01.01.05 (Probability theory and mathematical statistics)
Birth date:
15.09.1956
E-mail:
Keywords:
invariance principle,
strong approximation,
sums of independent random vectors,
infinitely divisible and compound Poisson approximation,
estimation of the rate of approximation,
Central Limit Theorem,
concentration functions,
inequalities.
UDC:
519.21, 519.2
Subject:
At the beginning of his scientific career A.Yu. Zaitsev worked on solving a problem posed in the mid 50s by A.N. Kolmogorov. He managed to get the correct order of the accuracy of infinitely divisible approximation of distributions of sums of independent random variables, the distribution of which are concentrated on the short intervals of length $ \tau $ to within a small probability $ p $. It was found that the accuracy of approximation in the Lévy metric has order $ p + \tau \log ( 1 / \tau) $, which is much more precise than the initial result of Kolmogorov $ p ^ { 1/5 } + \tau ^ { 1/2 } \log ^{1/4}( 1 / \tau) $, and also of the latest results obtained by other authors. As approximating, the so-called accompanying infinitely divisible compound Poisson distributions were used. Moreover, as was shown by T. Arak, the estimates are correct in order. In 1986, a joint monograph by T. Arak and A.Yu. Zaitsev, containing a summary of these results, was published. Later A.Yu. Zaitsev (1989) showed that a similar estimate holds in the multidimensional case, and an absolute constant factor is replaced a $ c (d) $, depending only on the dimension $ d $. While proving it was found that for $ p = 0 $ (i.e., when the norms of the terms are bounded by a constant $ \tau $ with probability one) then for any $ \lambda> 0 $ a random vector $ X $, having the same distribution as this sum, may be constructed on the same probability space with the corresponding Gaussian vector $ Y $, so that
$ {\mathbf P} (\| X - Y \|> \lambda) \le c_1 (d) \exp (- \lambda / c_2 (d) \tau) $. Moreover, A.Yu. Zaitsev (1986) proved that the same result holds for vectors with distributions from a certain class $ A_d (\tau) $ of distributions with sufficiently slowly growing cumulants containing, in particular, arbitrary infinitely divisible distributions with spectral measures concentrated on the ball of radius $ c\tau $ centered at the origin. Another important special case of estimating the accuracy of infinitely divisible approximation is obtained for $ \tau = 0 $, the right-hand side of the estimate of Kolmogorovs uniform distance between distribution functions $ \rho (\, \cdot \,, \, \cdot \,) $ has the form $ c (d) p $. In a paper published in 2003 in Zapiski nauchnyh seminarov POMI, this result is interpreted as a general estimate for the accuracy of approximation of the sample composed of non-i.i.d. rare events by a Poisson point process.
In other papers, some optimal bounds for the Kolmogorov distance were also obtained in the general case.
In particular, in the one-dimensional case, it succeeded to obtain the statements of results which imply simultaneously as (optimal in order) estimates for the rate of approximation of convolutions by accompanying infinitely divisible compound Poisson distributions, and rather general bounds in the CLT. Since tails of distributions of summands are arbitrary, the results cover the popular in the recent time case of the so called heavy tails of distributions of summands.
Similar methods were also used to obtain the following paradoxical result. There exists depending only on the dimension $ d $ value $ c (d) $, such that for any symmetric distribution $ F $ and any natural $ n $ uniform distance between the degrees in the convolution sense $ F ^ n $ admits the estimates $ \rho (F ^ n, F ^ {n +1}) \le c (d) n ^ {-1 / 2} $ and $ \rho (F ^ n, F ^ {n +2}) \le c (d) n ^ {-1 } $, and both estimates are unimprovable in order.
In recent joint papers, most of the results mentioned above have been carried over to the values of distributions in Hilbert space on convex polyhedra. In this case, the constants depend only on the number of half-spaces involved in the definition of the polyhedron.
In a recent paper, the following general related result was obtained.
Denote
$
\rho_{\mathcal{C}_d}(F,G) = \sup_A |F\{A\} - G\{A\}|
$,
where the supremum is taken over all convex subsets of $\mathbb R^d$. For any nontrivial distribution $F$
there is $c_1(F)$ such that
$$
\rho_{\mathcal{C}_d}(F^n, F^{n+1})\leq \frac{c_1(F)}{\sqrt n}
$$
for any natural $n$. The distribution $F$ is called trivial if
it is concentrated on a hyperplane that does not contain the origin.
