2018
Том 70
№ 9

All Issues

Matsak I. K.

Articles: 24
Article (Ukrainian)

Limit theorems for the maximum of sums of independent random processes

Matsak I. K., Plichko A. M., Sheludenko A. S.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 506-518

We study the conditions for the weak convergence of the maximum of sums of independent random processes in the spaces $C[0, 1]$ and $L_p$ and present examples of applications to the analysis of statistics of the type $\omega 2 $.

Article (Ukrainian)

Asymptotic behavior of the extreme values of random variables. Discrete case

Matsak I. K.

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 7. - pp. 945-956

We study the exact asymptotics of almost surely extreme values of discrete random variables.

Brief Communications (Ukrainian)

Asymptotic Behavior of a Counting Process in the Maximum scheme

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2013. - 65, № 11. - pp. 1575–1579

We determine the exact asymptotic behavior of the logarithm of a counting process in the maximum scheme.

Brief Communications (Ukrainian)

One improvement of the law of the iterated logarithm for the maximum scheme

Akbash K. S., Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2012. - 64, № 8. - pp. 1132-1137

A lower bound is found in the law of the iterated logarithm for the maximum scheme.

Article (Ukrainian)

The order law of large numbers of the Marcinkiewicz - Zygmund

Akbash K. S., Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1587-1597

The Marcinkiewicz - Zygmund order law of large numbers is established for random variables in Banach lattices. Similar results are obtained also for the maximum scheme.

Article (Ukrainian)

On the Marcinkiewicz–Zygmund law of large numbers in Banach lattices

Matsak I. K., Plichko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2010. - 62, № 4. - pp. 504–513

We strengthen the well-known Marcinkiewicz–Zygmund law of large numbers in the case of Banach lattices. Examples of applications to empirical distributions are presented.

Article (Ukrainian)

On some limit theorems for the maximum of sums of independent random processes

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2008. - 60, № 12. - pp. 1664–1674

We study conditions of weak convergence of maximum of sums of independent random processes in the space $L_p.$ We present a number of applications to asymptotic analysis of some $\omega^2$-type statistics.

Article (Ukrainian)

One moment estimate for the supremum of normalized sums in the law of the iterated logarithm

Matsak I. K., Plichko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2006. - 58, № 5. - pp. 653–665

For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their application to the law of the iterated logarithm in Banach lattices is given.

Article (Ukrainian)

A Limit Theorem for Integral Functionals of an Extremum of Independent Random Processes

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2005. - 57, № 2. - pp. 214–221

We prove a theorem on the convergence of integral functionals of an extremum of independent stochastic processes to a degenerate law of distributions.

Article (Ukrainian)

Limit Theorems for Random Elements in Ideals of Order-Bounded Elements of Functional Banach Lattices

Matsak I. K., Plichko A. M.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 2001. - 53, № 1. - pp. 41-49

For a sequence of independent random elements belonging to an ideal of order-bounded elements of a Banach lattice, we investigate the asymptotic relative stability of extremal values, the law of large numbers for the pth powers, and the central limit theorem.

Article (Ukrainian)

Convergence of distributions of integral functionals of extremal random functions

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1999. - 51, № 9. - pp. 1201–1209

We study the convergence of distributions of integral functionals of random processes of the formU n (t)=b n (Z n (t)-a n G(t)),tT, where {X=X(t), tT} is a random process,X n ,n≥1, are independent copies ofX, andZ n (t)=max1≤k≤n X k (t).

Article (Ukrainian)

Asymptotic properties of the norm of the extremum of a sequence of normal random functions

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1998. - 50, № 10. - pp. 1359–1365

Under additional conditions on a bounded normally distributed random function X = X( t), t ∈ T, we establish a relation of the form $$\mathop {\lim }\limits_{n \to \infty } P(b_n (||Z_n || - a_n ) \leqslant x) = \exp ( - e^{ - x} )\forall x \in R^1 $$ where \(Z_n = Z_n (t) = \mathop {\max }\limits_{1 \leqslant k \leqslant n} X_k (t),(X_n )\) are independent copies of \(X,||x(t)|| = \mathop {\sup }\limits_{1 \in T} |x(t)|\) , and (a n) and (b n) are numerical sequences.

Article (Ukrainian)

Asymptotic properties of the norm of extremum values of normal random elements in the space C[0, 1]

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1998. - 50, № 9. - pp. 1227–1235

We prove that $$\mathop {\lim }\limits_{n \to \infty } \left( {\left\| {Z_n } \right\| - (2 ln (n))^{1/2} \left\| \sigma \right\|} \right) = 0 a.s.,$$ where X is a normal random element in the space C [0,1], MX = 0, σ = {(M¦X(t2)1/2 t∈[0,1}, (X n ) are independent copies of X, and \(Z_n = \mathop {\max }\limits_{l \leqslant k \leqslant n} X_k \) . Under additional restrictions on the random element X, this equality can be strengthened.

Article (Ukrainian)

Weak convergence of the extreme values of independent random variables in banach spaces with unconditional bases

Matsak I. K.

↓ Abstract   |   Full text (.pdf)

Ukr. Mat. Zh. - 1996. - 48, № 6. - pp. 805-812

We generalize well-known results concerning the weak convergence of maxima of real independent random variables to the case of random variables taking values in the Banach spaces with unconditional bases.

Article (Ukrainian)

Asymptotic estimate for sums of independent random variables in a Banach space

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1991. - 43, № 2. - pp. 270–273

Article (Ukrainian)

Khinchin's inequality and the asymptotic behavior of the sums ??nxn in Banach lattices

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1990. - 42, № 5. - pp. 639–644

Article (Ukrainian)

A note on the central limit theorem in a Banach space

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 5. - pp. 649-651

Article (Ukrainian)

Central limit theorem in a Banach space

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1988. - 40, № 2. - pp. 234-239

Article (Ukrainian)

On the law of the iterated logarithm

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1987. - 39, № 2. - pp. 264–267

Article (Ukrainian)

A certain problem for a random walk on the plane

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1985. - 37, № 5. - pp. 664–668

Article (Ukrainian)

Fernique's condition and Gaussian processes

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1984. - 36, № 2. - pp. 262 - 265

Article (Ukrainian)

Asymptotic behavior of nonrecurrent random walks

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1981. - 33, № 3. - pp. 341–347

Article (Ukrainian)

Asymptotic properties of Gaussian processes

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1980. - 32, № 3. - pp. 332 - 339

Article (Ukrainian)

Regularity of sampling distribution functions of a random process

Matsak I. K.

Full text (.pdf)

Ukr. Mat. Zh. - 1978. - 30, № 2. - pp. 241–247