Conditions for finiteness and bounds on moments of record values from iid continuous life distributions

Abstract

We consider the standard and kth record values arising in sequences of independent identically distributed continuous and positive random variables with finite expectations. We determine necessary and sufficient conditions on the type of record k, its number n and moment order r so that the rth moment of the n value of kth record is finite for every parent distribution. Under the conditions we present the optimal upper bounds on these moments expressed in the scale units being the respective powers of the first population moment. The theoretical results are illustrated by some numerical evaluations.

1 Introduction

The notion of records in probabilistic setup was defined by Chandler (1952). Given a sequence of random variables $X_1,X_2,\ldots$, we say that the upper record value occurs in the sequence at time j and becomes equal to $X_j$ if $X_j > X_i$, $i <j$. Using the notion of order statistics $X_{1:j} \le \ldots \le X_{j:j}$ arising by ordering $X_1,\ldots ,X_j$ in the ascending order, we obtain the record at time j if $X_{j:j} > X_{j-1:j-1}$. Dziubdziela and Kopociński (1976) generalized the notion introducing so called kth records which appear when the new value of kth maximum becomes greater than the previous one. Formally, for $n \ge 1$ the sequences of consecutive kth record times $T_n^{(k)}$ and kth record values $R_n^{(k)}$ are defined as follows

$$\begin{aligned}T_1^{(k)} = &\;k, \\T_n^{(k)} = & \min \{ j > T_{n-1}^{(k)}: X_{j+1-k:j} > X_{j-k:j-1} \}\\= & \min \{ j > T_{n-1}^{(k)}: X_{j+1-k:j} > X_{T_{n-1}^{(k)}+1-k:T_{n-1}^{(k)}} \}, \qquad j=2,3,\ldots, \\ \\R_1^{(k)} = &\;X_{1:k}, \\ R_n^{(k)} = & \;X_{T_{n}^{(k)}+1-k:T_{n}^{(k)}} ,\qquad \qquad \qquad \qquad \qquad \qquad \qquad \;\; j=2,3,\ldots , \end{aligned}$$

For $k=1$ the above definitions describe the classic record times and values of Chandler (1952).

If the joint distribution of $X_1,X_2,\ldots$ does not admit the repetitions almost surely (e.g., $X_i$ are iid with a continuous parent distribution), then the sequences of kth records are infinite. The kth records with $k\ge 2$, although seemingly less natural and intuitive than the first ones, attract increasing interest of statistical investigators of extreme events. The main reason is that they occur much more often. Indeed, if a new first record appears in the sequence, the all the kth records change: the old kth records become new $(k+1)$th records. If a new observation becomes the $\ell$th maximum, then we get new kth records for $k \ge \ell$, but the other kth records $k < \ell$ remain unchanged.

The theory of records is best developed in the model of iid continuously distributed random variables. In particular, the distribution function of the nth value of kth record is the composition $G_n^{(k)}\;\circ\;F$, where

$$\begin{aligned} G_n^{(k)}(x) = 1 - (1-x)^k \sum\limits_{i=0}^{n-1} \frac{[-k\ln (1-x)]^i}{i!}, \qquad 0<x<1, \end{aligned}$$

(1)

and F is the parent distribution function of the original observations. Moreover, if F has a density function f with respect to the Lebesgue measure, then the density function of $R_n^{(k)}$ exists and has the form $g_n^{(k)}\;\circ\;F \cdot f$, where

$$\begin{aligned} g_n^{(k)}(x) = \frac{k^n}{(n-1)!} (1-x)^{k-1} [-\ln (1-x)]^{n-1}, \qquad 0<x<1, \end{aligned}$$

(2)

The above establishments can be found, for example, in the monographs by Arnold et al (1998) and Nevzorov (2001), devoted to the record value theory.

In the paper we consider the classic model of iid random variables $X_1,X_2,\ldots$ with a continuous distribution function F. We additionally assume that $X_1$ is positive and has a finite expectation $\mu >0$. Our main results are following. For every type $k \in \mathbb {N}$ and number $n \in \mathbb {N}$ of the record value $R_n^{(k)}$ we determine the necessary and sufficient conditions on the raw moment order $r>0$ such that $\mathbb {E}\big [ R_n^{(k)}\big ]^r < \infty$ for all parent distribution functions F satisfying the assumptions. Our conditions are following: $r \le k$ if $n=1$, and $r<k$ if $n \ge 2$. Moreover, under these conditions we establish the sharp upper bounds on $\mathbb {E}\big [ R_n^{(k)}\big ]^r$ expressed in the scale units being the respective powers $\mu ^r$ of the mean of the parent distribution which are valid for all continuous F supported on $\mathbb {R}_+$. The bounds are presented in Sect. 2. In particular, these results provide the sufficiency proof of our conditions for finiteness of moments of record values. The necessity of the conditions are proven in Sect. 3. In Sect. 4 we present and briefly discuss some numerical examples of the bounds determined in Sect. 2. Some conclusions of our results are presented in Sect. 5.

The problem of existence of kth record values has been hitherto solved only for some special cases of k and n. For $k=n=1$ it is trivial because $R_1^{(1)} =X_1$. For $k=1 <n$ the solution was presented by Nagaraja (1978), Sect. 2). It follows that both $\mathbb {E} \big [ R_n^{(1)} \big ] ^r < \infty$ for all baseline distributions with finite mean iff $r <1$. For $n=1 <k$, the conclusion can be deduced from Sen (1959) who considered order statistics. This was explicitly stated by Papadatos (2021): if $0< \mathbb {E} X_1 < \infty$ then $\mathbb {E}\big [ R_1^{(k)}\big ]^r < \infty$ for all parent distributions iff $r \le k$. Here we focus on the remaining cases $k,n \ge 2$. To the best of our knowledge, the most general sufficient conditions were presented by Cramer et al. (2002), Theorem 2.4 and the following Remark (v). They proved that finiteness of $\mathbb {E}X_1$ assures that $\mathbb {E}\big [R_n^{(k)}\big ]^r < \infty$ if $r < k$ and $r \le n$.