Clearly, for such $F$
$
\rho_{\mathcal{C}_d}(F^n, F^{n+1}) = 1
$.
A similar result for the Prokhorov distance is also obtained.
For any $d$-dimensional distribution $F$ there is a $c_2(F)>0$ that depends only on $F$ and such that
$$
(F^n)\{A\}\le (F^{n+1})\{A^{c_2(F)}\}+\frac{c_2(F)}{\sqrt{n}}\text{ and } (F^{n+1})\{A\}\leq (F^n)\{A^{c_2(F)}\}+\frac{c_2(F)} {\sqrt{n}}
$$
for any
Borel set $ A $ and for all positive integers $n$. Here $A^{\varepsilon }$ is the $ \varepsilon $-neighborhood of a set $ A $.
Using the Strassen-Dudley theorem, one can derive the following statement.
For any distribution $F\in\mathfrak F_d$ there is a value $c_3(F)$,
depending only on $F$ and such that
for any natural $n$ one
can construct on the same probability
space random vectors $\xi_n $ and $\eta_n $ with
$\mathcal{L}(\xi_n )=F^{n+1}$ and $\mathcal{L}(\eta_n )=F^n$, so that
$$\mathbf{P}\left\{ \Vert \xi_n -\eta_n \Vert >c_3(F) \right\} \le
\frac{c_3(F)}{\sqrt{n}}.
$$
Hence, the following bound for the Prokhorov distance holds:
$\pi(\mathcal{L}(\xi_n/\sqrt{n} ), \mathcal{L}(\eta_n/\sqrt{n} ))\leqslant
{c_3(F)}/{\sqrt{n}}$.
A negative answer was also given to the question A.N. Kolmogorov and Yu.V. Prokhorov about a possibility of infinitely divisible approximation of distributions of sums of independent identically distributed random variables in the sense of the distance in variation. A one-dimensional probability distribution was constructed, such that all its $ n $-fold convolutions are uniformly separated from the set of infinitely divisible laws in the sense of the distance in variation up to the distance $ 1 / 14$.
The most significant result obtained in the 90s is a multi-dimensional version of the classic one-dimensional result of Komlós, Major and Tusnády (1975) about the strong Gaussian approximation of sums of independent identically distributed random variables under the existence of exponential moments of terms. The dependence of the constants on the dimension and distribution of summands is indicated explicitly. Thus, the problem, standing more than 20 years was solved. Later, the result was generalized to the case of non-identically distributed summands and a full one-dimensional multivariate analogue of a result of A.I. Sakhanenko (1984) was obtained. These results were presented in an invited talk at the International Congress of Mathematicians in Beijing (2002). Relatively recently, estimates of strong Gaussian approximation of sums of independent $ d $-dimensional random vectors $ X_j $ with finite moments of the form $ {\mathbf E} H (\| X_j \|) $, where $ H $ is a monotone function growing no slower than $ x ^ 2 $ and not faster than $ \exp (cx) $, were obtained. These results may be considered as multidimensional generalizations and improvements of the corresponding results of Komlós, Major and Tusnády (1975), Sakhanenko (1985) and U. Einmahl (1989). In the special case, where $ H (x) = x ^ \gamma $, $ \gamma> 2 $, in a joint paper with F. Götze, estimates of optimal order were obtained for identically distributed random vectors. In 2011, in a joint paper the infinite-dimensional case was considered too.
In a paper of A.Yu. Zaitsev (1994), for any $ \varepsilon> 0 $ some pairs of bivariate distributions were constructed such that the distance in variation between their projections on an arbitrary one-dimensional direction does not exceed $ \varepsilon $, even though the distance $\rho$ between the two-dimensional distribution function is $ 1 / 2 $.
In 2003–2005 A.Yu. Zaitsev obtained new estimates of strong approximation of the $ L_1 $-norm of centered and normalized kernel density estimators. It was assumed that the kernel is bounded and has a bounded support. The different natural classes of densities with restrictions on the smoothness, growth, decay, and support size were considered. Estimates for the Prokhorov distance and for the size of zones, where the normal approximation is valid for large deviations, were also obtained. In a joint work with E. Giné and D.M. Mason (2003), the Central Limit Theorem for the $ L_1 $-norm of centered and normalized kernel density estimators of an arbitrary density was transferred to processes indexed by kernels.
