Solving functional integrodifferential equations with Liouville-Caputo fractional derivatives by fixed point techniques

1.   Introduction

Order statistic (OS) plays an important role in nonparametric statistics. Under the assumption of large sample size, relative investigations are mainly focused on asymptotic distributions of some functions of these OSs. Among these studies, the elegant one provided by Bahadur in 1966 (see ^[1]) is the central limit theorem on OSs. As was revealed there, under the situation of an absolute continuous population, the sequence of some normalized OSs usually has an asymptotic standard normal distribution. That is useful in the construction of a confidence interval for estimating some certain quantile of the population. Comparatively, study on some moment convergence of the mentioned sequence is also significant, for instance, if we utilize a sample quantile as an asymptotic unbiased estimator for the corresponding quantile of the population, then the analysis of the second moment convergence of the sequence is significant if we want to make an approximation of the mean square error of the estimate.

However, the analysis of moment convergence of OSs is usually very difficult, the reason, as was interpreted by Thomas and Sreekumar in ^[2], may lie in the fact that the moment of OS is usually very difficult to obtain.

For a random sequence, although it is well-known that the convergence in distribution does not necessarily guarantee the corresponding moment convergence, usually, that obstacle can be sufficiently overcome by the additional requirement of the uniform integrability of the sequence. For instance, we can see ^[3] as a reference dealing with some extreme OSs under some populations. In that article Wang et al. discussed uniform integrability of the sequence of some normalized extreme OSs and derived equivalent moment expressions there.

Here in the following theorem we discuss the moment convergence for some common OSs rather than extreme ones.

Theorem 1. For a population $X$ distributed according to a continuous probability density function ($pdf$) $f(x)$, let $p \in (0, 1)$ and $x_p$ be the $p-$quantile of $X$ satisfying $f({x_p}) > 0$. Let $({X_1}, ..., {X_n})$ be a random sample arising from $X$ and $X_{i:n}$ be the $i-$th OS. If the cumulative distribution function (cdf) $F(x)$ of $X$ has an inverse function $G(x)$ satisfying

for some constants $B > 0, q\geq0$ and all $x\in (0, 1)$, then for arbitrary $\delta > 0$, we have

provided $\mathop {\lim }\limits_{n\to+\infty}i/n = p$ or equivalently rewritten as $i/n = p+o(1)$.

Remark 1. Now we use the symbol $\lceil z\rceil$ for the integer part of a positive number $z$ and $m_{n, p}$ for the p-quantile of a random sample $(X_1, \cdots, X_n)$, namely, $m_{n, p} = (X_{pn:n}+X_{pn+1:n})/2$ if $pn$ is an integer and $m_{n, p} = X_{\lceil pn+1\rceil:n}$ otherwise. As both limiting conclusions $\mathop {\lim }\limits_{n \to \infty } E{X_{\lceil pn\rceil:n}^\delta } = {x_p^\delta }$ and $\mathop {\lim }\limits_{n \to \infty } E{X_{\lceil pn+1\rceil:n}^\delta } = {x_p^\delta }$ hold under the conditions of Theorem 1 and $m_{n, p}^\delta$ is always squeezed by $X_{\lceil pn\rceil:n}^\delta$ and $X_{\lceil pn+1\rceil:n}^\delta$, according to the Sandwich Theorem, we have $\mathop {\lim }\limits_{n \to \infty } E{m_{n, p}^\delta } = {x_p^\delta }$.

Remark 2. For a continuous function $H(x)$ where $x\in(0, 1)$, if

then there is a constant $C > 0$ such that the inequality $|H(x)|\leq C$ holds for all $x\in (0, 1)$. By that reason, the condition (1.1) can be replaced by the statement that there exists some constant $V\geq0$ such that

