**6. Convergence of the distributions of asymptotically normal statistics to the multivariate Student distribution**

The multivariate Student distribution is described, for example, in [30] (also see [31]). Consider an *r*-dimensional normal random vector *Y* with zero vector of expectations and covariance matrix Σ. Assume that a random variable *W<sup>γ</sup>* has the chi-square distribution with parameter (the "number of degrees of freedom") *γ* > 0 (not necessarily integer) and is independent of *Y*. The distribution *Pγ*,<sup>Σ</sup> of the random vector

$$Q\_{\mathbf{y},\Sigma} = \sqrt{\chi/\mathcal{W}\_{\mathbf{y}}} \cdot \mathbf{Y} \tag{23}$$

takes place if and only if

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

scale parameters equal to *γ=*2.

and Lemma 1.

*<sup>C</sup>k*�<sup>1</sup> *<sup>m</sup>*þ*k*�<sup>2</sup> is defined as

well known that

so that E*Np*,*<sup>m</sup>* ! ∞ as *p* ! 0.

Lemma 4. *For any fixed m* > 0

lim *p*!0

the probability of the success in a trial is 1 � *p*.

*s upx*<sup>∈</sup> ∣P

coinciding with the scale parameter and equal to *m*.

*Tn be asymptotically normal in the sense of* ð Þ 16 *. Then*

<sup>L</sup> ffiffiffiffiffiffiffi

Proof. By Lemma 4 we have

E*Nn* � E*Nn nm* <sup>¼</sup> *Nn* E*Nn* �

assertion directly follows from Theorem 6.

*Nn*

**181**

*nm* <sup>¼</sup> *Nn*

*Nn has the negative binomial distribution with parameters p* <sup>¼</sup> <sup>1</sup>

Pð Þ) *Nn* <*dnx G<sup>γ</sup>=*2,*γ=*2ð Þ *x* , *n* ! ∞,

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

where *G<sup>γ</sup>=*2,*γ=*2ð Þ *x* is the gamma-distribution function with coinciding shape and

Proof. This statement is a direct consequence of Theorem 3, representation (23)

*k*�1

ð Þ *<sup>k</sup>* � <sup>1</sup> ! � <sup>Γ</sup>ð Þ *<sup>m</sup> :*

*<sup>p</sup>* ,

� *Gm*,*<sup>m</sup>*ð Þ *x* ∣ ¼ 0,

*<sup>n</sup> and m. Let a statistic*

) *Um*

, *k* ¼ 1, 2, … (24)

Let *Np*,*<sup>m</sup>* be a random variable with the negative binomial distribution

Here *m* > 0 and *p*∈ð Þ 0, 1 are parameters; for non-integer *m*, the quantity

*<sup>C</sup><sup>k</sup>*�<sup>1</sup> *<sup>m</sup>*þ*k*�<sup>2</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>m</sup>* <sup>þ</sup> *<sup>k</sup>* � <sup>1</sup>

<sup>E</sup>*Np*,*<sup>m</sup>* <sup>¼</sup> *<sup>m</sup>*ð Þþ <sup>1</sup> � *<sup>p</sup> <sup>p</sup>*

As is known, the negative binomial distribution with natural *m* admits an illustrative interpretation in terms of Bernoulli trials. Namely, the random variable with distribution (24) is the number of the Bernoulli trials held up to the *m*th failure, if

> < *x* � �

where *Gm*,*<sup>m</sup>*ð Þ *x* is the gamma-distribution function with the shape parameter

The proof is a simple exercise on characteristic functions; for more details, see [8]. Corollary 3. *Let m* > 0 *be arbitrary. Assume that for each n*≥ 1 *the random variable*

*mn* <sup>p</sup> ð Þ *TNn* � *<sup>t</sup>* � � ) *<sup>P</sup>*2*m*,<sup>Σ</sup> ð Þ *<sup>n</sup>* ! <sup>∞</sup>

where *P*2*m*,<sup>Σ</sup> is the *r*-variate Student distribution with parameters *γ* ¼ 2*m* and Σ.

