 0) and record values (R n , n ≥ 0), respectively. For a deeper discussion of records and kth records we refer the reader to the monographs of Arnold et al. (1998) and Nevzorov (2001).
If the underlying distribution is continuous, then the sequence of kth records from a cdf F is distributed as the sequence of records from a cdf F 1:k = 1 − (1 − F) k : where d = denotes equality in distribution; see Nevzorov (2001, Theorem 22.6). This fact implies that in the continuous case distributional properties of kth records can be derived in a direct manner from those of records. However, there are some results concerning kth records which do not follow from the corresponding ones for records. An example of such non-trivial results are various bounds on moments of kth records, see for example Raqab and Rychlik (2002, Rychlik (2004, 2005), Charalambides and Rychlik (2008), Goroncy andRychlik (2011), Jasiński (2016), Kozyra and Rychlik (2017).
The situation becomes more complicated for discrete distributions. López-Blázquez et al. (2005) and Dembińska and López-Blázquez (2005b) showed that then (1.1) is not valid. Moreover, in the last-mentioned paper definitions of strong and weak kth records were introduced.
Properties of weak kth records, (ordinary) kth records and strong kth records are extensively studied in the literature. We refer to Dembińska and López-Blázquez (2005a), Dembińska (2007aDembińska ( , b, 2008, Dembińska and Danielak (2008) and Oncel and Aliev (2016). However, many problems concerning kth records from discrete distributions are still open. The aim of this paper is to fill this gap in the literature. Precisely, we derive recurrence relations satisfied by product moments of kth records arising from discrete distributions. These results together with the corresponding ones for single moments obtained by Dembińska and Danielak (2008) allow us to establish direct formulas for covariances and correlation coefficients between R (k) m and W (k) n or S (k) m and S (k) n , 0 ≤ m < n in some specific cases, when the underlying distribution is geometric.

Basic properties of discrete kth records
In this section, we recall some results concerning the distribution of discrete kth records, which can be found in Dembińska and López-Blázquez (2005b) and Dembińska and Danielak (2008). They will be useful in Sect. 3.
From now on we assume that X 1 , X 2 , . . . are iid rv's with support on non-negative integers, probability mass function p j = P(X 1 = j) and q j = P(X 1 ≥ j), j ≥ 0.
The following lemma describes how the distributional properties of discrete strong kth records can be derived from properties of records (R n , n ≥ 0).

Lemma 1 For any
(2.1) For ordinary and weak kth records the analogs of (2.1) are not valid. To analyze their properties, it is convenient to define new rv's η To pass from dependent ordinary kth records and weak kth records to independent rv's we use the lemma and representation given below.
Now, we are in a position to derive formulas for product moments and covariances of kth records, weak kth records and strong kth records. All these results will be evaluated for the geometric underlying distribution ge( p) with p j = pq j , j = 0, 1, . . . , where q = 1 − p ∈ (0, 1). We will use the recurrence relations for the first moment of any discrete ordinary and weak kth records established by Dembińska and Danielak (2008). They showed that for k = 1, 2, . . .
and η (k) i are described in Lemma 2. Moreover, in the geometric case, they obtained simple expressions for means and variances as follows where ( f (q)) ( j) stands for the jth derivative of the function f (q). In the next section, (2.4) and (2.5) will be used to get covariances and correlation coefficients for geometric kth records.

Product moments and covariances
To derive formulas expressing product moments of kth records arising from discrete distributions we will use a multivariate extension of Hoeffding's (1940) lemma. Block and Fang (1988) showed that if Z 1 , . . . , Z d are non negative rv's then If Z 1 , . . . , Z d take values in the set of non-negative integers, the above expression can be written as (3.1) In particular, we obtain well-known formulas for the first two moments of a single rv Z 1 :

kth records
We begin with a general result on ordinary kth records. We use the convention that R (k) −1 = 0 for k ≥ 1. Theorem 1 For any k = 1, 2, . . . , and 0 ≤ m < n: Proof Using (3.1) for d = 2 and Representation 1 we get By similar arguments, for m 2 ≥ m 1 , we get Now, applying (3.3) and (3.4) to (3.2), we complete the proof.
Theorem 1 and formula (2.2) give the following recurrence relations for covariances between kth records.
Corollary 1 For any k = 1, 2, . . . , and 0 ≤ m < n: The geometric distribution seems to be the only one for which the recurrence relations given in Theorem 1 and Corollary 1 lead to simple expressions.
Theorem 2 Assume that the parent distribution is geometric ge( p). Then for k = 1, 2, . . . and 0 ≤ m < n: (3.6) and Var(R for m ≥ 0, after some algebra, we obtain m 1 has a binomial distribution with parameters (m 2 −m 1 )k and p. Consequently, by (3.7) we have Using again (3.7), we can rewrite the last series as , m ≥ 1.
Recurrence relations (3.5) and (3.6) can be further simplified for k = 1 and k = 2. An easy computation shows that if k = 1 then we get (3.9) Note that the above conclusion can be also derived by use of the well-known represen- Arnold et al. (1998).

Weak kth records
The following theorem is analogous to Theorem 1 and may be proved in much the same way with the convention that W (k) −1 = 0 for k ≥ 1, so its proof is omitted.
Corollary 2 For any k = 1, 2, . . . , and 0 ≤ m < n: The next result shows, that the conclusion of Theorem 3 and Corollary 2 can be simplified in the geometric case.
Theorem 4 Assume that the parent distribution is geometric ge( p). Then for k = 1, 2, . . . and 0 ≤ m < n: Proof In the geometric case, the rv M (k) m 1 is a sum of iid rv's ξ (k) Using the formulas . (3.14) m 1 has a negative binomial distribution nb ((m 2 − m 1 )k, 1− p) and by (3.13), we obtain where the last equality is a consequence of (3.13). Combining it with (3.14) and Theorem 3 we obtain (3.11). Moreover, analogously to the proof of (3.6), formulas (3.11) and (2.4) imply immediately (3.12). In particular, for m = 0 we get Cov(W We can simplify recurrence relations (3.11) and (3.12) for k = 1 and k = 2. When k = 1, simple algebra yields Note that the above expressions can be also derived by use of the well-known representation W n d = n+1 j=1 X j , where X 1 , X 2 , . . . are original iid ge( p) rv's; see e.g. Dembińska (2007b).
Setting k = 2 and m = 0 in (3.11) and (3.12), we obtain Observe that for odd j: while for even j: Combining the above relations with (3.10) and we get for j ≥ 0: which, after some algebra, gives Remark 2 Note that (3.12) implies that for the geometric distribution we have n , n, k = 1, 2 . . . .

Strong kth records
By Lemma 1, to derive expressions for the product moments of strong kth records from cdf F, it suffices to find the corresponding ones of records from cdf F 1:k : This relation enables us to obtain easily the desired formulas for geometric strong kth records. Formulas given in (3.9) with the fact that F 1:k is geometric ge(1 − q k ) provided that F is geometric ge( p), imply the following theorem.
Theorem 5 For any 0 ≤ m < n and k = 1, 2, . . .: Moreover, it is known that the correlation coefficient between geometric records R m and R n is equal to √ (m + 1)/(n + 1), m < n (see, for example, Arnold et al. 1998, p. 40). This immediately implies that in the geometric case we get (3.15)

Tables
In this section, we present Tables 1, 2, 3 with numerical values of correlation coefficients between strong 2nd records S (2) m and S (2) n , 2nd records R (2) m and R (2) n and weak 2nd records W (2) n given in (3.15), which is also the maximum correlation possible between any continuous square-integrable kth records (see, Nevzorov 1992). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.