Remark 3. As the conclusion is on moment convergence of OSs, one may think that the moment of the population $X$ in Theorem 1 should exist. That is a misunderstanding because the existence of the moment of the population is actually unnecessary. We can verify that by a population according to the well-known Cauchy distribution $X\sim f(x) = \frac{1}{\pi (1+x^2)}$ where $x\in(-\infty, +\infty)$, in this case, the moment $EX$ of the population does not exist whereas the required conditions in Theorem 1 are satisfied. Even for some population without any moment of positive order, the conclusion of Theorem 1 still holds, for instance, if $f(x) = \frac{1}{x(\ln(x))^{2}}I_{[e, \infty)}(x)$ (where the symbol $I_{A}(x)$ or $I_{A}$ stands for the indicator function of a set $A$), then we have the conclusion

which leads to

and therefore the condition (1.1) holds, thus we can see that Theorem 1 is workable. That denies the statement in the final part of paper ^[4] exclaiming that under the situation $X\sim f(x) = \frac{1}{x(\ln(x))^{2}}I_{[e, \infty)}(x)$ any OS does not have any moment of positive order.

According to Theorem 1, we known that the OS $X_{i:n}$ of interest is an asymptotic unbiased estimator of the corresponding population quantile $x_p$. Now we explore the infinitesimal type of the mean error of the estimate and derive

Theorem 2. Let $({X_1}, ..., {X_n})$ be a random sample from $X$ who possesses a continuous $pdf$ $f(x)$. Let $p \in (0, 1)$ and $x_p$ be the $p-$quantile of $X$ satisfying $f({x_p}) > 0$ and $X_{i:n}$ be the $i-$th OS. If the cdf $F(x)$ of $X$ has an inverse function $G(x)$ with a continuous derivative function $G'''(x)$ in $(0, 1)$ and there is a constant $U\geq0$ such that

then under the assumption $i/n = p+O(n^{-1})$ which indicates the existence of the limit $\mathop {\lim }\limits_{x\to 0+}\frac{i/n-p}{1/n}$, the following proposition stands

Remark 4. Obviously we can see that $\vert E(m_{n, p} - {x_p})\vert = O(1/n)$ under the conditions of Theorem 2.

For i.i.d random variables(RVs) $X_1, ..., X_n$ with an identical expectation $\mu$ and a common finite standard deviation $\sigma > 0$, the famous Levy-Lindeberg central limit theorem reveals that the sequence of normalized sums

converges in distribution to the standard normal distribution $N(0, 1^2)$ which we denote that as

In 1964, Bengt presented his work ^[5] showing that if it is further assumed that $E\vert X_1\vert^k < +\infty$ for some specific positive $k$, then the m-th moment convergence conclusion

holds for any positive $m$ satisfying $m\leq k$. Here and throughout our paper, we denote $Z$ a RV of standard normal distribution $N(0, 1^2)$.

Let $f(x)$ be a continuous pdf of a population $X$ and $x_r$ be the $r-$quantile of $X$ satisfying $f(x_r) > 0$. Like the Levy-Lindeberg central limit theorem, Bahadur interpreted in ^[1] (1966) that for the OS $X_{i:n}$, following convergence conclusion holds

provided $i/n\to r$ as $n\to\infty$.

Later in 1967, Peter studied moment convergence on similar topic. He obtained in ^[6] that for some $\varepsilon > 0$, $r\in(0, 1)$ and $p_{n} = i/n$, if the limit condition

holds, then the conclusion

is workable for positive integer $k$ and $r n\leq i\leq (1-r)n$ as $n\to+\infty$.

In addition to the mentioned reference dealing with moment convergence on OSs, we find some more desirable conclusions on similar topic provided by Reiss in reference ^[7] in 1989, from which we excerpt the one of interest as what follows.

Theorem 3. Respectively let $f(x)$ and $F(x)$ be the pdf and cdf of a population $X$. Let $p\in(0, 1)$ and ${x_p}$ be the $p-$quantile of $X$ satisfying $f({x_p}) > 0$. Assume that on a neighborhood of $x_p$ the cdf $F(x)$ has $m+1$ bounded derivatives. If a positive integer $i$ satisfies $i/n = p+O(n^{-1})$ and $E\vert X_{s:j}\vert < \infty$ holds for some positive integer $j$ and $s\in\{1, ..., j\}$ and a measurable function $h(x)$ meets the requirement $\vert h(x)\vert\leq \vert x\vert^k$ for some positive integer k, then

