Geodesic
Repetitions of the values of a function of segments of a sequence of independent trials
Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 49-59
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the problem on the number $\xi(N)$ of $k$-fold matchings of symbols that occur in a sequence of random variables resulting from aggregating the states of a Markov–Bruns chain generated by $n$-tuples $(X_i,\dots,X_{i+n-1})$ consisting of elements of the sequence $(X_1,\dots,X_{N+n-1})$ of independent realisations of a random variable $X$. The aggregation of states consists of applying a given function $f$, which takes a finite number of values, to the $n$-tuples $(X_i,\dots,X_{i+n-1})$. We prove a theorem on convergence to a multivariate normal law for the joint distribution of $\xi(N)$ under various aggregation functions, and give sufficient conditions on convergence of the distribution of $\xi(N)$ to the chi-square law as $N\to\infty$. These results are applied to the problem on $k$-fold imperfect matchings of $n$-tuples in a sequence of polynomial trials. In particular, we prove a theorem on convergence to a multivariate normal law for the vector of numbers of $k$-fold imperfect matchings of various lengths and ranks. 
@article{DM_2000_12_3_a2,
     author = {A. M. Shoitov},
     title = {Repetitions of the values of a function of segments of a sequence of independent trials},
     journal = {Diskretnaya Matematika},
     pages = {49--59},
     year = {2000},
     volume = {12},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2000_12_3_a2/}
}
TY  - JOUR
AU  - A. M. Shoitov
TI  - Repetitions of the values of a function of segments of a sequence of independent trials
JO  - Diskretnaya Matematika
PY  - 2000
SP  - 49
EP  - 59
VL  - 12
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/DM_2000_12_3_a2/
LA  - ru
ID  - DM_2000_12_3_a2
ER  - 
%0 Journal Article
%A A. M. Shoitov
%T Repetitions of the values of a function of segments of a sequence of independent trials
%J Diskretnaya Matematika
%D 2000
%P 49-59
%V 12
%N 3
%U http://geodesic.mathdoc.fr/item/DM_2000_12_3_a2/
%G ru
%F DM_2000_12_3_a2
A. M. Shoitov. Repetitions of the values of a function of segments of a sequence of independent trials. Diskretnaya Matematika, Tome 12 (2000) no. 3, pp. 49-59. http://geodesic.mathdoc.fr/item/DM_2000_12_3_a2/

[1] Abdalimov B., “Asimptoticheskaya normalnost $U$-statistik dlya $s$-zavisimykh velichin”, Sluchainye protsessy i smezhnye voprosy, Fan, Tashkent, 1970, 26–30 | MR

[2] Mikhailov V. G., “Tsentralnaya predelnaya teorema dlya chisla nepolnykh dlinnykh povtorenii”, Teoriya veroyatnostei i ee primeneniya, 20:4 (1975), 880–884 | MR

[3] Mikhailov V. G., “Tsentralnaya predelnaya teorema dlya neodnorodnykh $U$-statistik ot konechno zavisimykh sluchainykh velichin”, Matem. sb., 98:4 (1975), 624–634 | MR

[4] Gharib M., “A uniform estimate for the convergence rate in the central limit theorem of the weighted sum of random vectors forming a homogeneous Markov chain”, Kobe J. Math., 1998, 1–15 | MR | Zbl