The first result concerning the bounds on the expectations of record values was presented by Nagaraja (1978) who determined the sharp evaluations of the expectations of the standard record values from the population means expressed in the population standard deviation units. Raqab (1997) derived analogous bounds for the kth records. These results were generalized by Raqab (2000) and Raqab and Rychlik (2002), respectively, by considering more general scale units based on various central absolute moments of the original variables. Klimczak (2007) considered kth records from bounded populations expressing the expectation bounds in terms of the length of the support interval. There are known some optimal mean-variance bounds on the expectations of record values from restricted families of distributions. The most general results are due to Bieniek (2007) and Goroncy (2019) who considered the families with monotone generalized failure rates. We also mention the bounds on differences of record values studied in Danielak (2005), and Danielak and Raqab (2004). Evaluations of record raw moments different from the first ones have not been presented by now in the literature. We can only mention analogous results for the order statistics determined in Papadatos (2021).

2 Bounds

In this section we present the sharp bounds on the ratios $\frac{\mathbb {E}\big [R_n^{(k)}\big ]^r }{\mu ^r}$ for $r<k \ge 1$ and $n \ge 2$. We exclude from our considerations the first values of kth records, because these are minima of the first k observations. The bounds for the moments of order statistics, and sample minima in particular were determined by Papadatos (2021). He proved that

$$\begin{aligned} \mathbb {E}X_{1:k}^r \le \mu ^r, \qquad 0 < r \le k. \end{aligned}$$

(3)

For $r <k$ the bound is attained by any degenerate parent distribution (see Papadatos (2021), Remark 4). If $r =k$, we get the equality in Eq. (3) iff the parent distribution is supported on two points 0 and $\frac{\mu }{p}$ with probabilities $1-p$ and p for arbitrary $0<p< 1$ (see Papadatos (2021), Corollary 3). A classic result by Sen (1959) says that if $r > k$, then there exist parent life distributions such that for iid $X_1,\ldots , X_k$ we have $\mathbb {E}X_1 < \infty$ and $\mathbb {E}X_{1:k}^{r}=+\infty$. Since in our model continuity of random variables is desired, we can replace $\mathbb {E}X_{1:k}^{r}$ by $\mathbb {E}[R_1^{(k)}]^r$ in the above relations, but the attainability conditions should be modified. The bounds are attained in the limit by sequences of continuous parent distributions converging weakly to the corresponding optimal discrete ones.

In the sequel, we use the following lemma.

Lemma 1

Fix $k,n \ge 2$ and $1 \le r <k$. Then the function

$$\begin{aligned} h(x) = h_n^{(k)}(x;\;r) =\frac{1-G_n^{(k)}(x)}{(1-x)^r } , \qquad 0\le x < 1, \end{aligned}$$

(4)

is maximized by the unique point $0< \alpha = \alpha _n^{(k)}(r)< 1 - \exp \left( - \frac{n-1}{r+k-2}\right)$ that is the unique solution to the equation

$$\begin{aligned} r[1- G_n^{(k)}(x)] = (1-x) g_n^{(k)}(x), \qquad 0<x<1. \end{aligned}$$

(5)

We denote the maximal value of function Eq. (4) by

$$\begin{aligned} B_n^{(k)}(r) =\frac{1-G_n^{(k)}(\alpha )}{(1-\alpha )^r }= \frac{g_n^{(k)}(\alpha )}{r(1-\alpha )^{r-1}} = \frac{k^n}{r(n-1)!} (1-\alpha )^{k-r} [-\ln (1-\alpha )]^{n-1}. \end{aligned}$$

The middle formula follows from Eq. (5). The last one provides the explicit form of it.

The proof of Lemma 1 is postponed to the Appendix.

We first take into account the moments of record values of orders $0< r <1$ for $k,n \ge 2$. To this end we use a simplified version of Theorem 1 in Moriguti (1953) (see Lemma 2) and the Hölder inequality (see Lemma 3).

Lemma 2

Suppose that a real function g defined on [a, b] has a finite integral. Let $\underline{g}$ denote (the right-continuous version of, say) the derivative of the greatest convex minorant $\underline{G}$ of the antiderivative $G(x) = \int _{a}^{x} g(t)dt$, $a \le x \le b$, of g. Then for every nondecreasing function $h : [a,b]\mapsto \mathbb {R}$ we have

$$\begin{aligned} \int _{a}^{b} g(x)h(x)\,dx \le \int _{a}^{b} \underline{g}(x)h(x) \,dx \end{aligned}$$

(6)

under the assumption that both the integrals exist. The equality in Eq. (6) is attained it h is constant on every interval contained in the set $\{ x \in [a,b]: \, \underline{G}(x) < G(x)\}$.

We use the above construction for the density functions Eq. (2). For $k, n \ge 2$ the derivative of the greatest convex minorant of the antiderivative Eq. (1) of Eq. (2) has the form

$$\begin{aligned} \underline{g}_n^{(k)}(x) = g_n^{(k)}(\min \{ x, \alpha _n^{(k)}(1) \}), \end{aligned}$$

where $\alpha _n^{(k)}(1)$ is the unique solution to Eq. (5) with $r =1$. The greatest convex minorant is equal to the antiderivative $G_n^{(k)}$ (see Eq. 1) on the interval $(0,\alpha _n^{(k)}(1))$ and less than $G_n^{(k)}$ on $(\alpha _n^{(k)}(1), 1)$. These facts were established and applied in a number of papers, see, e.g., Raqab (1997) and Raqab and Rychlik (2002). The Hölder inequality can be found, e.g., in Mitrinović (1970).