Here the function $\varphi(x)$ and $\Phi(x)$ are respectively the pdf and cdf of a standard normal distribution while $S_{i, n}(x)$, a polynomial of $x$ with degree not more than $3i-1$ and coefficients uniformly bounded over $n$, especially

Remark 5. By putting $h(x) = x^2$ and $m = 2$, we derive under the conditions of Theorem 3 that as $n\to+\infty$,

Therefore, we see that the sequence

is uniformly bounded over $n\geq N_0$. Here $N_0$ is the positive integer number that the moment $EX_{i:n}^2$ exists when $n\geq N_0$. In accordance with the inequality $\vert E\xi\vert\leq \sqrt{E\xi^2}$ if only the moment $E\xi^2$ exists, the sequence

is also uniformly bounded, say, by a number $L$ over $n\in\{N_0, N_0+1, ...\}$. Now that

we have

Under the conditions in Theorem 2, when we estimate a population quantile $x_p$ by an OS $X_{i:n}$, usually the estimate is not likely unbiased, compared with the two conclusions (1.3) and (1.6), the result (1.3) in Theorem 2 is more accurate.

Remark 6. For a random sample $(Y_1, Y_2, ..., Y_n)$ from a uniformly distributed population $Y\sim U[0, 1]$, we write $Y_{i:n}$ the $i-$th OS. Obviously, conditions in Theorem 3 are fulfilled for any positive integer $m\geq2$. That yields

and

where for each $i = 1, 2, ..., 5$, $\alpha_{i}(n)$ is uniformly bounded over $n$.

As is above analyzed, we conclude that under the assumption $i/n = p+O(n^{-1})$,

Based on Theorems 1 and 3, here we give some alternative conditions to those in Theorem 3 to embody its range of applications including situations even when the population $X$ in Theorem 3 has no definite moment of any positive order. We obtain:

Theorem 4. Let $({X_1}, ..., {X_n})$ be a random sample derived from a population $X$ who has a continuous pdf $f(x)$. Let $p\in(0, 1)$ and ${x_p}$ be the $p-$quantile of $X$ satisfying $f({x_p}) > 0$ on a neighborhood of $x_p$ and the following three conditions hold,

(i) The cdf $F(x)$ of $X$ has an inverse function $G(x)$ satisfying

for some constants $B > 0, Q\geq0$ and all $x\in (0, 1)$.

(ii) $F(x)$ has $m+1$ bounded derivatives where $m$ is a positive integer.

(iii) Let $i/n = p+O(n^{-1})$ and $a_{i:n} = x_p+O(n^{-1})$ as $n\to+\infty$.

Then the following limiting result holds as $n\to+\infty$

Remark 7. For the mean $\overline{X}_n$ of the random sample $(X_1, ..., X_n)$ of a population $X$ whose moment $EX^m$ exists, according to conclusion (1.4), we see

which indicates that the $m-$th central moment of sample mean $E(\overline{X}_n-\mu)^m$ is usually of infinitesimal $O(n^{-m/2})$.

Here under the conditions of Theorem 4, if $EX_{i, n} = x_p+O(n^{-1})$ (we will verify in later section that for almost all continuous populations we may encounter, this assertion holds according to Theorem 2), then by Eq (1.9), we are sure that the central moment $E(X_{i:n}-EX_{i:m})^m$ is also of an infinitesimal $O(n^{-m/2})$. Moreover, by putting $a_{i:n} = x_p$, we derive under the assumptions of Theorem 4 that

Similar to Remark 1, we can also show by Sandwich Theorem that

indicating that if we use the sample $p-$quantile $m_{n, p}$ to estimate $x_p$, the corresponding population $p-$quantile, then $E(m_{n, p}-x_p)^m = O(n^{-m/2})$.