*m n*ð Þþ � 1 1

as *n* ! ∞ where *Um* is the random variable having the gamma-distribution function with coinciding shape and scale parameters equal to *m*. Now the desired

*mr* <sup>¼</sup> *Nn*

E*Nn*

<sup>1</sup> <sup>þ</sup> *<sup>O</sup>* <sup>1</sup> *n* � � � �

*Np*,*<sup>m</sup>* E*Np*,*<sup>m</sup>*

In particular, for *m* ¼ 1, relation (24) determines the geometric distribution. It is

<sup>P</sup> *Np*,*<sup>m</sup>* <sup>¼</sup> *<sup>k</sup>* � � <sup>¼</sup> *<sup>C</sup>k*�<sup>1</sup> *<sup>m</sup>*þ*k*�<sup>2</sup>*pm*ð Þ <sup>1</sup> � *<sup>p</sup>*

is called the *multivariate Student distribution* (with parameters *γ* and Σ). For any **x**∈ *<sup>r</sup>* the distribution density of *Z* has the form

$$p\_{\boldsymbol{\gamma},\boldsymbol{\Sigma}}(\mathbf{x}) = \frac{\Gamma(r+\boldsymbol{\gamma})/2)}{|\boldsymbol{\Sigma}|^{1/2}\Gamma(\boldsymbol{\gamma}/2)(\boldsymbol{\pi}\boldsymbol{\gamma})^{r/2}} \cdot \frac{1}{\left(\mathbf{1} + \frac{1}{r}\mathbf{x}^T\boldsymbol{\Sigma}^{-1}\mathbf{x}\right)^{(r+\boldsymbol{\gamma})/2}}.$$

According to Theorem 3, the multivariate Student distribution is the resulting transformation of the limit distribution of an asymptotically normal (in the sense of (16)) statistic under the replacement of the sample size by a random variable whose asymptotic distribution is chi-square. Consider this case in more detail.

Let *Gm*,*<sup>m</sup>*ð Þ *x* be the gamma-distribution function with the shape parameter coinciding with the scale parameter and equal to *m*:

$$G\_{m,m}(\boldsymbol{\chi}) \;= \begin{cases} 0 & \text{if } \boldsymbol{x} \le \mathbf{0}, \\\frac{m^m}{\Gamma(m)} \int\_0^\chi e^{-m\boldsymbol{y}} \boldsymbol{\mathcal{y}}^{m-1} d\boldsymbol{y} & \text{if } \boldsymbol{x} > \mathbf{0}. \end{cases}$$

Theorem 6. *Let γ* > 0 *be arbitrary,* Σ *be a positive definite matrix and let d*f g*<sup>n</sup> <sup>n</sup>*≥<sup>1</sup> *be an infinitely increasing sequence of positive numbers. Assume that Nn* ! ∞ *in probability as n* ! ∞*. Let a statistic Tn be asymptotically normal in the sense of* ð Þ 16 *. Then the convegence*

$$\mathcal{L}\left(\sqrt{d\_n}(T\_{N\_n} - t)\right) \implies P\_{\gamma, \Sigma} \qquad (n \to \infty),$$

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

takes place if and only if

*<sup>L</sup> SNn* ffiffiffiffiffi *dn* p � �

**to the multivariate Student distribution**

**x**∈ *<sup>r</sup>* the distribution density of *Z* has the form

ciding with the scale parameter and equal to *m*:

*Gm*,*<sup>m</sup>*ðÞ ¼ *x*

<sup>L</sup> ffiffiffiffiffi *dn* <sup>p</sup> ð Þ *TNn* � *<sup>t</sup>* � �

j j <sup>Σ</sup> <sup>1</sup>*=*<sup>2</sup>

*pγ*,Σð Þ¼ **x**

*α*∈ ð � 0, 2 , if and only if

*Probability, Combinatorics and Control*

as *n* ! ∞.

relation (22).

random vector

*convegence*

**180**

) Lð Þ *Lr*,*α*,<sup>Σ</sup> ð Þ *n* ! ∞

with some infinitely increasing sequence of positive numbers f g *dn <sup>n</sup>*≥<sup>1</sup> and some

) 2*M<sup>α</sup>=*<sup>2</sup>

Proof. This theorem is a direct consequence of Theorem 2 with the account of

**6. Convergence of the distributions of asymptotically normal statistics**

[31]). Consider an *r*-dimensional normal random vector *Y* with zero vector of expectations and covariance matrix Σ. Assume that a random variable *W<sup>γ</sup>* has the chi-square distribution with parameter (the "number of degrees of freedom") *γ* > 0 (not necessarily integer) and is independent of *Y*. The distribution *Pγ*,<sup>Σ</sup> of the

*Qγ*,<sup>Σ</sup> ¼

Γð Þ *r* þ *γ =*2Þ

asymptotic distribution is chi-square. Consider this case in more detail.