Lemma 3

Let g and h be non-negative non-zero elements of the Banach spaces $L^p([a,b],dx)$ and $L^q([a,b],dx)$, respectively, for some $1<p <\infty$ and $1< q = \frac{p}{p-1} < \infty$. Then

$$\begin{aligned} \int _{a}^{b} g(x)h(x)\,dx \le \left[ \int _{a}^{b} g^p(x) \,dx \right] ^{1/p} \left[ \int _{a}^{b} h^q(x) \,dx \right] ^{1/q}, \end{aligned}$$

(7)

and the equality in Eq. (7) holds if

$$\begin{aligned} g(x)= c \, h^{q/p}(x) \end{aligned}$$

almost everywhere on [a, b] for some positive c.

Theorem 1

Let $X_1, X_2, \ldots$, be an infinite sequence of iid continuous positive random variables with a mean $0< \mu < \infty$. Then for $k ,n \ge 2$ and $0< r <1$ we have

$$\begin{aligned} \frac{\mathbb {E}\big [R_n^{(k)}\big ]^r}{\mu ^r} \le B = B_n^{(k)}(r), \end{aligned}$$

(8)

where

$$\begin{aligned} \begin{aligned} B_n^{(k)}(r)&=\frac{k^{n}}{(n-1)!} \left[ \left( \frac{1-r}{k-r} \right) ^\frac{n-r}{1-r} \gamma \left( \frac{n-r}{1-r}, - \frac{k-r}{1-r} \ln (1-\alpha ) \right) \right. \\&+ (1-\alpha )^\frac{k-r}{1-r} [-\ln (1-\alpha )]^\frac{n-1}{1-r}\Bigg ]^{1-r}, \end{aligned} \end{aligned}$$

(9)

$\gamma (t, x) = \int _{0}^{x} y^{t-1} e^{-y}\,dy$ is the lower incomplete gamma function, and $\alpha = \alpha _n^{(k)}(1)$ is defined in Lemma 1 (see Eq. 5).

The bound in Eq. (8) is attained in the limit by continuous parent distributions tending weakly to the distribution function

$$\begin{aligned} \begin{aligned} F(x)=F_n^{(k)}(x;\;r)=\left\{ \begin{array}{ll}0&{}x\le 0,\\ {(g_n^{(k)})}^{-1}\left( B_n^{(k)}(r)\left( \frac{x}{\mu }\right) ^{1-r}\right) ,&{}0< x<\mu \left[ \frac{g_n^{(k)}(\alpha )}{B_n^{(k)}(r)}\right] ^{\frac{1}{1-r}},\\ 1,&{}x>\mu \left[ \frac{g_n^{(k)}(\alpha )}{B_n^{(k)}(r)}\right] ^\frac{1}{1-r},\end{array}\right. \end{aligned} \end{aligned}$$

(10)

($\big (g_n^{(k)}\big )^{-1}$ denotes here the inverse of the increasing part of Eq. 2) with the atom of size $1-\alpha _n^{(k)}(1)$ at the right end-point of the support, and absolutely continuous part between 0 and the atom.

The proof of Theorem 1 can be found in the Appendix.

Now we consider the cases $k=1$ with $n\ge 2$ and $0< r <1$. The idea of getting the bounds is analogous to that of the previous theorem. However, for $k=1$ we can obtain explicit formulae and we decided to present them below. The reason is that in this case the method presented in Lemma 1 generates $\underline{g}_n^{(1)}(x) = g_n^{(1)}(x)$, $0<x<1$, because Eq. (2) for $k=1$ is then increasing, its antiderivative is convex, and so it coincides with its greatest convex minorant. This substantially simplifies the evaluations.

Theorem 2

For $X_1,X_2,\ldots$ being iid random variables with an expectation $0< \mu < \infty$, and the respective classic records with $k=1$, $n\ge 2$ and $0< r <1$ we get

$$\begin{aligned} \frac{\mathbb {E}\big [R_n^{(1)}\big ]^r}{\mu ^r} \le B_n^{(1)}( r) = \frac{\Gamma ^{1-r} \left( \frac{n-r}{1-r} \right) }{(n-1)!}, \end{aligned}$$

(11)

where $\Gamma (t)= \gamma (t, +\infty ) = \int _{0}^{\infty } y^{t-1} e^{-y}\,dy$ is the standard gamma function. The equality is attained in Eq. (11) by the rescaled Weibull parent distribution function

$$\begin{aligned} F(x) = \left\{ \begin{array}{ll} 0, &{} x \le 0, \\ 1- \exp \left( - \left[ \Gamma \left( \frac{n-r}{1-r} \right) \frac{x}{\mu } \right] ^\frac{1-r}{n-1} \right) , &{} x \ge 0, \end{array} \right. \end{aligned}$$

with the shape parameter $\frac{1-r}{n-1}$.

We postpone the proof of Theorem 2 to the Appendix.

If $\frac{n-r}{1-r}$ is an integer number m, say, i.e. $r = \frac{i+1}{n+i}$ for some $i=0,1,\ldots$, then the right-hand side of Eq. (11) has yet nicer form $\frac{[(m-1)!]^\frac{n-1}{m-1}}{(n-1)!}$.

We now concentrate on evaluations of rth moments of kth record values for $r \ge 1$. At the beginning, we exclude from our investigations the classic moments $R_n^{(1)}$, $n \ge 2$, because there exist parent distributions such that $\mathbb {E}X_1 = \mu < \infty$ and $\mathbb {E}R_2^{(1)} = +\infty$ (see Nagaraja (1978)). Otherwise we use the following lemma which was presented in Papadatos (2021), Corollary 4.

Lemma 4

Let F be the distribution function of a non-negative random variable with a positive and finite mean. Then for all $r >1$

$$\begin{aligned} r \int _{0}^{\infty } x^{r-1} [1-F(x)]^r dx \le \left[ \int _0^\infty [1-F(x)]\,dx \right] ^r, \end{aligned}$$

(12)

and the equality is attained if F is a two-point distribution function supported on 0 and a positive number.

Observe that for $r =1$ relation Eq. (12) becomes a trivial equality with no assumptions on F.

