**8. Conclusion**

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

*<sup>P</sup>* <sup>∞</sup> ð Þ *<sup>t</sup>* ! <sup>∞</sup> .

to the distribution of some random variable *Z* as *t* ! ∞:

*N t*ð Þ

Λð Þ*t*

normalized, converge to the same distribution:

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

*Q n*ð Þ¼ ð Þ *Q*1ð Þ *n* , … , *Qr*ð Þ *n* , *ξ* ¼ *ξλ*<sup>1</sup> , … , *ξλ<sup>r</sup>*

itive function. Set

*quantiles ξλ<sup>i</sup> and p ξλ<sup>i</sup>*

where Σ ¼ *σij*

and

**184**

Theorem 8. *Let* ΛðÞ!*t*

random variable *U* such that

� �,

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>*

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

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

2.One-dimensional distributions of the controlling process Λð Þ*t* , appropriately

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

Now we proceed to the main results of this section. In addition to the notation

*d t*ð Þ <sup>p</sup> ð Þ *QNt* ð Þ� ð Þ *<sup>ξ</sup> :*

� �. Let *d t*ð Þ be an infinitely increasing pos-

*<sup>P</sup>* <sup>∞</sup> *as t* ! <sup>∞</sup>*. If p x*ð Þ *is differentiable in neighborhoods of the*

<sup>Σ</sup>ð Þ *<sup>A</sup>* , *<sup>A</sup>* <sup>∈</sup> <sup>B</sup> *<sup>r</sup>* ð Þ,

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

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*,

� � 6¼ <sup>0</sup>*, i* <sup>¼</sup> 1, … ,*r, then the convergence*

*Z t*ðÞ) *Z t*ð Þ ! ∞ ,

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

� �

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

*Z t*ðÞ¼ ffiffiffiffiffiffiffiffi

Pð Þ¼ *Z* ∈ *A* EΦ*<sup>U</sup>*�<sup>1</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>*

Λð Þ*t*

were obtained.

∞ ð Þ *t* ! ∞ *if and only if* ΛðÞ!*t*

*Probability, Combinatorics and Control*

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 resulting in the sample size having the mixed Poisson distribution.
