Minimum of heavy-tailed random variables is not heavy tailed

1.   Introduction

We say that distribution $F$ is heavy-tailed and write $F\in \mathscr{H}$ if

If $F(x) = \mathbb{P}(X\leqslant x)$, then random variable $X$ is called heavy-tailed. It is well known (see, for instance, Theorem 2.6 in ^[10]) that $F\in\mathscr{H}$ if and only if

Here $\overline F(x) = 1-F(x)$ denotes the right tail of $F(x)$. We say that distribution $F$ is strongly heavy-tailed and write $F\in \mathscr{H}^*$ if

Obviously, $\mathscr{H}^*\subset\mathscr{H}$ and one can check that $\mathscr{H}\backslash\mathscr{H}^*\ne \varnothing$. For discussion on classes $\mathscr{H}$, $\mathscr{H}^*$ and examples $F\in \mathscr{H}\setminus\mathscr{H}^*$ see ^[2,15,16] among others.

Concerning other properties of heavy-tailed distribution class, it is easy to see that $\mathscr{H}$ is closed under convolution, mixing, maximum and product-convolution.

Let us denote the convolution of distributions $F_1$ and $F_2$ by

We say that some class of distributions $\mathscr{B}$ is closed under convolution if for any two distributions $F_1$ and $F_2$ it holds that

The relation (1.1) for class of distributions $\mathscr{B} = \mathscr{H}$ follows immediately from definition of $\mathscr{H}$. Namely, by supposing that $F_1$, $F_2$ are distributions of independent random variable $X_1$ and $X_2$, we get

Similarly, we say that a class of distributions $\mathscr{B}$ is closed under mixing if for $p\in(0, 1)$

Since for any $\lambda > 0$

we get a stronger assertion

It is said that class of distributions $\mathscr{B}$ is closed under maximum if $F_1, F_2\in\mathscr{B}$ implies

Like in the case of convolution, a stronger assertion on closure under maximum follows

because

Considering the closure under the product-convolution, we present the following result:

where symbol $\otimes$ denotes the product-convolution, i.e., $F_1\otimes F_2(x) = \mathbb{P}(X_1X_2\leqslant x)$ for independent random variables $X_1$ and $X_2$ with distributions $F_1$ and $F_2$. For the proof of (1.2) it suffices to observe that

where $\lambda > 0$ is an arbitrary constant, and $a > 0$ is such that $\mathbb{P}(X_2 > a) > 0$.

Studies of other interesting properties of heavy-tailed distributions can be found in ^{[2,3,4,7,8,9,10]} among others.

The problem whether class $\mathscr{H}$ is closed with respect to minimum is much more difficult and, to our knowledge, was not solved. In this paper, we prove that class $\mathscr{H}$ is not closed under minimum. We construct two independent random variables $X$ and $Y$ with the corresponding distributions $F\in\mathscr{H}$ and $G\in\mathscr{H}$, such that their minimum $X\wedge Y = \min\{X, Y\}$ is not heavy tailed, i.e., $F_{X\wedge Y} = 1-\overline{F}\ \overline{G} = F+G-FG\notin\mathscr{H}$.

2.   Main results

Consider the distribution tail $\overline F(x)$ of the following form:

This distribution and distribution in (2.5) below will be used for the main result on the minimum of heavy-tailed r.v.s. Our first result yields several properties of the distribution $F$.

Theorem 2.1. Assume that $F$ is defined in (2.1). Then $F\in\mathscr{H}$, $F\notin\mathscr{H}^*$ and

The property in (2.2) defines the class of generalized long-tailed distributions, $\mathscr{OL}$, introduced in ^[13]. Recall that a distribution $F$ on ${\mathbb{R}}$ belongs to the class $\mathscr{OL}$, if for any (or some) $y > 0$

Thus, Theorem 2.1 says that

By Proposition 2.2(ii) in ^[13], $F\in \mathscr{OL}$ implies that $\lim_{x\to\infty}{\mathrm e}^{\delta x} \overline{F}(x) = \infty$ for some $\delta > 0$, and $\mathscr{OL}$ also admits some light-tailed distributions. Various results related to class $\mathscr{OL}$ can be found in ^{[1,5,6,19,20]}. In particular, authors of ^[20] showed that $\mathscr{H}^*\backslash\mathscr{OL}\ne \varnothing$, cf. (2.4). Note that class $\mathscr{OL}$ was also introduced in ^[14], where it was called a Semi-$\mathscr{L}$ class of distributions.

Consider now another distribution with the tail $\overline G(x)$ of the following form:

Analogously to the result in Theorem 2.1, it holds that $G\in\mathscr{H}$, $G\notin\mathscr{H}^*$ and $G\in\mathscr{OL}$.

The main result of the paper says that the distribution $F_{X\wedge Y}(x) = 1-\overline F(x) \overline G(x)$ is light-tailed. Indeed, by construction of $\overline F$ and $\overline G$, we have

and we obtain the following assertion.

Theorem 2.2. Assume that $X$ and $Y$ are independent r.v.s with distribution tails $\overline F$ in (2.1) and $\overline G$ in (2.5), respectively. Then

Remark 2.1. We mention two related results, which follow easily from definitions. First result says that, although class $\mathscr{H}$ is not closed under minimum, it is closed in the class $\mathscr{H}^*$, i.e.,

where $X_1$ and $X_2$ are random variables with corresponding distributions $F_1$ and $F_2$. Second result says that class $\mathscr{OL}$ is closed under minimum:

The study of the minimum of random variables is important for problems related to various stochastic models. For example it concerns the order statistics $X_{1:n}\leqslant X_{2:n}\leqslant\ldots\leqslant X_{n:n}$ of random variables $X_1, X_2, \ldots, X_n$. It is obvious that

in the case of independent and identically distributed random variables with common distribution $F_X$. We can see from this expression that properties of order statistics are related to the closure property of random variables under minimum. The order statistics properties for various subclasses of $\mathscr{H}$ were considered in ^{[11,12,17,18]}, for instance. The definition of the class $\mathscr{H}$ implies immediately the following assertion.

Theorem 2.3. Let $X_1, X_2, \ldots, X_n$ be independent and identically distributed random variables with common distribution $F_X$. Then $F_{X_{k, n}}\in\mathscr{H}$ for $k\in\{1, 2, \ldots, n\}$ if and only if $F_X\in\mathscr{H}$.

While, it follows from Theorem 2.2 that the analogous statement to Theorem 2.3 fails even in the case $n = 2$ if the random variables $X_1, X_2, \ldots, X_n$ are independent but possibly differently distributed.

3.   Proof of Theorem 2.1

Take the sequence $x_n = (2n)!$, $n\ge 1$. For any $\lambda > 0$ we have

as $n\to\infty$. Hence,

implying $F\in \mathscr{H}$.

To show that $F\notin \mathscr{H}^*$, define the sequence $y_n = ((2n)!+(2n+1)!)/2$, $n\geqslant 1$. Then

as $n\to\infty$ for $0 < \lambda < 1$. Hence, for such $\lambda$,

It remains to prove that $F\in\mathscr{OL}$. Take $x\in [(2n)!, (2n+2)!)$ and consider the following four cases:

In case (a) we have

In case (b),

and, therefore,

In case (c),

In case (d),

and, because $x < (2n)!+1$, it holds

These four estimates yield

Thus, $F\in \mathscr{OL}$.

Acknowledgments

The authors would like to thank the anonymous reviewers for the careful reading of the manuscript, and for all the suggestions which contributed to improve the quality and presentation of the paper.

Conflict of interest

The authors declare that they have no conflicts of interest.