Theorem 3

Let $X_1,X_2, \ldots$ satisfy the assumptions of Theorem 1. For $n,k \ge 2$ and $1\le r < k$ we have

$$\begin{aligned} \frac{\mathbb {E}\big [R_n^{(k)}\big ]^r}{\mu ^r} \le B_n^{(k)} (r) , \end{aligned}$$

(13)

where

$$\begin{aligned} B_n^{(k)} (r) = \frac{1-G_n^{(k)}(\alpha _n^{(k)}(r))}{(1-\alpha _n^{(k)}(r))^r} = \frac{g_n^{(k)}(\alpha _n^{(k)}(r))}{r(1-\alpha _n^{(k)}(r))^{r-1}} \end{aligned}$$

with $\alpha _n^{(k)}(r)$ determined by the Eq. (5).

The equality in Eq. (13) is attained in the limit by continuous parent distributions tending weakly to the two-point distribution

$$\begin{aligned} \mathbb {P}(X_1=0) = \alpha _n^{(k)}(r) = 1- \mathbb {P}\left( X_1 = \frac{\mu }{1-\alpha _n^{(k)}(r)}\right) . \end{aligned}$$

The proof of Theorem 3 is presented in the Appendix.

3 Conditions for moment finiteness

Here we present the necessary and sufficient conditions for finiteness of rth moments of nth values of kth records for arbitrary continuous life distributions of the baseline sequence with a finite mean. As we mentioned in the introduction, the conditions are known in some special cases. If $n=1$ then $R_1^{(k)}= X_{1:k}$, and the result is immediately concluded from the classic paper by Sen (1959) on order statistics (see also Papadatos (2021)). If $r\le k$, then for every parent distribution with $\mathbb {E}X_1 < \infty$ we have $\mathbb {E}[R_1^{(k)}]^r < \infty$ as well. For $r>k$ there exist parent distribution functions such that $\mathbb {E}X_1 < \infty$ and $\mathbb {E}[R_1^{(k)}]^r = \infty$. An example of such distribution function is

$$\begin{aligned} F(x) = 1- \frac{e}{x(\ln x)^2}, \qquad x > e, \end{aligned}$$

(14)

(see Papadatos (2021), Remark 3). If $n \ge 2$ and $k=1$, the respective condition $r<1$ was established in Nagaraja (1978). In Lemma 2.1 of the paper he proved that finiteness of $\mathbb {E}X_1$ and $\mathbb {E}[R_n^{(1)}]^r < \infty$ for all $n \ge 2$ and $0<r<1$. On the other hand, for the distribution function Eq. (14) we have $\mathbb {E}X_1 =2e < \infty$ and $\mathbb {E}R_n^{(1)}= \infty$ for all $n\ge 2$.

In Theorem 4 below we treat all the remaining cases with $n,k \ge 2$. Cramer et al. (2002) proved that finiteness of $\mathbb {E}X_1 =\mu$ assures that $\mathbb {E}\big [R_n^{(k)}\big ]^r < \infty$ if $r < k$ and $r \le n$. Our conditions are essentially weaker.

Theorem 4

Let $X_1,X_2, \ldots$ be positive iid random variables with a continuous marginal distribution and finite mean. The necessary and sufficient condition for finiteness of $\mathbb {E}\big [R_n^{(k)}\big ]^r$ for all parent distributions and some $n,k \ge 2$ is $r<k$.

We present the proof of Theorem 4 in the Appendix. It consists in delivering an example of a single absolutely continuous life marginal distribution of the elements of the sequence $X_1,X_2,\ldots ,$ such that $\mathbb {E}X_1 < + \infty$ and $\mathbb {E}\big [ R^{(k)}_n\big ]^r = + \infty$ for all $k \ge 2$, $n \ge 2$, and $r \ge k$. We finally present a simple generalization of the results of this section.

Corollary 1

Consider a sequence of positive and continuous iid random variables $X_1, X_2,\ldots$. Then for arbitrary parent distribution with finite pth moment we have $\mathbb {E} \big [ R_n^{(k)}\big ]^r < \infty$ iff either $n=1$ and $r \le pk$ or $n \ge 2$ and $r < pk$.

4 Numerical results

Here we illustrate the results of Sect. 2 by some numerical examples. In Tables 1, 2 and 3 we present the bounds on the ratios of rth moments of kth records and the rth powers of the population means for some values of the first, second and third records, respectively. In all the cases we consider the second, third, fifth and tenth values of kth records. Table 1 is based on Theorem 2, and contains the evaluations of moments of orders $r=0.25$, 0.5, and 0.9. In Tables 2 and 3 we present the bounds for the moments of orders $r=0.5\,i$, $i=1,\ldots , 2k-1,$ and $r=k-0.1$, where $k=2$ and 3, respectively. The results of the first columns of Tables 2 and 3 are concluded from Theorem 1, and the other ones are calculated with use of Theorem 3.

Table 1 Bounds $B_n^{(1)}(r)$ on the ratios $\frac{\mathbb {E}\big [R_n^{(1)}\big ]^r}{\mu ^r}$ for the first records

Table 2 Bounds $B_n^{(2)}(r)$ on the ratios $\frac{\mathbb {E}\big [R_n^{(2)}\big ]^r}{\mu ^r}$ for the second records

Table 3 Bounds $B_n^{(3)}(r)$ on the ratios $\frac{\mathbb {E}\big [R_n^{(3)}\big ]^r}{\mu ^r}$ for the third records

All the bounds are greater than 1, which is a consequence of the sharp bounds

$$\begin{aligned} \frac{\mathbb {E}\big [R_1^{(k)}\big ]^r}{\mu ^r} \le 1 \end{aligned}$$

proved by Papadatos (2021). We also note that the bounds are increasing in n and decreasing in k, which is also obvious, because we have the relations $R_n^{(k)} < R_{n+1}^{(k)}$ and $R_n^{(k+1)} > R_n^{(k)}$ following from the definitions. Moreover, they increase with respect to r and tend to infinity as r approaches k. In the case $k=1$ we can apply Eq. (11) with the factorial approximation of the Gamma function and Stirling formula in order to obtain

$$\begin{aligned} \lim _{r \rightarrow 1} (1-r) \frac{B_{n+1}^{(1)}(r)}{B_n^{(1)}(r)} = \frac{1}{e} \, \left( \frac{n}{n-1} \right) ^{n-1}. \end{aligned}$$

The right-hand side expression converges to 1 as n increases. We cannot provide such approximations for $\frac{B_{n+1}^{(k)}(r)}{B_n^{(k)}(r)}$ with $r \rightarrow k \ge 2$, because $\alpha _n^{(k)}(r)$ do not have analytic representations, and the bounds, dependent on them, do not have explicit formulae. Anyway, we can see that for $k=2,3$ and $r=k-0.1$ the bounds $B_{10}^{(k)}(r)$ are greater than $10^{9}$.

