**4. Some remarks on the heavy-tailedness of scale mixtures of normals**

The class of scale mixtures of normal laws is very rich and involves distributions with various character of decrease of tails. For example, this class contains Student distributions with arbitrary (not necessarily integer) number of degrees of freedom (and the Cauchy distribution included), symmetric stable distributions (see the "multiplication theorem" 3.3.1 in [15]), symmetric fractional stable distributions (see [16]), symmetrized gamma distributions with arbitrary shape and scale parameters (see [10]), and symmetrized Weibull distributions with shape parameters belonging to the interval 0, 1 ð � (see [17, 18]). As an example, in the next section, we will discuss the conditions for the convergence of the distributions of the statistics constructed from samples with random sizes to the multivariate Student

*From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums…*

**5. Convergence of the distributions of random sums of random vectors with finite covariance matrices to multivariate elliptically contoured**

**5.1 Convergence of the distributions of random sums of random vectors to**

<sup>⊤</sup>*<sup>X</sup>* � � <sup>¼</sup> exp � **<sup>t</sup>**

By *ζα*, we will denote a positive random variable with the one-sided stable

Let *α*∈ ð � 0, 2 . It is known that, if *Y* is a random vector such that Lð Þ¼ *Y* ΦΣ

As in Section 3, let *X*1, *X*2, … be independent *r*-valued random vectors. For *n*∈ , denote *Sn* ¼ *X*<sup>1</sup> þ … þ *Xn*. Consider a sequence of integer-valued positive

*<sup>d</sup>* ffiffiffiffiffiffiffiffi *ζα<sup>=</sup>*<sup>2</sup> q

*Z<sup>α</sup>*,<sup>Σ</sup> ¼

2 *<sup>i</sup>παsignt* � � � � , *<sup>t</sup>*<sup>∈</sup> ,

Univariate stable distributions are popular examples of heavy-tailed distributions. Their moments of orders *δ*≥*α* do not exist (the only exception is the normal law corresponding to *α* ¼ 2). Stable laws and only they can be limit distributions for sums of a non-random number of independent identically distributed random variables with infinite variance under linear normalization. Here it will be shown that they also can be limiting for *random* sums of random vectors with *finite covariance matrices*. The result of this subsection generalizes the main theorem of [19] to a

teristic exponent *α*, if its characteristic function g*<sup>α</sup>*,Σð Þ**t** has the form

Let Σ be a positive definite ð Þ *r* � *r* -matrix, *α* ∈ð � 0, 2 . A random vector *Zα*,<sup>Σ</sup> is said to have the (centered) elliptically contoured stable distribution *Gα*,<sup>Σ</sup> with charac-

<sup>⊤</sup>Σ**<sup>t</sup>** � �*<sup>α</sup>=*<sup>2</sup> n o, **<sup>t</sup>**<sup>∈</sup> *<sup>r</sup>*

*:*

� *Y* (17)

*G<sup>α</sup>*,<sup>Σ</sup> ¼ EΦ*ζα<sup>=</sup>*2<sup>Σ</sup>*:* (18)

distribution.

multivariate case.

**177**

**stable and Linnik distributions**

*DOI: http://dx.doi.org/10.5772/intechopen.89659*

g*<sup>α</sup>*,Σð Þ� **t** E exp *i***t**

distribution corresponding to the characteristic function

with 0< *α*≤1 (for more details see [15] or [4]).

(see Proposition 2.5.2 in [4]). In other words,

independent of the random variable *ζα=*2, then

<sup>g</sup>*α*ðÞ¼ *<sup>t</sup>* exp �j j*<sup>t</sup> <sup>α</sup>* exp � <sup>1</sup>

**multivariate stable laws**

The one-dimensional marginals of the multivariate limit law in Theorems 2 and 3 are scale mixtures of normals with zero means of the form EΦð Þ *x=U* , *x*∈ , where Φð Þ *x* is the standard normal distribution function and *U* is a nonnegative random variable. It turns out, although absolutely not so evident, that these distributions are *always* leptokurtic having sharper vertex and heavier tails than the normal law itself.

