**7. The asymptotic distribution of sample quantiles in samples with sizes generated by a Cox process**

Sometimes, when the performance of a technical or financial system is analyzed, a forecast of main characteristics is made on the basis of data accumulated during a certain period of the functioning of the system. As a rule, data are accumulated as a result of some "informative events" that occur during this period. For example, inference concerning the distribution of insurance claims, which is very important for the estimation of, say, the ruin probability of an insurance company, is usually performed on the basis of the statistic *W*1,*W*2, … ,*WN T*ð Þ of the values of insurance claims arrived within a certain time interval 0, ½ � *T* (here *N T*ð Þ denotes the number of claims arrived during the time interval 0, ½ � *T* ). Moreover, this inference is typically used for the prediction of the value of the ruin probability for the next period ½ � *T*, 2*T* . But it is obvious (at least in the example above) that the observed number of informative events occurred during the time interval 0, ½ � *T* is actually a realization of a random variable, because both the number of insurance claims arrived within this interval follow a stochastic counting process. If the random character of the number of available observations is not taken into consideration, then all what can be done is the *conditional* forecast. To obtain a complete prediction with the account of the randomness of the number of "informative events," we should use the results similar to Theorems 2 and 3. One of rather realistic and general assumptions concerning *N t*ð Þ, the number of observations accumulated by the time *t*, is that *N t*ð Þ is a Cox process. In this section, as an example, we will consider the asymptotic behavior of sample quantiles constructed from a sample whose size is determined by a Cox process. As we have already noted in the introduction, this problem is very *From Asymptotic Normality to Heavy-Tailedness via Limit Theorems for Random Sums… DOI: http://dx.doi.org/10.5772/intechopen.89659*

important for the proper application of such risk measures as VaR (Value-at-Risk) in, say, financial engineering.

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 *W*<sup>1</sup> will be denoted *ξλ<sup>i</sup>* , *i* ¼ 1, … ,*r*. The sample quantiles of orders *λ*1, … , *λ<sup>r</sup>* are the 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 classical. Denote

$$Y\_{n,j}^\* = \sqrt{n} \left( W\_{\left( \left[ \lambda\_j n \right] + 1 \right)} - \xi\_{\lambda\_j} \right), \qquad j = 1, \ldots, r.$$

Theorem 7 [32]. *If p x*ð Þ *is differentiable in some neighborhoods of the quantiles ξλ<sup>i</sup> and p ξλ<sup>i</sup>* � � 6¼ <sup>0</sup>*, i* <sup>¼</sup> 1, … ,*r, then, as n* ! <sup>∞</sup>*, the joint distribution of the normalized sample quantiles Y* <sup>∗</sup> *<sup>n</sup>*,1, … , *Y* <sup>∗</sup> *<sup>n</sup>*,*<sup>r</sup> weakly converges to the r-variate normal distribution with zero vector of expectations and covariance matrix* <sup>Σ</sup> <sup>¼</sup> *<sup>σ</sup>ij* � �,

$$\sigma\_{ij} = \frac{\lambda\_i \left(1 - \lambda\_j\right)}{p\left(\xi\_{\lambda\_i}\right)p\left(\xi\_{\lambda\_j}\right)}, \ i \le j.$$

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 distribution density *p x*ð Þ.

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

$$N(t) = N\_1(\Lambda(t)), \qquad t \ge 0.$$

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 the negative binomial distribution.

Cox processes are widely used as models of inhomogeneous chaotic flows of events, see, for example, [2].

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 as *t* ! ∞.

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

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 distri-

Remark 3. In the case where the sample size *Nn* has the negative binomial

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

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

**7. The asymptotic distribution of sample quantiles in samples with sizes**

Sometimes, when the performance of a technical or financial system is analyzed, a forecast of main characteristics is made on the basis of data accumulated during a certain period of the functioning of the system. As a rule, data are accumulated as a result of some "informative events" that occur during this period. For example, inference concerning the distribution of insurance claims, which is very important for the estimation of, say, the ruin probability of an insurance company, is usually performed on the basis of the statistic *W*1,*W*2, … ,*WN T*ð Þ of the values of insurance claims arrived within a certain time interval 0, ½ � *T* (here *N T*ð Þ denotes the number of claims arrived during the time interval 0, ½ � *T* ). Moreover, this inference is typically used for the prediction of the value of the ruin probability for the next period ½ � *T*, 2*T* . But it is obvious (at least in the example above) that the observed number of informative events occurred during the time interval 0, ½ � *T* is actually a realization of a random variable, because both the number of insurance claims arrived within this interval follow a stochastic counting process. If the random character of the number of available observations is not taken into consideration, then all what can be done is the *conditional* forecast. To obtain a complete prediction with the account of the randomness of the number of "informative events," we should use the results similar to Theorems 2 and 3. One of rather realistic and general assumptions

concerning *N t*ð Þ, the number of observations accumulated by the time *t*, is that *N t*ð Þ is a Cox process. In this section, as an example, we will consider the asymptotic behavior of sample quantiles constructed from a sample whose size is determined by a Cox process. As we have already noted in the introduction, this problem is very

, and *n* is large.