5 Conclusions

We consider the moments of kth upper record values in the classic model of sequences of independent and identically continuously distributed positive random variables. Our purpose is to determine the necessary and sufficient conditions for the moment orders $r>0$ such that the rth moment $\mathbb {E}\big [ R_n^{(k)}\big ]^r$ of the nth value, $n \ge 1$, of kth record, $k \ge 1$, is finite for all parent distributions with a finite mean. Since the solution is known in the literature for the particular cases $k=1 \le n$ and $n=1 \le k$, we focus on the remaining cases $k,n \ge 2$ proving that the necessary and sufficient conditions are $r< k$ for arbitrary $n \ge 2$. The necessity proof consists in constructing the parent distribution with a finite expectation such that $\mathbb {E}\big [ R_n^{(k)}\big ]^r = + \infty$ for all $n \ge 2$ and $r \ge k \ge 2$. Instead of the sufficiency proof, we provide a stronger result: for every $n \ge 2$ and $r<k \ge 1$ we determine the sharp upper bounds on $\mathbb {E}\big [ R_n^{(k)}\big ]^r$ over all the parent distributions with finite means, expressed in the scale units being the rth powers of the mean of the single observation. Numerical exemplary evaluations show that the bounds are extremely large when the moment orders r are close to the border value k even for moderate n.

Our findings allow the researchers not to bother about existence of moments of kth records of orders less than pk if they consider sequences of random variables with arbitrary parent life distributions with finite pth moments. We hope that the tools presented in the paper shall be useful in determining the necessary and sufficient conditions for finiteness of moments in other models of ordered random variables, e.g., for progressively type II censored order statistics and generalized order statistics.

Appendix

The Appendix contains the proofs of Lemma 1 and Theorems 1-4.

Proof of Lemma 1. The function Eq. (4) is positive and satisfies $h(0)=1$.

By the de l’Hospital rule

$$\begin{aligned} \lim _{x \rightarrow 1} h(x) = \lim _{x \rightarrow 1} \frac{g_n^{(k)}(x)}{r(1-x)^{r-1}} = \frac{ k^n}{r(n-1)!} \lim _{x \rightarrow 1} (1-x)^{k-r}[-\ln (1-x)]^{k-1} =0, \end{aligned}$$

(15)

and

$$\begin{aligned} h'(x) = \frac{r[1-G_n^{(k)}(x)]- (1-x)g_n^{(k)}(x)}{(1-x)^{r+1}}. \end{aligned}$$

(16)

The numerator N(x), say, has the derivative

$$\begin{aligned} N'(x) = (1-r)g_n^{(k)}(x) -(1-x) [g_n^{(k)}]'(x). \end{aligned}$$

Since

$$\begin{aligned}{}[g_n^{(k)}]'(x)= \frac{k^n}{(n-1)!} (1-x)^{k-2} [-\ln (1-x)]^{n-2}\{n-1-(k-1)[-\ln (1-x)]\}, \end{aligned}$$

we have

$$\begin{aligned} N'(x) = \frac{k^n}{(n-1)!} (1-x)^{k-1} [-\ln (1-x)]^{n-2}\{(r+k-2)[-\ln (1-x)]-n+1\}, \end{aligned}$$

which means that it is negative for $0<x< \beta = \beta _n^{(k)}(r) = 1-\exp \left( - \frac{n-1}{r+k-2}\right) <1$, and positive for $\beta< x <1$. It follows that N(x) is first decreasing and then increasing. Since $N(0)= r \ge 1$, and $N(1) =0$, the function is positive on some interval $\left[ 0, \alpha \right)$, $\alpha =\alpha _n^{(k)}(r) < \beta _n^{(k)}(r)$, and negative elsewhere. The same concerns Eq. (16), because its denominator is always positive. Since $h(0)=1$ and Eq. (15) holds, we conclude that Eq. (4) is increasing on $[0,\alpha )$, and decreasing on $(\alpha ,1)$. Obviously its unique maximum $\alpha = \alpha _n^{(k)}(r)$ is the only point that satisfies Eq. (5) (cf Eq. 16). It is also less than $\beta _n^{(k)}(r) = 1-\exp \left( - \frac{n-1}{r+k-2}\right)$. $\blacksquare$  

Proof of Theorem 1. We have

$$\begin{aligned} \begin{aligned} \mathbb {E}\big [R_n^{(k)}\big ]^r&=\int _0^1 [F^{-1}(x)]^r g_n^{(k)}(x) \,dx \\&\le \int _0^1 [F^{-1}(x)]^r g_n^{(k)}(\min \{ x, \alpha \}) \,dx \\&\le \left[ \int _0^1 \{[F^{-1}(x)]^r\}^\frac{1}{r} \,dx \right] ^r \left[ \int _0^1 [ g_n^{(k)}(\min \{ x, \alpha \})]^\frac{1}{1-r} \,dx\right] ^{1-r} \\&= \mu ^r \left[ \int _0^\alpha [ g_n^{(k)}(x)]^\frac{1}{1-r} \,dx + (1-\alpha ) \big [g_n^{(k)}(\alpha )\big ]^{\frac{1}{1-r}} \right] ^{1-r}. \end{aligned} \end{aligned}$$

(17)