For estimating a parameter of a population without an expectation, estimators based on functions of sample moments are always futile because of uncontrollable fluctuation. Alternatively, estimators obtained by some functions of OSs are usually workable. To find a desirable one of that kind, approximating some moment expressions of OSs is therefore significant. For instance, let a population $X$ be distributed according to a $pdf$

where constants $\theta_2 > 0$ and $\theta_1$ is unknown. Here $x_{0.56} = 0.19076\theta_2+\theta_1$ and $x_{0.56}+x_{0.44} = 2x_{0.5} = 2\theta_1$. To estimate $x_{0.5} = \theta_1$, we now compare estimators $m_{n, 0.5}$ and $(m_{n, 0.56}+m_{n, 0.44})/2$. Under large sample size, we deduce according to conclusion (1.10) that

whereas

Obviously, both estimators $m_{n, 0.5}$ and $(m_{n, 0.56}+m_{n, 0.44})/2$ are unbiased for $\theta_1$. For large $n$, the main part $0.2554\pi \theta_2 n^{-1}$ of the mean square error (MSE) $E[(m_{n, 0.56}+m_{n, 0.44})/2-\theta_1]^2$ is even less than one-third of $0.785\pi\theta_2 n^{-1}$, the main part of the MSE $E\left(m_{n, 0.5}-\theta_1\right)^2$. That is the fundamental reason why Sen obtained in ^[8] the conclusion that the named optimum mid-range $(m_{n, 0.56}+m_{n, 0.44})/2$ is more effective than the sample median $m_{n, 0.5}$ in estimating $\theta_1$.

By statistical comparison of the scores presented in following Table 1 standing for 30 returns of closing prices of German Stock Index(DAX), Mahdizadeh and Zamanzade reasonably applied the previously mentioned Cauchy distribution $(1.11)$ as a stock market return distribution with $\theta_1$ and $\theta_2$ being respectively estimated as $\widehat{\theta_1} = 0.0009629174$ and $\widehat{\theta_2} = 0.003635871$ (see ^[9]).

Now we utilize $(m_{n, 0.56}+ m_{n, 0.44})/2$ as a quick estimator of $\theta_1$ and derive a value $0.00105955$ which roughly closes to the estimate value 0.0009629174 in reference ^[9].

Even now there are many estimate problems (see ^[10] for a reference) dealing with situations when a population have no expectation, as above analysis, further study on moment convergence for some OSs may be promising.

2.   Preparation of main proof

Lemma 1. (see ^[11] and ^[12]) For a random sequence $\left\{ {{\xi _1}, {\xi _2}, \cdots } \right\}$ converging in distribution to a RV $\xi$ which we write as ${\xi _n}\mathop \to \limits^D \xi$, if $d > 0$ is a constant and the following uniform integrability holds

then $\mathop {\lim }\limits_{n \to \infty } E{\lvert {{\xi _n}} \lvert^d} = E{\lvert \xi \lvert^d}$ and accordingly $\mathop {\lim }\limits_{n \to \infty } E{ {{\xi _n}} ^d} = E{ \xi^d}$.

Remark 8. As discarding some definite number of terms from $\left\{ {{\xi _1}, {\xi _2}, \cdots } \right\}$ does not affect the conclusion $\mathop {\lim }\limits_{n \to \infty } E{\lvert {{\xi _n}} \lvert^d} = E{\lvert \xi \lvert^d}$, the above condition $\mathop {\lim }\limits_{s \to +\infty } \mathop {\sup }\limits_n E{\lvert {{\xi _n}} \lvert^d}{I_{{{\lvert {{\xi _n}} \lvert}^d} \ge s}} = 0$ can be replaced by $\mathop {\lim }\limits_{s \to +\infty } \mathop {\sup }\limits_{n \ge M} E\lvert {\xi _n^d} \lvert{I_{\lvert {\xi _n^d} \lvert \ge s}} = 0$ for any positive constant $M > 0$.

Lemma 2. For $p\in(0, 1)$ and a random sample $\left({{\xi _1}, {\xi _2}, \cdots, {\xi _n}} \right)$ from a population possessing a continuous pdf $f(x),$ if the $p$-quantile ${x_p}$ of the population satisfies $f({x_p}) > 0,$ then for the $i$-th OS ${\xi _{i:n}}$ where ${i / n} = p + o(1)$, we have ${\xi _{i:n}}\mathop \to \limits^D {x_p}$.