*<sup>n</sup>*), then, as *n* ! ∞, we obtain the limit *r*-variate Student

*<sup>n</sup>*, *m* ¼ 1 (that is, the geometric distribution

*<sup>n</sup>*, *<sup>m</sup>* <sup>¼</sup> <sup>1</sup> 2

bution with the parameters *<sup>p</sup>* <sup>¼</sup> <sup>1</sup>

*Probability, Combinatorics and Control*

with the parameter *<sup>p</sup>* <sup>¼</sup> <sup>1</sup>

degrees of freedom.

asymptotic approximation.

**182**

**generated by a Cox process**

distribution with the parameters *<sup>p</sup>* <sup>¼</sup> <sup>1</sup>

one-dimensional distributions of sample quantiles in samples with random sizes were obtained.

The proof is a simple combination of Lemmas 1, 5, and 6 and Theorem 3. Corollary 4. *Under the conditions of Theorem 8, the joint distribution of the normal-*

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

*r-variate normal law with zero expectation and covariance matrix* Σ*, if and only if*

This statement immediately follows from Theorem 8 with the account of

Corollary 5. *Under the conditions of Theorem 8, the joint distribution of the*

*d t*ð Þ <sup>p</sup> *<sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jN t*ð Þ <sup>þ</sup><sup>1</sup> � *ξλ <sup>j</sup>*

*to the r-variate Student distribution with parameters γ* > 0 *and* Σ *defined in Theorem 4,*

Pð Þ) Λð Þ*t* <*xd t*ð Þ *Gγ=*2,*γ=*2ð Þ *x* , *t* ! ∞,

where *Gγ=*2,*γ=*2ð Þ *x* is the gamma-distribution function with coinciding shape and

Let 0 <*λ*<1 and let *ξλ* be the *λ*-quantile of the random variable *W*1. As above,

The purpose of the chapter was to give a possible explanation of the emergence

of heavy-tailed distributions that are often observed in practice instead of the expected normal laws. As the base for this explanation, limit theorems for random sums and statistics constructed from samples with random sizes were considered. Within this approach, it becomes possible to obtain arbitrarily heavy tails of the data distributions without assuming the non-existence of the moments of the observed characteristics. Some comments were made on the heavy-tailedness of scale mixtures of normal distributions. Two general theorems presenting necessary and sufficient conditions for the convergence of the distributions of random sums of random vectors and multivariate statistics constructed from samples with random sizes were proved. As examples of the application of these general theorems, conditions were presented for the convergence of the distributions of random sums of independent random vectors with finite covariance matrices to multivariate elliptically contoured stable and Linnik distributions. An alternative definition of the latter was proposed. Also, conditions were presented for the convergence of the distributions of asymptotically normal (in the traditional sense) statistics to multivariate elliptically contoured Student distributions when the sample size is replaced by a random variable. The joint asymptotic behavior of sample quantiles in samples with random sizes was considered. Special attention was paid to the continuoustime case assuming that the sample size increases in time following a Cox process

*d t*ð Þ ) <sup>1</sup> ð Þ *<sup>t</sup>* ! <sup>∞</sup> *:*

� �*, j* <sup>¼</sup> 1, … ,*r, weakly converges to the*

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

*d t*ð Þ <sup>p</sup> *<sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jN t*ð Þ <sup>þ</sup><sup>1</sup> � *ξλ <sup>j</sup>*

Λð Þ*t*

the standard normal distribution function will be denoted Φð Þ *x* .

resulting in the sample size having the mixed Poisson distribution.

Supported by Russian Science Foundation, project 18-11-00155.

*ized sample quantiles* ffiffiffiffiffiffiffiffi

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

*normalized sample quantiles* ffiffiffiffiffiffiffiffi

scale parameters equal to *γ=*2.

Lemma 1.

*if and only if*

**8. Conclusion**

**Acknowledgements**

**185**

Our aim here is to give necessary and sufficient conditions for the weak convergence of the *joint* distributions of sample quantiles constructed from samples with random sizes driven by a Cox process and to describe the *r*-variate limit distributions emerging here, thus extending Mosteller's Theorem 4 to samples with random sizes. The results of this section extend those of [36] to the continuous-time case.