Assuming that i.i.d. multidimensional random terms have zero expectations and finite moments of the fourth order, A.Yu. Zaitsev (2010, 2014 together with F. Götze) showed that, for sets bounded by surfaces of the second order, the accuracy of approximation by short asymptotic expansions in the Central Limit Theorem is of the order $ O ( 1 / N) $, where $ N $ is the number of summands, provided that the dimension is not less than five. Earlier, similar statements were obtained in 1997 in a paper by F. Götze and V. Bentkus provided that the dimension of not less than nine. In the joint paper of F. Götze and A.Yu. Zaitsev, nine is replaced by five, and a further reduction of dimension is impossible. There were also obtained new explicit expressions for the simple power dependence of the corresponding constants of the fourth moments and the eigenvalues of the covariance operator of summands. Estimates are uniform with respect to isometric operators involved in the definition of surfaces.
In recent years, several joint papers of A.Yu. Zaitsev were published about estimating the concentration functions of distributions of sums of independent random variables.
Biography
A.Yu. Zaitsev is a specialist in the field of probability theory and mathematical statistics, the author of more than 100 publications, including a monograph. His main results are related to the study of sums of independent variables.
In September 1973 A.Yu. Zaitsev entered the Mathematics and Mechanics Faculty of the Leningrad State University. In June 1978 he graduated in mathematics. In August 1978 was hired by the Leningrad branch of the Steklov Mathematical Institute of the USSR Academy in the laboratory of statistical methods. In January 1981 he defended his candidate thesis on "Approximation of distributions of sums of independent random vectors infinitely divisible distributions" under the direction of I.A. Ibragimov. In January 1989 he defended his doctoral thesis on "Uniform limit theorems for sums of independent random vectors." In December 1992, A.Yu. Zaitsev was elected to the position of leading scientific researcher of PDMI RAS. From March 2001 to March 2006 he worked as a scientific secretary of PDMI. Since March 2006 he is again leading researcher PDMI. From January 2005 to June 2006 and from January 2010 to the present time A.Yu. Zaitsev is working as a professor of the chair of probability theory and mathematical statistics at the St. Petersburg State University.
A.Yu. Zaitsev is a member of the specialized council D 002.202.01 for doctoral dissertations, a member of the editorial boards of "Journal of Statistical Planning and Inference", "European Journal of Mathematics" and "Notes of scientific seminars PDMI".
In 2009 A.Yu. Zaitsev was awarded by the A.A. Markov Prize of RAS for a collection of works "Estimates for the accuracy of approximation of distributions of sums of independent variables".
Main publications:
T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables”, Proc. Steklov Inst. Math., 174 (1988), 1–222
A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors”, Russian Math. Surveys, 68:4 (2013), 721–761
A. Yu. Zaitsev, “Multidimensional version of a result of Sakhanenko in the invariance principle for vectors with finite exponential moments. I, II, III”, Theory Probab. Appl., 45:4 (2001), 624–641; 46:3 (2002), 490–514; 46:4 (2002), 676–698
F. Götze, A. Yu. Zaitsev, “On alternative approximating distributions in the multivariate version of Kolmogorov's second uniform limit theorem”, Theory Probab. Appl., 67:1 (2022), 1–16
A. Yu. Zaitsev, “An example of a distribution whose set of $n$-fold convolutions is uniformly separated from the set of infinitely divisible laws in the sense of the variation distance”, Theory Probab. Appl., 36:2 (1991), 419–425
A. Yu. Zaitsev, “On the proximity of distributions of successive sums in the Prokhorov distance”, Theory Probab. Appl., 69:2 (2024), 217–226
2023
2.
A. Yu. Zaitsev, “Estimates of stability with respect to the number of summands for distributions of successive sums of i.i.d. vectors”, Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, POMI, St. Petersburg, 2023, 86–95
3.
Veroyatnost i statistika. 34, Posvyaschaetsya yubileyu Andreya Nikolaevicha BORODINA, Zap. nauchn. sem. POMI, 525, ed. A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2023 , 189 pp.
4.
Veroyatnost i statistika. 35, Posvyaschaetsya yubileyu Yany Isaevny BELOPOLSKOI, Zap. nauchn. sem. POMI, 526, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2023 , 211 pp.