8 < :

*m<sup>m</sup>* Γð Þ *m* ð*x* 0 *e* �*my ym*�<sup>1</sup>

Theorem 6. *Let γ* > 0 *be arbitrary,* Σ *be a positive definite matrix and let d*f g*<sup>n</sup> <sup>n</sup>*≥<sup>1</sup> *be an infinitely increasing sequence of positive numbers. Assume that Nn* ! ∞ *in probability as n* ! ∞*. Let a statistic Tn be asymptotically normal in the sense of* ð Þ 16 *. Then the*

The multivariate Student distribution is described, for example, in [30] (also see

ffiffiffiffiffiffiffiffiffiffiffiffi *γ=W<sup>γ</sup>* q

is called the *multivariate Student distribution* (with parameters *γ* and Σ). For any

<sup>Γ</sup>ð Þ *<sup>γ</sup>=*<sup>2</sup> ð Þ *πγ <sup>r</sup>=*<sup>2</sup> � <sup>1</sup>

According to Theorem 3, the multivariate Student distribution is the resulting transformation of the limit distribution of an asymptotically normal (in the sense of (16)) statistic under the replacement of the sample size by a random variable whose

Let *Gm*,*<sup>m</sup>*ð Þ *x* be the gamma-distribution function with the shape parameter coin-

<sup>1</sup> <sup>þ</sup> <sup>1</sup>

0 if *x*≤0,

) *P<sup>γ</sup>*,<sup>Σ</sup> ð Þ *n* ! ∞ ,

*<sup>γ</sup>* **x***Τ*Σ�<sup>1</sup> **x** � �ð Þ *<sup>r</sup>*þ*<sup>γ</sup> <sup>=</sup>*<sup>2</sup> *:*

*dy* if *x* > 0*:*

� *Y* (23)

*Nn dn*

$$\mathbf{P}(N\_n < d\_n \mathbf{x}) \Rightarrow G\_{\mathbf{y}/2, \mathbf{y}/2}(\mathbf{x}), \quad n \to \infty, 1$$

where *G<sup>γ</sup>=*2,*γ=*2ð Þ *x* is the gamma-distribution function with coinciding shape and scale parameters equal to *γ=*2.

Proof. This statement is a direct consequence of Theorem 3, representation (23) and Lemma 1.

Let *Np*,*<sup>m</sup>* be a random variable with the negative binomial distribution

$$\mathbf{P}(N\_{p,m} = k) \quad = \ \mathbf{C}\_{m+k-2}^{k-1} p^m (\mathbf{1} - p)^{k-1}, \quad k = \mathbf{1}, \mathbf{2}, \ldots \tag{24}$$

Here *m* > 0 and *p*∈ð Þ 0, 1 are parameters; for non-integer *m*, the quantity *<sup>C</sup>k*�<sup>1</sup> *<sup>m</sup>*þ*k*�<sup>2</sup> is defined as

$$\mathcal{C}\_{m+k-2}^{k-1} = \frac{\Gamma(m+k-1)}{(k-1)! \cdot \Gamma(m)}.$$

In particular, for *m* ¼ 1, relation (24) determines the geometric distribution. It is well known that

$$\mathbb{E}N\_{p,m} = \frac{m(1-p) + p}{p},$$

so that E*Np*,*<sup>m</sup>* ! ∞ as *p* ! 0.

As is known, the negative binomial distribution with natural *m* admits an illustrative interpretation in terms of Bernoulli trials. Namely, the random variable with distribution (24) is the number of the Bernoulli trials held up to the *m*th failure, if the probability of the success in a trial is 1 � *p*.

Lemma 4. *For any fixed m* > 0

$$\lim\_{p \to 0} s \, \upmu p\_{\ge \mp \mathbb{R}} |\mathbf{P} \left( \frac{N\_{p,m}}{\mathbf{E} N\_{p,m}} < \infty \right) - \mathbf{G}\_{m,m}(\infty)|\, \, = \,\, \mathbf{0}, \,\, \theta$$

where *Gm*,*<sup>m</sup>*ð Þ *x* is the gamma-distribution function with the shape parameter coinciding with the scale parameter and equal to *m*.