Lemma 5. *Let N t*ð Þ *be a Cox process controlled by the process* <sup>Λ</sup>ð Þ*<sup>t</sup> . Then N t*ðÞ!*<sup>P</sup>* ∞ ð Þ *t* ! ∞ *if and only if* ΛðÞ!*t <sup>P</sup>* <sup>∞</sup> ð Þ *<sup>t</sup>* ! <sup>∞</sup> .

Lemma 6. *Let N t*ð Þ *be a Cox process controlled by the process* Λð Þ*t . Let d t*ð Þ > 0 *be a function such that d t*ðÞ! ∞ ð Þ *t* ! ∞ *. Then the following conditions are equivalent:*

1.One-dimensional distributions of the normalized Cox process weakly converge to the distribution of some random variable *Z* as *t* ! ∞:

$$\frac{N(t)}{d(t)} \Rightarrow Z \qquad (t \to \infty).$$

2.One-dimensional distributions of the controlling process Λð Þ*t* , appropriately normalized, converge to the same distribution:

$$\frac{\Lambda(t)}{d(t)} \Rightarrow Z \qquad (t \to \infty).$$

For the proof of Lemmas 5 and 6 see [37].

Now we proceed to the main results of this section. In addition to the notation introduced above, for positive integer *<sup>n</sup>* set *<sup>Q</sup> <sup>j</sup>*ð Þ¼ *<sup>n</sup> <sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jn* <sup>þ</sup><sup>1</sup> , *<sup>j</sup>* <sup>¼</sup> 1, … ,*r*, *Q n*ð Þ¼ ð Þ *Q*1ð Þ *n* , … , *Qr*ð Þ *n* , *ξ* ¼ *ξλ*<sup>1</sup> , … , *ξλ<sup>r</sup>* � �. Let *d t*ð Þ be an infinitely increasing positive function. Set

$$Z(t) = \sqrt{d(t)} (Q(N(t)) - \xi) \dots$$

Theorem 8. *Let* ΛðÞ!*t <sup>P</sup>* <sup>∞</sup> *as t* ! <sup>∞</sup>*. If p x*ð Þ *is differentiable in neighborhoods of the quantiles ξλ<sup>i</sup> and p ξλ<sup>i</sup>* � � 6¼ <sup>0</sup>*, i* <sup>¼</sup> 1, … ,*r, then the convergence*

$$Z(t) \Rightarrow Z \quad (t \to \infty),$$

to some random vector *Z* takes place, if and only if there exists a nonnegative random variable *U* such that

$$\mathbb{P}(Z \in A) = \quad \mathbf{E} \Phi\_{U^{-1}\Sigma}(A), \ A \in \mathcal{B}(\mathbb{R}^r),$$

where <sup>Σ</sup> <sup>¼</sup> *<sup>σ</sup>ij* � �,

$$
\sigma\_{\vec{\eta}} = \frac{\lambda\_i \left(1 - \lambda\_j\right)}{p\left(\xi\_{\vec{\lambda}\_i}\right) p\left(\xi\_{\vec{\lambda}\_j}\right)}, \ i \le j,
$$

and

$$\frac{\Lambda(t)}{d(t)} \Rightarrow U \quad (t \to \infty).$$

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

The proof is a simple combination of Lemmas 1, 5, and 6 and Theorem 3. Corollary 4. *Under the conditions of Theorem 8, the joint distribution of the normalized sample quantiles* ffiffiffiffiffiffiffiffi *d t*ð Þ <sup>p</sup> *<sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jN t*ð Þ <sup>þ</sup><sup>1</sup> � *ξλ <sup>j</sup>* � �*, j* <sup>¼</sup> 1, … ,*r, weakly converges to the r-variate normal law with zero expectation and covariance matrix* Σ*, if and only if*

$$\frac{\Lambda(t)}{d(t)} \Rightarrow \mathbf{1} \quad (t \to \infty).$$

This statement immediately follows from Theorem 8 with the account of Lemma 1.

Corollary 5. *Under the conditions of Theorem 8, the joint distribution of the normalized sample quantiles* ffiffiffiffiffiffiffiffi *d t*ð Þ <sup>p</sup> *<sup>W</sup>*ð Þ ½ � *<sup>λ</sup> jN t*ð Þ <sup>þ</sup><sup>1</sup> � *ξλ <sup>j</sup>* � �*, j* <sup>¼</sup> 1, … ,*r, weakly converges to the r-variate Student distribution with parameters γ* > 0 *and* Σ *defined in Theorem 4, if and only if*

$$\mathbf{P}(\Lambda(t) < \mathbf{x}d(t)) \Rightarrow \mathbf{G}\_{\mathbf{r}/2,\mathbf{r}/2}(\mathbf{x}), \quad t \to \infty,$$

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

Let 0 <*λ*<1 and let *ξλ* be the *λ*-quantile of the random variable *W*1. As above, the standard normal distribution function will be denoted Φð Þ *x* .
