## Abstract

We consider the standard and *k*th 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 *r*th moment of the *n* value of *k*th 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 KopociĆski (1976) generalized the notion introducing so called *k*th records which appear when the new value of *k*th maximum becomes greater than the previous one. Formally, for \(n \ge 1\) the sequences of consecutive *k*th record times \(T_n^{(k)}\) and *k*th record values \(R_n^{(k)}\) are defined as follows

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 *k*th records are infinite. The *k*th 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 *k*th records change: the old *k*th records become new \((k+1)\)th records. If a new observation becomes the \(\ell\)th maximum, then we get new *k*th records for \(k \ge \ell\), but the other *k*th 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 *n*th value of *k*th record is the composition \(G_n^{(k)}\;\circ\;F\), where

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

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.Â 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 *k*th 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),Â 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 etÂ al. (2002),Â 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 *k*th 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 *k*th 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 *k*th 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

For \(r <k\) the bound is attained by any degenerate parent distribution (see Papadatos (2021),Â 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),Â 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

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

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

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 HĂ¶lder inequality (see Lemma 3).

### Lemma 2

Suppose that a real function *g* defined on [*a*,Â *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

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

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)Â and Raqab and RychlikÂ (2002). The HĂ¶lder inequality can be found, e.g., in MitrinoviÄ (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

and the equality in Eq. (7) holds if

almost everywhere on [*a*,Â *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

where

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

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

(\(\big (g_n^{(k)}\big )^{-1}\) denotes here the inverse of the increasing part of Eq.Â 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 1Â 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

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

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

We postpone the proof of Theorem 2Â 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 *r*th moments of *k*th 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),Â 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 \(r >1\)

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

where

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

The proof of Theorem 3Â is presented in the Appendix.

## 3 Conditions for moment finiteness

Here we present the necessary and sufficient conditions for finiteness of *r*th moments of *n*th values of *k*th records for arbitrary continuous life distributions of the baseline sequence with a finite mean.Â As we mentioned in the introduction, the conditions are known in some special cases.Â 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)).Â 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

(see Papadatos (2021),Â 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 4Â below we treat all the remaining cases with \(n,k \ge 2\). Cramer etÂ 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 4Â 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 *p*th 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 *r*th moments of *k*th records and the *r*th 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 *k*th 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.

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

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

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 *k*th 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 *r*th moment \(\mathbb {E}\big [ R_n^{(k)}\big ]^r\) of the *n*th value, \(n \ge 1\), of *k*th 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 *r*th 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 *k*th records of orders less than *pk* if they consider sequences of random variables with arbitrary parent life distributions with finite *p*th 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.

## Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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## Acknowledgements

The authors thank two anonymous referees whose valuable comments allowed them to improve the presentation of the paper.

## Funding

The second author has been partially supported by PUT under Grant 0211/SBAD/0122.

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The authors have no competing interests to declare that are relevant to the content of this article.

## Appendix

### 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 lâHospital rule

and

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

Since

we have

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.Â 16). It is also less than \(\beta _n^{(k)}(r) = 1-\exp \left( - \frac{n-1}{r+k-2}\right)\). \(\blacksquare\)Â Â

Proof of Theorem 1. We have

The first inequality holds by Lemma 2Â due to the facts that \([F^{-1}(x)]^r\) is nondecreasing, and Eq. (7)Â holds. The latter the an application of the HĂ¶lder 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

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

It represents a life distribution function with the mean \(\mu\). It satisfies the equality condition for the HĂ¶lder 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).Â \(\blacksquare\)

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

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

The relations

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

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

and a family of disjoint intervals

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

The mixture is a proper probability measure, because

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

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

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

for \(j=1,\, 2,\ldots .\) Remind that the corresponding distribution function of the second value of *k*th record is

The distribution function of Eq. (23) satisfies

Since

we have

and

We are now in a position to prove that

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

Note that

for all \(p \in \mathbb {R}\), because

Therefore

Moreover,

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

It follows that

By similar arguments

because both the factors tend to 0.

Accordingly,

and

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.Â 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 4Â tends to infinity at an extremely fast rate. The approximate values of its first five elements are

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

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 \(n \ge k+1\).

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### Cite this article

Rychlik, T., Szymkowiak, M. Conditions for finiteness and bounds on moments of record values from iid continuous life distributions.
*Extremes* **26**, 83â100 (2023). https://doi.org/10.1007/s10687-022-00449-3

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DOI: https://doi.org/10.1007/s10687-022-00449-3

### Keywords

- Independent identically distributed random variables
- Continuous life distribution
- Record value
- Moment
- Sharp bound

### Mathematical Subject Classification

- 60E15
- 62G32