It is easy to see that

$$\mathbf{E}\Phi(\mathbf{x}/U) = \mathbf{P}(X \cdot U < \mathbf{x}), \ \mathbf{x} \in \mathbb{R}, \ \mathbf{x}$$

where *X* is a standard normal variable independent of *U*. First, as a measure of leptokurtosity, consider the excess coefficient which is traditionally used in (descriptive) statistics. Recall that for a random variable *Y* with E*Y*<sup>4</sup> < ∞, the excess coefficient (kurtosis) *κ*ð Þ *Y* is defined as

$$\kappa(Y) = \operatorname{E}\left(\frac{Y - \operatorname{E}Y}{\sqrt{\operatorname{D}Y}}\right)^4.$$

If Pð Þ¼ *X* <*x* Φð Þ *x* , then *κ*ð Þ¼ *X* 3. Densities with sharper vertices (and, respectively, with heavier tails) than the normal density, have *κ* > 3, and *κ* <3 for densities with more flat vertices.

Lemma 2. *Let X and U be independent random variables with finite fourth moments; moreover, let* E*X* ¼ 0 *and* Pð Þ¼ *U* ≥0 1*. Then*

$$
\kappa(XU) \ge \kappa(X).
$$

*Furthermore, κ*ð Þ¼ *XU κ*ð Þ *X if and only if* Pð Þ¼ *U* ¼ const 1*.* For the proof see [10].

So, if *X* is a standard normal random variable and *U* is a nonnegative random variable with E*U*<sup>4</sup> <sup>&</sup>lt; <sup>∞</sup> independent of *<sup>X</sup>*, then *<sup>κ</sup>*ð Þ *<sup>X</sup>* � *<sup>U</sup>* <sup>≥</sup>3 and *<sup>κ</sup>*ð Þ¼ *<sup>X</sup>* � *<sup>U</sup>* 3 if and only if *U* is non-random.

Using the Jensen inequality, we can easily obtain one more inequality directly connecting the tails of the normal mixtures with the tails of the normal distribution.

Lemma 3. *Assume that the random variable U satisfies the normalization condition* <sup>E</sup>*U*�<sup>1</sup> <sup>¼</sup> <sup>1</sup>*. Then*

$$\mathbf{1} - \mathbf{E}\Phi(\mathbf{x}/U) \ge \mathbf{1} - \Phi(\mathbf{x}), \qquad \mathbf{x} > \mathbf{0}.$$

From Lemma 3, it follows that if *X* is the standard normal random variable and *<sup>U</sup>* is a nonnegative random variable independent of *<sup>X</sup>* with E*U*�<sup>1</sup> <sup>¼</sup> 1, then for any *x*≥0

$$\mathbb{P}(|X \cdot U| \ge \mathbf{x}) \; \; \; \; \mathbb{P}(|X| \ge \mathbf{x}) \; \; (= \mathbf{2}[\mathbf{1} - \Phi(\mathbf{x})]),$$

that is, scale mixtures of normal laws are always more leptokurtic and have heavier tails than normal laws themselves.

*From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums… DOI: http://dx.doi.org/10.5772/intechopen.89659*

The class of scale mixtures of normal laws is very rich and involves distributions with various character of decrease of tails. For example, this class contains Student distributions with arbitrary (not necessarily integer) number of degrees of freedom (and the Cauchy distribution included), symmetric stable distributions (see the "multiplication theorem" 3.3.1 in [15]), symmetric fractional stable distributions (see [16]), symmetrized gamma distributions with arbitrary shape and scale parameters (see [10]), and symmetrized Weibull distributions with shape parameters belonging to the interval 0, 1 ð � (see [17, 18]). As an example, in the next section, we will discuss the conditions for the convergence of the distributions of the statistics constructed from samples with random sizes to the multivariate Student distribution.