The first inequality holds by Lemma 2 due to the facts that $[F^{-1}(x)]^r$ is nondecreasing, and Eq. (7) holds. The latter the an application of the Hölder inequality with parameters $p = \frac{1}{r}>1$ and $q= \frac{p}{p-1} = \frac{1}{1-r}>1$. Note that changing the variables twice $y=-\ln (1-x)$ and $z = \frac{k-r}{1-r}\,y$, we obtain

$$\begin{aligned} \begin{aligned} \int _0^\alpha \big [ g_n^{(k)}(x)\big ]^\frac{1}{1-r} \,dx&= \left[ \frac{k^n}{(n-1)!}\right] ^\frac{1}{1-r} \int _0^\alpha (1-x)^\frac{k-1}{1-r}[-\ln (1-x)]^\frac{n-1}{1-r}\,dx \\&= \left[ \frac{k^n}{(n-1)!}\right] ^\frac{1}{1-r} \int _{0}^{-\ln (1-\alpha )} y^\frac{n-1}{1-r} e^{-\frac{k-r}{1-r}y}\,dy \\&= \left[ \frac{k^n}{(n-1)!}\right] ^\frac{1}{1-r} \left( \frac{1-r}{k-r} \right) ^\frac{n-r}{1-r} \int _{0}^{-\frac{k-r}{1-r}\ln (1-\alpha )} z^\frac{n-1}{1-r} e^{-z}\,dz \\&= \left[ \frac{k^n}{(n-1)!}\right] ^\frac{1}{1-r} \left( \frac{1-r}{k-r} \right) ^\frac{n-r}{1-r} \gamma \left( \frac{n-r}{1-r}, - \frac{k-r}{1-r} \ln (1-\alpha ) \right) \!, \end{aligned} \end{aligned}$$

(18)

which together with Eq. (2) allows us to rewrite the last line of Eq. (17) as $\mu ^r B_n^{(k)}(r)$.

Consider the relation

$$\begin{aligned} F^{-1}(x) = \mu \, \frac{\big [g_n^{(k)}(\min \{ x, \alpha \})\big ]^\frac{1}{1-r}}{\int _0^1 \big [ g_n^{(k)} (\min \{ x, \alpha \})\big ]^\frac{1}{1-r}\,dx} = \mu \, \frac{\big [g_n^{(k)}(\min \{ x, \alpha \})\big ]^\frac{1}{1-r}}{[B_n^{(k)}(r)]^\frac{1}{1-r}} \end{aligned}$$

(19)

It represents a life distribution function with the mean $\mu$. It satisfies the equality condition for the Hölder inequality in Eq. (17). It is constant on the interval $(\alpha ,1)$, where the antiderivative Eq. (1) of Eq. (2) is greater that its greatest convex minorant which implies that the first inequality in Eq. (17) becomes the equality as well. The explicit representation of the distribution function satisfying Eq. (19) is given in Eq. (10). $\blacksquare$

Proof of Theorem 2. Mimicking Eq. (17) we obtain

$$\begin{aligned} \mathbb {E}\big [R_n^{(1)}\big ]^r \le \mu ^r \left[ \int _0^1 [ g_n^{(1)}(x)]^{\frac{1}{1-r}} \,dx\right] ^{1-r}. \end{aligned}$$

(20)

For $k=1$ and $\alpha =1$ Eq. (18) takes on the form

$$\begin{aligned} \int _0^1 \big [ g_n^{(1)}(x)\big ]^\frac{1}{1-r} \,dx = \frac{\Gamma \left( \frac{n-r}{1-r} \right) }{[(n-1)!]^\frac{1}{1-r}}. \end{aligned}$$

(21)

The relations

$$\begin{aligned} F^{-1}(x) = \mu \, \frac{\big [g_n^{(1)}(x)\big ]^\frac{1}{1-r}}{\int _0^1 \big [ g_n^{(1)} x)\big ]^\frac{1}{1-r}\,dx} = \mu \, \frac{\big [-\ln (1-x)\big ]^\frac{n-1}{1-r}}{\Gamma \left( \frac{n-r}{1-r} \right) } \end{aligned}$$

assuring the equality in Eq. (20) can be rewritten as Eq. (21). This completes the proof. $\blacksquare$

Proof of Theorem 3. The inequality follows from the relations

$$\begin{aligned} \begin{aligned} \mathbb {E}\big [R_n^{(k)}\big ]^r&=\int _{0}^{1} [F^{-1}(x)]^r g_n^{(k)}(x) \,dx \\&= r \int _{0}^{\infty } x^{r-1} [1- G_n^{(k)}(F(x)]^r \,dx \\&\le \sup _{0 < u < 1} \frac{1-G_n^{(k)}(u)}{(1-u)^r}\,\, r \int _{0}^{\infty } x^{r-1} [1- F(x)]^r \,dx \\&\le \frac{1-G_n^{(k)}(\alpha _n^{(k)}(r))}{(1-\alpha _n^{(k)}(r))^r}\left[ \int _0^\infty [1-F(x)]\,dx \right] ^r \\&= B_n^{(k)} (r)\, \mu ^r.\end{aligned} \end{aligned}$$

By Lemma 1, the equality in the former inequality occurs if the only value of F on $(0,\infty )$ different from 1 is $\alpha _n^{(k)}(r)$. By Lemma 4, for $r >1$, we get the equality in the latter inequality if F is supported on 0 and a positive number. Combining the conditions, we deduce that the only distribution that attains the equality in Eq. (13) is one that assigns probability $\alpha _n^{(k)}(r)$ to 0 and $1-\alpha _n^{(k)}(r)$ to a positive number. This distribution satisfies the moment condition if this positive number amounts to $\frac{\mu }{1-\alpha _n^{(k)}(r)}$. This ends the proof. $\blacksquare$

Proof of Theorem 4. The sufficiency of the condition is deduced from Theorem 3. To verify its necessity we should present examples of the parent distributions with finite means such that $\mathbb {E}\big [R_n^{(k)}\big ]^r = \infty$ for given $n\ge 2$ and $r \ge k \ge 2$. In fact, it suffices to construct distributions such that $\mathbb {E}\big [R_2^{(k)}\big ]^k = \infty$ for $k \ge 2$. Our construction works for all $k\ge 2$.