The proof is a simple exercise on characteristic functions; for more details, see [8]. Corollary 3. *Let m* > 0 *be arbitrary. Assume that for each n*≥ 1 *the random variable Nn has the negative binomial distribution with parameters p* <sup>¼</sup> <sup>1</sup> *<sup>n</sup> and m. Let a statistic Tn be asymptotically normal in the sense of* ð Þ 16 *. Then*

$$\mathcal{L}\left(\sqrt{mn}(T\_{N\_n} - t)\right) \implies P\_{2m,\Sigma} \qquad (n \to \infty).$$

where *P*2*m*,<sup>Σ</sup> is the *r*-variate Student distribution with parameters *γ* ¼ 2*m* and Σ. Proof. By Lemma 4 we have

$$\frac{N\_n}{nm} = \frac{N\_n}{\text{EN}\_n} \cdot \frac{\text{EN}\_n}{nm} = \frac{N\_n}{\text{EN}\_n} \cdot \frac{m(n-1)+1}{mr} = \frac{N\_n}{\text{EN}\_n} \left[1 + O\left(\frac{1}{n}\right)\right] \Rightarrow U\_m$$

as *n* ! ∞ where *Um* is the random variable having the gamma-distribution function with coinciding shape and scale parameters equal to *m*. Now the desired assertion directly follows from Theorem 6.

Remark 2. The *r*-variate Cauchy distribution (*γ* ¼ 1) appears in the situation described in Corollary 2 when the sample size *Nn* has the negative binomial distribution with the parameters *<sup>p</sup>* <sup>¼</sup> <sup>1</sup> *<sup>n</sup>*, *<sup>m</sup>* <sup>¼</sup> <sup>1</sup> 2 , and *n* is large.

important for the proper application of such risk measures as VaR (Value-at-Risk)

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

random variables *W*ð Þ ½ �þ *<sup>λ</sup>in* <sup>1</sup> , *i* ¼ 1, … ,*r*, with ½ � *a* denoting the integer part of a number *a*. The following result due to Mosteller [32] (also see [33], Section 9.2) is

*<sup>n</sup>* <sup>p</sup> *<sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jn* <sup>þ</sup><sup>1</sup> � *ξλ <sup>j</sup>*

*<sup>σ</sup>ij* <sup>¼</sup> *<sup>λ</sup><sup>i</sup>* <sup>1</sup> � *<sup>λ</sup> <sup>j</sup>*

*p ξλ<sup>i</sup>* � �*<sup>p</sup> ξλ <sup>j</sup>*

*t* > 0. Assume that the processes Λð Þ*t* and *N*1ð Þ*t* are independent. Set

Let *W*1, … ,*Wn*, *n*≥ 1, be independent identically distributed random variables with common distribution density *p x*ð Þ and *W*ð Þ<sup>1</sup> , … ,*W*ð Þ *<sup>n</sup>* be the corresponding order statistics, *W*ð Þ<sup>1</sup> ≤*W*ð Þ<sup>2</sup> ≤ … ≤ *W*ð Þ *<sup>n</sup>* . Let *r*∈ *λ*1, … , *λ<sup>r</sup>* be some numbers such that 0< *λ*<sup>1</sup> <*λ*<sup>2</sup> < … < *λ<sup>r</sup>* <1. The quantiles of orders *λ*1, … , *λ<sup>r</sup>* of the random variable

, *i* ¼ 1, … ,*r*. The sample quantiles of orders *λ*1, … , *λ<sup>r</sup>* are the

*<sup>n</sup>*,*<sup>r</sup> weakly converges to the r-variate normal distribution with*

� �, *<sup>j</sup>* <sup>¼</sup> 1, … ,*r:*

� � , *<sup>i</sup>* <sup>≤</sup> *<sup>j</sup>:*

Theorem 7 [32]. *If p x*ð Þ *is differentiable in some neighborhoods of the quantiles ξλ<sup>i</sup>*

� �

To take into account the randomness of the sample size, consider the sequence *W*1, *W*<sup>2</sup> … of independent identically distributed random variables with common

