2019
Том 71
№ 11

# Matsak I. K.

Articles: 26
Article (Ukrainian)

### Limit theorems for the maximum of sums of independent random processes

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

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

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

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

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

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

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

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

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

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

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

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]

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

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.

Brief Communications (Ukrainian)

### The limit theorem for the maximum ot $C$-valued Gaussian random variables

Ukr. Mat. Zh. - 1995. - 47, № 7. - pp. 1006-1008

The well known asymptotic equality for the maximum of real Gaussian random variables is generalized to the case of random variables with the values taken in the space $C$.

Article (Ukrainian)

### On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space

Ukr. Mat. Zh. - 1993. - 45, № 9. - pp. 1225–1231

Assume that (X n) are independent random variables in a Banach space, (b n) is a sequence of real numbers, Sn= ∑ 1 n biXi, and Bn=∑ 1 n b i 2 . Under certain moment restrictions imposed on the variablesX n, the conditions for the growth of the sequence (bn) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (Sn/B n ln ln Bn)1/2).

Article (Ukrainian)

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

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

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

Article (Ukrainian)

### A note on the central limit theorem in a Banach space

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

Article (Ukrainian)

### Central limit theorem in a Banach space

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

Article (Ukrainian)

### On the law of the iterated logarithm

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

Article (Ukrainian)

### A certain problem for a random walk on the plane

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

Article (Ukrainian)

### Fernique's condition and Gaussian processes

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

Article (Ukrainian)

### Asymptotic behavior of nonrecurrent random walks

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

Article (Ukrainian)

### Asymptotic properties of Gaussian processes

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

Article (Ukrainian)

### Regularity of sampling distribution functions of a random process

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