5.
I. A. Ibragimov, A. Yu. Zaitsev, D. N. Zaporozhets, M. A. Lifshits, “To the 70th anniversary of A. N. BORODIN”, Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, POMI, St. Petersburg, 2023, 5–6
2022
6.
F. Gettse, A. Yu. Zaitsev, “Ob alternativnykh approksimiruyuschikh raspredeleniyakh v mnogomernom variante vtoroi ravnomernoi predelnoi teoremy Kolmogorova”, Teoriya veroyatn. i ee primen., 67:1 (2022), 3–22 , arXiv: 2006.01942
Friedrich Götze, Andrei Yu. Zaitsev, “A new bound in the Littlewood–Offord problem”, This article belongs to the Special Issue Limit Theorems of Probability Theory, Mathematics, 10:10 (2022), 1740 , 6 pp., arXiv: 2112.12574
Ya. S. Golikova, A. Yu. Zaitsev, “On the accuracy of infinitely divisible approximation of $n$-fold convolutions of probability distributions”, Probability and statistics. Part 33, Zap. Nauchn. Sem. POMI, 515, POMI, St. Petersburg, 2022, 83–90
9.
Veroyatnost i statistika. 32, Posvyaschaetsya yubileyu Ildara Abdullovicha IBRAGIMOVA, Zap. nauchn. sem. POMI, 510, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2022 , 287 pp.
10.
Veroyatnost i statistika. 33, Zap. nauchn. sem. POMI, 515, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2022 , 238 pp.
2023
11.
F. Götze, A. Yu. Zaitsev, “Convergence to infinite-dimensional compound Poisson distributions on convex polyhedra”, J. Math. Sci. (N. Y.), 273:5 (2023), 732–737 , arXiv: 2109.11845
2021
12.
Veroyatnost i statistika. 30, Zap. nauchn. sem. POMI, 501, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2021 , 343 pp.
13.
Veroyatnost i statistika. 31, Zap. nauchn. sem. POMI, 505, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2021 , 330 pp.
2020
14.
Veroyatnost i statistika. 29, Zap. nauchn. sem. POMI, 495, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2020 , 322 pp.
15.
A. N. Borodin, A. Yu. Zaitsev, I. A. Ibragimov, M. A. Lifshits, V. N. Solev, “In memory of M. S. Nikulin”, Probability and statistics. Part 29, Zap. Nauchn. Sem. POMI, 495, POMI, St. Petersburg, 2020, 7–8
2019
16.
A. Yu. Zaitsev, A. M. Kagan, Ya. Yu. Nikitin, “Toward the History of the St. Petersburg School of Probability and Statistics. IV. Characterization of Distributions and Limit Theorems in Statistics”, Vestnik St Petersburg University: Mathematics, 52:1 (2019), 36–53
2021
17.
F. Götze, A. Yu. Zaitsev & D. Zaporozhets, “An Improved Multivariate Version of Kolmogorov’s Second Uniform Limit Theorem”, J. Math. Sci. (N. Y.), 258 (2021), 782–792 , arXiv: 1912.13296
2019
18.
Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2019 , 309 pp.
2018
19.
F. Götze, A. Yu. Zaitsev, “New applications of Araks inequalities to the Littlewood–Offord problem”, European Journal of Mathematics, 4:2 (2018), 10.1007/s40879-018-0215-3 , 25 pp. http://rdcu.be/Gb4B, arXiv: 1611.00831
M. A. Lifshits, Y. Y. Nikitin, V. V. Petrov, A. Y. Zaitsev, A. A. Zinger,, “Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random Variables”, Vestnik St. Petersburg University: Mathematics, 51:2 (2018), 144–163
2020
21.
F. Götze, A. Yu. Zaitsev, “Estimates for Closeness of Convolutions of Probability Distributions on Convex Polyhedra”, J. Math. Sci. (N. Y.), 251 (2020), 67–73 , arXiv: 1812.07473
2018
22.
Veroyatnost i statistika. 27, Zap. nauchn. sem. POMI, 474, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2018 , 247 pp.
23.
F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Arak inequalities for concentration functions and the Littlewood–Offord problem”, Theory Probab. Appl., 62:2 (2018), 196–215 , arXiv: 1506.09034
2020
24.