Proof. Obviously, the sequence $\{\frac{{f({x_p})({\xi _{i:n}} - {x_p})}}{{\sqrt {p(1 - p)/n} }}, n = 1, 2, ...\}$ has an asymptotic standard normal distribution $N(0, 1^2)$, thus we see that the statistic $\xi_{i:n}$ converges to $x_p$ in probability. That leads to the conclusion ${\xi _{i:n}}\mathop \to \limits^D {x_p}$ by the reason that, for a sequence of RVs, the convergence to a constant in probability is equivalent to the convergence in distribution.

3.   Proof of theorems

Clarification before presenting the proof:

● Under the assumption ${i / n} = p + o(1)$ when $n\to \infty$, we would better think of $i$ as a function of $n$ and use the symbol ${a_n}$ instead of $i$. Nevertheless, for simplicity concern, we prefer no adjustment.

● Throughout our paper, ${C_1}$, ${C_2}, \cdots$ are some suitable positive constants.

3.1.   Proof of Theorem 1

As $\frac{i}{n}\to p\in(0, 1)$ when $n\to \infty$, we only need care large numbers $n, i$ and $n-i$.

Let an integer $K > \delta q$ be given and $M > 0$ be such a number that if $n\geq M$, then all the following inequalities $i-1-\delta q > 0$, $n-i-\delta q > 0$, $n-i-K > 0$ and $\frac{i+K}{n} < v = \frac{1+p}{2}$ hold simultaneously. Here the existence of $v$ in the last inequality is ensured by the fact $\frac{i+K}{n}\to p$ as $n\to \infty$.

According to Lemmas 1 and 2 as well as Remark 8, to prove Theorem 1 we only need to show that

That is

To show that equation, it suffices for us to prove respectively

and

Equivalently by putting $x = F(u)$, we need to prove respectively

as well as

As both proofs are similar in fashion, we chose to prove the Eq (3.2) only. Actually, according to the given condition $\lvert {G(x)} \lvert \le B{x^{ - q }}{(1 - x)^{ - q }}$, we see

Here the positive number ${C_1} > 0$ exists because $n/i = 1/p + o(1)$ where $p \in (0, 1)$.

Now applying the Stirling's formula $n! = \sqrt {2\pi n} {\left({n/e} \right)^n}{e^{\frac{\theta }{{12n}}}}$ where $\theta \in (0, 1)$ (see ^[13]), we have

Noting that

as $n \to \infty$, we see that there exists a positive constant, say $Q > 0$ such that

for all $n$. Consequently,

Due to the assumptions $\frac{i+K}{n} < v = \frac{1+p}{2} < 1$ as $n\geq M$, we derive

Finally, by the fact that if $u > 0$ is given sufficiently small, then the first term of the sequence $\left\{ {{u^n}\sqrt n, n \ge 1} \right\}$ is the maximum, thus we can confirm

Combining the five conclusions numbered from (3.3) to (3.7), we obtain Eq (3.2).

3.2.   Proof of Theorem 2

Here we would like to assume $U > 1$ (or we may use $U+2$ instead of $U$).

By the reason interpreted in Remark 2 and according to condition (1.2), we see that there is a constant $A > 0$ satisfying

Now we define $Y = F(X)$ and $Y_{i:n} = F(X_{i:n})$ or equivalently $X = G(Y)$ and $X_{i:n} = G(Y_{i, n})$, we have $G(p) = x_p$. Obviously, the conclusions in Remark 6 are workable here.

By the Taylor expansion formula we have

where

Noting that almost surely $0 < min(Y_{i:n}, p) < \xi < max(Y_{i:n}, p) < 1$, we obtain

by Eq (3.8). Here the last step is in accordance to (1.7).

Now we can draw the conclusion that

That is

provided $i/n = p+O(n^{-1})$.