Let *N t*ð Þ, *t*≥ 0, be a Cox process controlled by a process Λð Þ*t* . Recall the definition of a Cox process. Let *N*1ð Þ*t* , *t*≥ 0, be a standard Poisson process (i.e., a homogeneous Poisson process with unit intensity). Let Λð Þ*t* , *t*≥0, be a random process with non-decreasing right-continuous trajectories, Λð Þ¼ 0 0, Pð Þ¼ Λð Þ*t* < ∞ 1 for all

*N t*ðÞ¼ *N*1ð Þ Λð Þ*t* , *t*≥ 0*:*

Cox processes are widely used as models of inhomogeneous chaotic flows of

As we have already noted, it was B. V. Gnedenko who drew attention to the

essential distinction between the asymptotic properties of sample quantiles constructed from samples with random sizes and the analogous properties of sample quantiles in the standard situation. Briefly recall the history of the problem under consideration. B. V. Gnedenko, S. Stomatovič, and A. Shukri [34] obtained sufficient conditions for the convergence of distribution of the sample median constructed from sample of random size. In the candidate (PhD) thesis of A. K. Shukri, these conditions were extended to quantiles of arbitrary orders. In [35],

necessary and sufficient conditions for the weak convergence of the

Assume that all the involved random variables and processes are independent. In this section, under the assumption that ΛðÞ!*t* ∞ in probability, the asymptotics of the joint distribution of the random variables *W*ð Þ ½ �þ *<sup>λ</sup>iN t*ð Þ <sup>1</sup> , *i* ¼ 1, … ,*r* is considered

The process *N t*ð Þ is called a doubly stochastic Poisson process (or a Cox process) controlled by the process Λð Þ*t* . The one-dimensional distributions of a Cox process are mixed Poisson. For example, if Λð Þ*t* has the gamma distribution, then *N t*ð Þ has

� � 6¼ <sup>0</sup>*, i* <sup>¼</sup> 1, … ,*r, then, as n* ! <sup>∞</sup>*, the joint distribution of the normalized*

in, say, financial engineering.

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

*Y* ∗ *<sup>n</sup>*, *<sup>j</sup>* <sup>¼</sup> ffiffiffi

*<sup>n</sup>*,1, … , *Y* <sup>∗</sup>

*zero vector of expectations and covariance matrix* <sup>Σ</sup> <sup>¼</sup> *<sup>σ</sup>ij* � �,

*W*<sup>1</sup> will be denoted *ξλ<sup>i</sup>*

classical. Denote

*sample quantiles Y* <sup>∗</sup>

distribution density *p x*ð Þ.

the negative binomial distribution.

events, see, for example, [2].

as *t* ! ∞.

**183**

*and p ξλ<sup>i</sup>*

Remark 3. In the case where the sample size *Nn* has the negative binomial distribution with the parameters *<sup>p</sup>* <sup>¼</sup> <sup>1</sup> *<sup>n</sup>*, *m* ¼ 1 (that is, the geometric distribution with the parameter *<sup>p</sup>* <sup>¼</sup> <sup>1</sup> *<sup>n</sup>*), then, as *n* ! ∞, we obtain the limit *r*-variate Student distribution with parameters *γ* ¼ 2 and Σ. Moreover, if Σ ¼ *Ir* (that is, the *r*-variate Student distribution is spherically symmetric), then its one-dimensional marginals have the form (1). As we have already noted, distribution (1) was apparently for the first time introduced as a limit distribution for the sample median in a sample with geometrically distributed random size in [11]. It is worth noticing that in the cited paper [11], distribution (1) was not identified as the Student distribution with two degrees of freedom.

Thus, the main conclusion of this section can be formulated as follows. If the number of random factors that determine the observed value of a random variable is random itself with the distribution that can be approximated by the gamma distribution with coinciding shape and scale parameters (e.g., is negative binomial with probability of success close to one, see Lemma 4), then those functions of the random factors that are regarded as asymptotically normal in the classical situation are actually asymptotically Student with considerably heavier tails. Hence, since gamma-models and/or negative binomial models are widely applicable (to confirm this it may be noted that the negative binomial distribution is mixed Poisson with mixing gamma distribution, this fact is widely used in insurance), the Student distribution can be used in descriptive statistics as a rather reasonable heavy-tailed asymptotic approximation.