F. Götze, A. Yu. Zaitsev, “Rare Events and Poisson Point Processes”, J. Math. Sci. (N. Y.), 244 (2020), 771–778 , arXiv: 1802.06638
2017
25.
Veroyatnost i statistika. 25, Posvyaschaetsya pamyati Vladimira Nikolaevicha SUDAKOVA, Zap. nauchn. sem. POMI, 457, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2017 , 322 pp.
26.
Veroyatnost i statistika. 26, Zap. nauchn. sem. POMI, 466, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2017 , 338 pp.
2016
27.
F. Götze, Yu. S. Eliseeva, A. Yu. Zaitsev, “Arak’s inequalities for concentration functions and the Littlewood–Offord problem”, Doklady Mathematics, 93:2 (2016), 202–206 , arXiv: 1512.02938
2018
28.
A. Yu. Zaitsev, “Araks inequalities for the generalized arithmetic progressions”, J. Math. Sci. (N. Y.), 220:6 (2018), 698–701http://rdcu.be/HuDm
2016
29.
Veroyatnost i statistika. 24, Zap. nauchn. sem. POMI, 454, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2016 , 315 pp.
30.
A. Yu. Zaitsev, “A bound for the maximal probability in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 219:5 (2016), 743–746
2015
31.
Veroyatnost i statistika. 22, Zap. nauchn. sem. POMI, 441, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2015 , 324 pp.
32.
Veroyatnost i statistika. 23, Zap. nauchn. sem. POMI, 442, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2015 , 183 pp.
Yu. S. Eliseeva, A. Yu. Zaitsev, “On the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 214:4 (2016), 467–473
2014
35.
Veroyatnost i statistika. 21, Posvyaschaetsya yubileyu Mikhaila Iosifovicha GORDINA, Zap. nauchn. sem. POMI, 431, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2014 , 258 pp.
2013
36.
A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors”, Russian Math. Surveys, 68:4 (2013), 721–761
2015
37.
Yu. S. Eliseeva, F. Götze, A. Yu. Zaitsev, “Estimates for the concentration functions in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 206:2 (2015), 146–158 , arXiv: 1203.6763
2013
38.
Veroyatnost i statistika. 19, Zap. nauchn. sem. POMI, 412, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2013 , 278 pp.
39.
Veroyatnost i statistika. 20, Zap. nauchn. sem. POMI, 420, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2013 , 178 pp.
2014
40.
A. Yu. Zaitsev, “Approximation of convolutions by accompanying laws in the scheme of series”, J. Math. Sci. (N. Y.), 199:2 (2014), 162–167 , arXiv: 1312.5652
2013
41.
Yu. S. Eliseeva, A. Yu. Zaitsev, “Estimates of the concentration functions of weighted sums of independent random variables”, Theory Probab. Appl., 57:4 (2013), 670–678 , arXiv: 1203.5520
2012
42.
Veroyatnost i statistika. 18, Posvyaschaetsya yubileyu Ildara Abdullovicha IBRAGIMOVA, Zap. nauchn. sem. POMI, 408, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2012 , 330 pp.
2011
43.
F. Götze, A. Yu. Zaitsev, “Estimates for the rate of strong approximation in Hilbert space”, Siberian Math. J., 52:4 (2011), 628–638 , arXiv: 1203.5695
2013
44.
A. Yu. Zaitsev, “Optimal estimates for the rate of strong Gaussian approximation in the infinite dimensional invariance principle”, J. Math. Sci. (N. Y.), 188:6 (2013), 689–693
2011
45.
A. Yu. Zaitsev, “On the rate of decay of concentration functions of n-fold convolutions of probability distributions”, Vestnik St. Petersburg University: Mathematics, 44:2 (2011), 110–114
46.
Veroyatnost i statistika. 17, Posvyaschaetsya yubileyu Valentina Nikolaevicha SOLEVA, Zap. nauchn. sem. POMI, 396, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2011 , 262 pp.
47.
F. Götze, A. Yu. Zaitsev, “Uniform rates of approximation by short asymptotic expansions in the CLT for quadratic forms”, J. Math. Sci. (N. Y.), 176:2 (2011), 162–189
2010
48.
Veroyatnost i statistika. 16, Zap. nauchn. sem. POMI, 384, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2010 , 315 pp.
49.
F. Götze, A. Yu. Zaitsev, “Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments”, J. Math. Sci. (N. Y.), 167:4 (2010), 495–500
2009
50.