Still according to conclusion (1.7), we have

Finally, as $i/n = p+O(n^{-1})$ also guarantees $i/(n+1)-p = O(n^{-1})$, we can complete the proof of $E(X_{i:n}-x_p) = O(n^{-1})$ or equivalently

by the assertion of (3.11).

3.3.   Proof of Theorem 4

As $EZ = 0$, the proposition holds when $m = 1$, now we only consider the case of $m\geq2$. By Theorem 1, we see $EX_{i:n}^2\to x_{p}^{2}$, therefore $E\vert X_{s:j}\vert$ exists for some integer $j$ and $s\in\{1, ..., j\}$ and Theorem 3 is workable here when we put $h(x) = x^m$. We derive

Moreover, for given positive integer $m\geq2$, as the coefficients in polynomial $S_{i, n}(x)$ are uniformly bounded over $n$ and $\varphi'(x) = -x \varphi(x)$, the sequence of the integrals

is also uniformly bounded over $n$. That indicates that

according to conclusion (3.12).

As a consequence, we can conclude that for explicitly given $m\geq 2$ the sequence

is uniformly bounded over $n$. Moreover, due to the inequality

we see that the sequence

is also uniformly bounded over $n$.

Now that $a_{i:n} = x_p+O(n^{-1})$, we complete the proof by the following reasoning

4.   Some verifications and one application

4.1.   Verification examples

Now we consider the applicability of our theorems obtained so far. As other conditions can be trivially or similarly verified, here we mainly focus on the verification of condition (1.2).

Example 1: Let the population $X$ have a Cauchy distribution with a pdf $f(y) = \frac{1}{\pi(1+y^2)}, -\infty < y < +\infty,$ correspondingly the inverse function of the cdf of $X$ can be figured out to be

satisfying

Example 2: For $X\sim f(x) = \frac{1}{x(\ln(x))^{2}}I_{[e, \infty)}(x)$, we have

and

Example 3: For $X\sim N(0, 1^2)$, on that occasion, $f(y) = \frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}$, $f'(y) = -yf(y)$ and $y = G(x)\Leftrightarrow x = F(y) = \int_{ - \infty }^y {\frac{1}{{\sqrt {2\pi } }}{e^{ - \frac{{{{t }^2}}}{{2}}}}dt}$, therefore, as $x \to 0 +$, we have

Noting that as $x = F(y)\to 0+$ or equivalently $y\to-\infty$,

we have as $x\to 0+$,

By the same fashion, we can show as $x\to 1-$ that

In conclusion, for $x\to 0+$ as well as for $x\to 1-$,

Accordingly, there exists a positive $M > 0$ such that for all $x\in (0, 1)$,

No matter if $x\to 0+$ or $x\to 1-$, we get

Here the last step holds in accordance to Eq (4.2).

For $x\to 0+$ as well as for $x\to 1-$,

Thus we can see the achievement of condition (1.2) by

Remark 9. For a RV $X$ with a cdf $F(x)$ possessing an inverse function $G(x)$, we can prove that if $\sigma > 0$ and $\mu\in(-\infty, +\infty)$ are constants, then the cdf of the RV $\sigma X+\mu$ will have an inverse function $\sigma G(x)+\mu$. Thus for the general case $X\sim N(\mu, \sigma^2)$, we can still verify the condition (1.2).

Example 4: For a population $X\sim U[a, b]$, $G(x) = (b-a)x+a$ is the inverse function of the cdf of $X$. As $G^{'''}(x) = 0$, the assumption of condition (1.2) holds.

Generally, for any population distributed over an interval $[a, b]$ according to a continuous $pdf$ $f(x)$, if $G{'''}(0+)$ and $G{'''}(1-)$ exist, then the condition (1.2) holds.

For length concern, here we only point out without detailed proof that for a population $X$ according to a distribution such as Gamma distribution (including special cases such as the Exponential and the Chi-square distributions) and beta distribution and so on, the requirement of condition (1.2) can be satisfied.