We take a sequence of positive numbers $e_j$, $j=0,1,\ldots$, tending to $\infty$ by the recursive relation

$$\begin{aligned} e_0 = 1, \qquad e_j = e^{e_{j-1}}, \qquad j=1,2,\ldots , \end{aligned}$$

and a family of disjoint intervals

$$\begin{aligned} I_j = \left( e_j , e_j + 1 \right) , \qquad j=0,1,\ldots . \end{aligned}$$

Consider a mixture of distributions $\sum _{j=0}^{\infty }\alpha _j U_j$, where $U_j$ is the uniform distribution over the interval $I_j$, $j=0,1, \ldots$, and the weights are defined as

$$\begin{aligned}&\alpha _0 = 1- \frac{1}{e-1},\\&\alpha _j = \frac{1}{j^2(e_j-e_{j-1})} - \frac{1}{(j+1)^2(e_{j+1}-e_{j})}, \qquad j=1,2, \ldots .\end{aligned}$$

The mixture is a proper probability measure, because

$$\begin{aligned} \begin{aligned}\sum _{j=0}^{\infty } \alpha _j&= 1- \frac{1}{e-1}+ \sum _{j=1}^{\infty } \left[ \frac{1}{j^2(e_j-e_{j-1})} - \frac{1}{(j+1)^2(e_{j+1}-e_{j})}\right] \\&= \lim _{j \rightarrow \infty }\left[ 1- \frac{1}{(j+1)^2(e_{j+1}-e_{j})}\right] = 1. \end{aligned} \end{aligned}$$

The measure is obviously absolutely continuous with respect to the Lebesgue measure, and its density function has the form

$$\begin{aligned} \begin{aligned} f(x) = \left\{ \begin{array}{ll} 1- \frac{1}{e-1}, &{} x \in I_0, \\ \frac{1}{j^2(e_j-e_{j-1})} - \frac{1}{(j+1)^2(e_{j+1}-e_{j})}, &{} x \in I_j, \; j=1,2,\ldots , \\ 0, &{} x \in \mathbb {R} \setminus \bigcup _{j=0}^{\infty } I_j. \end{array} \right. \end{aligned} \end{aligned}$$

(22)

Suppose that the sequence of iid life random variables $X_1,X_2,\ldots$ has the common density function Eq. (22). It has a finite expectation

$$\begin{aligned} \begin{aligned} \mathbb {E}X_1&= \sum\limits _{j=0}^{\infty } \mathbb {E}(X_1|X_1 \in I_j) \mathbb {P}(X_1 \in I_j) = \sum\limits _{j=0}^{\infty }\left( e_j + \frac{1}{2} \right) \mathbb {P}(X_1 \in I_j)\\&= 1- \frac{1}{e-1} + \sum\limits_{j=1}^{\infty } \left[ \frac{e_j}{j^2(e_j-e_{j-1})} - \frac{e_j}{(j+1)^2(e_{j+1}-e_{j})} \right] + \frac{1}{2} \\&= \frac{3}{2} + \sum\limits _{j=1}^{\infty } \frac{1}{j^2} = \frac{3}{2} + \frac{\pi ^2}{6}.\end{aligned} \end{aligned}$$

(23)

For proving that $\mathbb {E}\big [R_2^{(k)}\big ]^k = \infty$ we need to calculate

$$\begin{aligned} \begin{aligned}\mathbb {P}(R_2^{(k)} \in I_j)&= G_2^{(k)}\left( F \left( e_j+1 \right) \right) - G_2^{(k)}\left( F \left( e_j \right) \right) \\&= G_2^{(k)}\left( F \left( e_{j+1} \right) \right) - G_2^{(k)}\left( F \left( e_{j} \right) \right) .\end{aligned} \end{aligned}$$

for $j=1,\, 2,\ldots .$ Remind that the corresponding distribution function of the second value of kth record is

$$\begin{aligned} G_2^{(k)}(x) = 1- [1-F(x)]^k \big (1- k \ln [1-F(x)]\big ). \end{aligned}$$

The distribution function of Eq. (23) satisfies

$$\begin{aligned} F\left( e_{j}\right) = F\left( e_{j-1}+ 1 \right) = \mathbb {P}\left( X_1 \in \bigcup _{i=0}^{j-1}I_i\right) = 1- \frac{1}{j^2(e_{j}-e_{j-1})}, \;j=1,2,\ldots\end{aligned}$$

Since

$$\begin{aligned} \begin{aligned} \left[ 1-F\left( e_j \right) \right] ^k = \frac{1}{j^{2k}(e_{j}-e_{j-1})^k}, \\ -\ln \left[ 1-F\left( e_j \right) \right] = 2 \ln j + \ln (e_{j}-e_{j-1}), \end{aligned} \end{aligned}$$

we have

$$\begin{aligned} G_2^{(k)}\left( F \left( e_j \right) \right) = 1- \frac{1+2k\ln j +k \ln (e_{j}-e_{j-1})}{j^{2k}(e_{j}-e_{j-1})^k}, \end{aligned}$$

and

$$\begin{aligned} \begin{aligned}\mathbb {P}(R_2^{(k)} \in I_j)&= G_2^{(k)}\left( F \left( e_{j+1} \right) \right) - G_2^{(k)}\left( F \left( e_{j} \right) \right) \\&= \frac{1+2k\ln j +k \ln (e_j-e_{j-1})}{j^{2k}(e_j-e_{j-1})^k}- \frac{1+2k\ln (j+1) +k \ln (e_{j+1}-e_{j})}{(j+1)^{2k}(e_{j+1}-e_{j})^k}.\end{aligned} \end{aligned}$$

We are now in a position to prove that

$$\begin{aligned} \mathbb {E}\big [R_2^{(k)}\big ]^k = \sum\limits _{j=0}^{\infty } \mathbb {E}\big ( \big [R_2^{(k)}\big ]^k \mathbf {1}_{I_j}(R_2^{(k)}) \big ) = \infty . \end{aligned}$$