A. Yu. Zaitsev, “Rate of strong Gaussian approximation for the sums of i.i.d. multidimensional random vectors”, J. Math. Sci. (N. Y.), 163:4 (2009), 399–408
51.
Veroyatnost i statistika. 15, Zap. nauchn. sem. POMI, 368, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, POMI, SPb., 2009 , 288 pp.
2010
52.
A. N. Borodin, A. Yu. Zaitsev, I. A. Ibragimov, M. A. Lifshits, “From editors”, J. Math. Sci. (N. Y.), 167:4 (2010), 435
2009
53.
F. Götze, A. Yu. Zaitsev, “Bounds for the Rate of Strong Approximation in the Multidimensional Invariance Principle”, Theory Probab. Appl., 53:1 (2009), 59–80
2008
54.
A. Yu. Zaitsev, “Estimates for the rate of strong Gaussian approximation for the sums of i.i.d. multidimensional random vectors”, J. Math. Sci. (N. Y.), 152:6 (2008), 875–884
2007
55.
A. Yu. Zaitsev, “Estimates for the rate of strong approximation in the multidimensional invariance principle”, J. Math. Sci. (N. Y.), 145:2 (2007), 4856–4865
2005
56.
A. Yu. Zaitsev, “Moderate deviations for the $L_1$-norm of kernel density estimators”, Vestnik St. Petersburg University: Mathematics, 38:4 (2005), 15–24
2006
57.
F. Götze, A. Yu. Zaitsev, “Approximation of convolutions by accompanying laws without centering”, J. Math. Sci. (N. Y.), 137:1 (2006), 4510–4515
2005
58.
A. Yu. Zaitsev, “On approximation of the sample by a Poisson point process”, J. Math. Sci. (N. Y.), 128:1 (2005), 2556–2563
2003
59.
E. Giné, D.M. Mason, A.Yu. Zaitsev, “The $L_1$-norm density estimator process”, Annals of Probability, 31:2 (2003), 719–768
A. Yu. Zaitsev, “Estimates of the rate of approximation in the Central Limit Theorem for $L_1$-norm of kernel density estimators”, High Dimensional Probability. III, Progress in Probability, 55, eds. E. Giné, M. Marcus, J.A. Wellner, Birkhäuser, Basel, 2003, 255–292http://arxiv.org/pdf/1402.1417v1.pdf
A. Yu. Zaitsev, “Estimates for the strong approximation in multidimensional Central Limit Theorem”, Proceedings of the International Congress of Mathematicians (Bejing 2002), Invited Lectures, III, eds. Li, Ta Tsien et al., Higher Ed. Press, Bejing, 2002, 107–116http://arxiv.org/abs/math/0304373
62.
A. Yu. Zaitsev, “Estimates of the rate of approximation in a de-Poissonization lemma”, En l'honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov, Annales de l'Institut Henri Poincaré. Probabilités et Statistiques, 38:6 (2002), 1071–1086
A. Yu. Zaitsev, “Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. III”, Theory Probab. Appl., 46:4 (2002), 676–698
64.
A. Yu. Zaitsev, “Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. II”, Theory Probab. Appl., 46:3 (2002), 490–514
2001
65.
F. Götze, A. Yu. Zaitsev, “Multidimensional Hungarian construction for vectors with almost Gaussian smooth distributions”, Asymptotic Methods in Probability and Statistics with Applications, eds. N. Balakrishnan et al., Birkhäuser, 2001, 101–132http://arxiv.org/pdf/1402.1420v1.pdf
A. Yu. Zaitsev, “On the strong Gaussian approximation in multidimensional case”, Annales de l'I.S.U.P. Publications de l'Institut de Statistique de l'Université de Paris, 45:2–3 (2001), 3–7
67.
A. Yu. Zaitsev, “Multidimensional Version of a Result of Sakhanenko in the Invariance Principle for Vectors with Finite Exponential Moments. I”, Theory Probab. Appl., 45:4 (2001), 624–641
2000
68.