4.2.   An application in obtaining a combination formula

For a random sample $(X_1, ..., X_n)$ derived from a population $X$ which is uniformly distributed over the interval $[0, 1]$, the moment of the $i-$th OS $EX_{i:n} = i/(n+1)\to p$ if $i/n\to p\in(0, 1)$ as $n\to \infty$. Let $a_{i:n} = i/n$. According to conclusion (1.9) where $f(x_p) = 1$ and $x_p = p\in(0, 1)$, we have for integer $m\geq2$,

That results in

or equivalently

Consequently we have the following equation

which yields

As $i/n\to p\in(0, 1)$ when $n\to+\infty$, the above equation indicates that

For convenience sake, now we denote $\sum_{k = u}^{v} = 0$ and $\prod_{k = u}^{v} = 1$ if $v < u$. Noting for given explicit integers $m\geq2$ and $j\in\{0, 1, ..., m\}$ the expression

is a multinomial of $i$ and $n$. We see that the nominator of the LHS of Eq (4.10) is also a multinomial which we now denote as

Equivalently, we derive

By Eq (4.10), we see for any given $p\in(0, 1)$, if $i/n\to p\in(0, 1)$ as $n\to+\infty$, then

Noting that

we see in accordance to (4.12) that

That indicates that if a non-negative integer $k$ satisfies $2m-k > 3m/2$, or equivalently $0\leq k < m/2$, then the coefficient of $n^{2m-k}$ in the nominator of LHS of Eq (4.13) must be zero for any given $p\in(0, 1)$, namely

holds for any $p\in (0, 1)$. Thereby, for the case of non-negative integers $s$ and $t$ satisfying $s+t = k < m/2$, we see that the equation $a_{s, t}^{(m)} = 0$ surely holds.

It is funny to notice that for big $m$, we immediately have the following three corresponding equations

and

according to the conclusions $a_{0, 0}^{(m)} = 0$, $a_{1, 0}^{(m)} = 0$ and $a_{1, 1}^{(m)} = 0$.

As for the structure of $a_{s, t}^{(m)}$ when $s\geq2$, $t\geq1$ and $m > 2(s+t)$, obviously $s < m-t$ holds on this occasion and the term $a_{s, t}^{(m)}i^{m-s}n^{m-t}$ in the multinomial

is also the term $a_{s, t}^{(m)}i^{m-s}n^{m-t}$ in the multinomial

Noting for given $j\in\{s, ..., m-t\}$, the monomial

is the term with degree $m-s$ in the polynomial of $i$

while the monomial

is the term with degree $m-t$ in the polynomial of $n$

we see for $s+t < m/2$,

Now that $a_{s, t}^{m} = 0$ holds provided $s+t = k < m/2$ according to Eq (4.13), we conclude the following Theorem.

Theorem 5. If $s$, $t$ and $m$ are integers satisfying $s\geq2$, $t\geq1$ and $m > 2(s+t)$, then

Example 5: For big integer $m$, according to Theorem 5, we have $a_{2, 1}^{(m)} = 0$ and $a_{2, 2}^{(m)} = 0$. Correspondingly, we obtain equations

and

Both equations can be verified by the aid of Maple software.

5.   Conclusions

Let real $\delta > 0$ and integer $m > 0$ be given. For a population satisfying condition (1.1), no matter if the population has an expectation or not, the moment of $X_{i:n}^{\delta}$ exists and the sequence $\{EX_{i:n}^{\delta}, n\geq1\}$ converges for large $i$ and $n$ satisfying $i/n\to p\in(0, 1)$. Under some further trivial assumptions, for large integer $n$ the $m-$th moment of the standardized sequence $\{X_{i:n}, n\geq1\}$ can be approximated by the $m-$th moment of a standard normal distribution $EZ^m$.

Due to the fact that the existence requirement of some expectation $X_{s:j}$ in Theorem 3 has always been hard to be verified for a population without an expectation, for a long time, real-life world data corresponding to that population of interest has been unavailable in the vast majority of references. Now that the alternative condition (1.8) is presented, maybe things will improve in the future and we still have a long way to go.

Acknowledgments

This work was supported by the Science and Technology Plan Projects of Jiangxi Provincial Education Department, grant number GJJ180891.

Conflict of interest

There exists no conflict of interest between authors.