It suffices to show that the summands do not tend to 0 as $j \rightarrow \infty$. We have

$$\begin{aligned} \begin{aligned} \mathbb {E}\big ( \big [R_2^{(k)}\big ]^k \mathbf {1}_{I_j}(R_2^{(k)}) \big )&\ge e_j^k\, \mathbb {P}(R_2^{(k)} \in I_j) \\&=\frac{ e_j^k \,[1+2k\ln j +k \ln (e_j-e_{j-1})]}{j^{2k}(e_j-e_{j-1})^k} \\&- \frac{ e_j^k\, [1+2k\ln (j+1) +k \ln (e_{j+1}-e_{j})]}{(j+1)^{2k}(e_{j+1}-e_{j})^k}. \end{aligned} \end{aligned}$$

Note that

$$\begin{aligned} \lim _{j \rightarrow \infty } \frac{e_j}{e_{j-1}^p}= \lim _{j \rightarrow \infty } \frac{e^{e_{j-1}}}{e_{j-1}^p}= \infty \end{aligned}$$

for all $p \in \mathbb {R}$, because

$$\begin{aligned} \lim _{x \rightarrow \infty } \frac{e^x}{x^p} = \infty . \end{aligned}$$

Therefore

$$\begin{aligned} \lim _{j \rightarrow \infty } \frac{ e_j^k }{(e_j-e_{j-1})^k} = \lim _{j \rightarrow \infty } \frac{1}{\left( 1-\frac{e_{j-1}}{e_j}\right) ^k} =1. \end{aligned}$$

Moreover,

$$\begin{aligned} \begin{aligned} \ln (e_j- e_{j-1})&= \ln \left( e^{e_{j-1}} - e^{e_{j-2}}\right) = \ln \left( e^{e_{j-1}-e_{j-2}}-1\right) + \ln e^{e_{j-2}} \\&= e_{j-1} [1+o(1)]\end{aligned} \end{aligned}$$

with o(1) denoting a sequence tending to 0 as $j \rightarrow \infty$, and in consequence

$$\begin{aligned} 1+ 2 k \ln j + k \ln (e_j-e_{j-1}) = k e_{j-1} (1+ o(1)). \end{aligned}$$

It follows that

$$\begin{aligned} \lim _{j \rightarrow \infty } \frac{ e_j^k \,[1+2k\ln j +k \ln (e_j-e_{j-1})]}{j^{2k}(e_j-e_{j-1})^k} = \lim _{j \rightarrow \infty } \frac{ke_{j-1}}{j^{2k}} = \infty . \end{aligned}$$

By similar arguments

$$\begin{aligned}\lim _{j \rightarrow \infty }&\frac{ e_j^k \,[1+2k\ln (j+1) +k \ln (e_{j+1}-e_{j})]}{(j+1)^{2k}(e_{j+1}-e_{j})^k} \\&= \lim _{j \rightarrow \infty } \frac{e_j^{k+1}}{(e_{j+1}-e_j)^k} \, \frac{k}{(j+1)^{2k}} \\& = \lim _{j \rightarrow \infty }\frac{1}{\left( \frac{e_{j+1}}{e_j^{(k+1)/k}}-\frac{1}{e_j^{1/k}}\right) ^k}\, \frac{k}{(j+1)^{2k}} = 0,\end{aligned}$$

because both the factors tend to 0.

Accordingly,

$$\begin{aligned} \lim _{j \rightarrow \infty } \mathbb {E}\big ( \big [R_2^{(k)}\big ]^k \mathbf {1}_{I_j}(R_2^{(k)}) \big ) =\infty , \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}\big [R_2^{(k)}\big ]^k =\infty . \end{aligned}$$

We finally note that taking the baseline distribution with the density function $\frac{\pi ^2+9}{6\mu } f\left( \frac{\pi ^2+9}{6\mu } \,x\right)$ (see Eq. 22) for arbitrary $\mu >0$ we obtain $\mathbb {E}X_1 =\mu$ and $\mathbb {E}\big [R_2^{(k)}\big ]^k =\infty$. $\blacksquare$

The sequence $e_j$, $j=0,1,\ldots$, used in the proof of Theorem 4 tends to infinity at an extremely fast rate. The approximate values of its first five elements are

$$\begin{aligned} e_0= &\;1,\! \\ e_1= &\;e\quad \approx 2.71828, \\ e_2= & \;e^e\ \ \approx 15.15426, \\ e_3= & \;e^{e^e}\ \approx 3\,814\,279.10476,\\ e_4= & \;e^{e^{e^e}} \approx 2.33150 \cdot 10^{1656520}. \end{aligned}$$

The integer part of $e_5$ has approximately $10^{1.012558\cdot 10^{1656520}}$ digits. It follows that the support intervals $I_j$, $j=0,1,\ldots$, are very dispersed on the positive half-axis, and the corresponding distribution function has a very heavy tail. One could decrease the dispersion by replacing e in the construction of above proof by some $1<a <e$, especially one close to 1 but this would generate more sophisticated formulae. Anyway, for any $a>1$ the tails are heavier than these of the distribution functions

$$\begin{aligned} F_{m,\varepsilon }(x) = \max \left\{ 1- \frac{1}{x \prod _{i=1}^{m-1}\ln _i x (\ln _m x )^{1+\varepsilon }}, 0 \right\} \end{aligned}$$

with $\ln _m x$ denoting m-fold composition of logarithm functions and any integer $m \ge 2$ and $\varepsilon >0$. It can be checked that these distribution functions have finite expectations and satisfy $\mathbb {E}_{m,\varepsilon } \big [ R_n^{(k)}\big ]^k < \infty$ iff $n \le k$ and $\mathbb {E}_{m,\varepsilon } \big [ R_n^{(k)}\big ]^r = \infty$ for all $r > k$ with $n \ge 2$ and r=k with $n \ge k+1$.