F. Götze, A. Yu. Zaitsev, “A multiplicative inequality for concentration functions of $n$-fold convolutions”, High dimensional probability II, Progress in Probability, 47, eds. E. Giné, D. Mason, J. Wellner, Birkhäuser, Boston-Basel-Berlin, 2000, 39–47http://arxiv.org/pdf/1402.6966v1.pdf
F. Götze, A. Yu. Zaitsev, “Estimates for the rapid decay of concentration functions of $n$-fold convolutions”, Journal of Theoretical Probability, 11:3 (1998), 715–731
A. Yu. Zaitsev, “Multidimensional version of the results of Komlos, Major and Tusnady for vectors with finite exponential moments”, ESAIM: Probability and Statistics, 2 (1998), 41–108
V. Bentkus, F. Götze, A. Yu. Zaitsev, “Approximation of quadratic forms of independent random vectors by accompanying laws”, Theory Probab. Appl., 42:2 (1998), 189–212
1997
72.
A. Yu. Zaitsev, “Multidimensional variant of the Komlós, Major and Tusnády results for vectors with finite exponential moments”, Dokl. Math. 56, No. 3, 935-937, 56:3 (1997), 935–937
1996
73.
A. Yu. Zaitsev, “Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle”, Siberian Math. J., 37:4 (1996), 706–729
1999
74.
A. Yu. Zaitsev, “Approximation of convolutions by accompanying laws under the existence of moment of low orders”, J. Math. Sci. (New York), 93:3 (1999), 336–340
1996
75.
A. Yu. Zaitsev, “An improvement of U. Einmahl estimate in the multidimensional invariance principle”, Probability Theory and Mathematical Statistics. Proceedings of the Euler Institute Seminars Deducated to the Memory of Kolmogorov. St. Petersburg, 1993, I., eds. I. Ibragimov, A. Zaitsev, Gordon & Breach, 1996, 109-116
76.
Sbornik statei, Probability theory and mathematical statistics. Lectures presented at the semester held in St. Petersburg, Russia, March 2–April 23, 1993., eds. I. A. Ibragimov, A.Yu. Zaitsev, Gordon and Breach Publishers, Amsterdam, 1996 , 321 pp.
1995
77.
A. Yu. Zaitsev, Multidimensional version of the results of Komlós, Major and Tusnády for vectors with finite exponential moments, Universität Bielefeld, Bielefeld, 1995 , 115 pp., Working papers by Bielefeld University. Series “Sonderforschungsbereich 343”, no. 95-055 http://www.mathematik.uni-bielefeld.de/sfb343/preprints/index95.html
1998
78.
A. Yu. Zaitsev, “Nonstability of the inversion of the Radon transform”, J. Math. Sci., New York, 88:1 (1998), 53–58
1994
79.
Problemy teorii veroyatnostnykh raspredelenii. 13, Zap. nauchn. sem. POMI, 216, ed. A. Yu. Zaitsev, V. N. Sudakov, Nauka, SPb., 1994 , 166 pp.
1993
80.
A. Yu. Zaitsev, “Approksimatsiya svertok soprovozhdayuschimi zakonami pri momentnykh ogranicheniyakh”, Koltsa i moduli. Predelnye teoremy teorii veroyatnostei, 3, SPbGU, 1993, 152–158
1995
81.
A. Yu. Zaitsev, “Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions”, J. Math. Sci. (N. Y.), 75:5 (1995), 1922–1930
1991
82.
A. Yu. Zaitsev, “An example of a distribution whose set of $n$-fold convolutions is uniformly separated from the set of infinitely divisible laws in the sense of the variation distance”, Theory Probab. Appl., 36:2 (1991), 419–425
83.
A. Yu. Zaitsev, “Approksimatsiya svertok bezgranichno delimymi zakonami pri momentnykh ogranicheniyakh.” (Mezhdunarodnyi Suzdalskii seminar 1991 goda po problemam ustoichivosti stokhasticheskikh modelei), Teoriya veroyatnostei i ee primeneniya, 36, vyp. 4, eds. V. M. Zolotarev, Nauka, 1991, 787–788
1994
84.
A. Yu. Zaitsev, “Certain class of nonuniform estimates in multidimensional limit theorems”, J. Math. Sci. (N. Y.), 68:4 (1994), 459–468
1990
85.
A. Yu. Zaitsev, “On the approximation of convolutions by infinitely divisible distributions”, Probability theory and mathematical statistics, Proc. 5th Vilnius Conf. (Vilnius, 1989), II, Mokslas, Vilnius, 1990, 602–608
1989
86.
A. Yu. Zaitsev, “Multivariate Version of the Second Kolmogorov's Uniform Limit Theorem”, Theory Probab. Appl., 34:1 (1989), 108–128
1992
87.
A. Yu. Zaitsev, “On the approximation of convolutions of multi-dimensional symmetric distributions by accompaning laws”, J. Soviet Math., 61:1 (1992), 1859–1872
1988
88.
A. Yu. Zaitsev, “Estimates for the closeness of successive convolutions of multidimensional symmetric distributions”, Probability Theory and Related Fields, 79:2 (1988), 175–200
A. Yu. Zaitsev, “On the approximation of distributions of sums of independent random vectors.”, Annales Academiae Scientiarum Fennicae. Ser. A. I. Mathematika, 13 (1988), 277–282
90.
A. Yu. Zaitsev, “O svyazi mezhdu dvumya klassami veroyatnostnykh raspredelenii”, Koltsa i moduli. Predelnye teoremy teorii veroyatnostei, 2, Izd-vo LGU, Leningrad, 1988, 153–158
1987
91.
A. Yu. Zaitsev, “On the Uniform Approximation of Distributions of Sums of Independent Random Variables”, Theory Probab. Appl., 32:1 (1987), 40–47
92.
A. Yu. Zaitsev, “On the Gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions”, Probability Theory and Related Fields, 74:4 (1987), 535–566
A. Yu. Zaitsev, “On the uniform approximation of distribution functions of sums of independent non-identically distributed random variables”, Probability theory and applications, Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986, Vol. 1, VNU Science Press, 1987, 697–700
1988
94.
A. Yu. Zaitsev, “Multidimensional generalized method of triangular functions”, J. Soviet Math., 43:6 (1988), 2797–2810
1987
95.
A. Yu. Zaitsev, “Letter to editors”, Theory Probab. Appl., 32:4 (1987), 750
A. Yu. Zaitsev, “On the logarithmic factor in the smoothing inequalities for Levi and Levi–Prohorov distances”, Theory Probab. Appl., 31:4 (1987), 691–693
1986
98.
A. Yu. Zaitsev, “Estimates for the Lévy-Prokhorov distance in the multivariate central limit theorem for random vectors with finite exponential moments”, Theory Probab. Appl., 31:2 (1986), 203–220
1987
99.
A. Yu. Zaitsev, “Approximation of convolutions of multidimensional distributions”, J. Soviet Math., 36:4 (1987), 482–489
1986
100.
A. Yu. Zaitsev, “Some remarks regarding the approximation of distributions of sums of independent terms”, J. Soviet Math., 33:1 (1986), 728–733
1984
101.
A. Yu. Zaitsev, “On approximation by Gaussian distributions under realization of multidimensional analogues of Bernstein conditions”, Sov. Math., Dokl., 29 (1984), 624–626
1985
102.
A. Yu. Zaitsev, “On the approximation of distributions of sums of independent random vectors”, Theory Probab. Appl., 29:4 (1985), 847–848
1984
103.
A. Yu. Zaitsev, “Approximation by Gaussian distributions with satisfaction of multidimensional analogues of Bernstein's conditions”, Dokl. Akad. Nauk SSSR, 276:5 (1984), 1046–1048
1985
104.
A. Yu. Zaitsev, “Letter to the editors”, Theory Probab. Appl., 29:1 (1985), 199
1984
105.
A. Yu. Zaitsev, “On the accuracy of approximation of distributions of sums of independent random variables – which are nonzero with a small probability – by means of accompanying laws”, Theory Probab. Appl., 28:4 (1984), 657–669
106.
A. Yu. Zaǐtsev, T. V. Arak, “On the rate of convergence in the second Kolmogorov's uniform limit theorem”, Theory Probab. Appl., 28:2 (1984), 351–374
107.
A. Yu. Zaitsev, “Approximation by infinitely divisible distributions in the multidimensional case”, J. Soviet Math., 27:6 (1984), 3227-3237
108.
A. Yu. Zaitsev, “Estimates for the closeness of successive convolutions of symmetric distributions”, Summary of Reports Presented at Sessions of the Probability and Mathematical Statistics Seminar at the Leningrad Section of the Mathematical Institute of the USSR Academy of Sciences 1981, Theory Probab. Appl., 28:1 (1984), 194–195
109.
A. Yu. Zaitsev, “Estimates for the Levy-Prokhorov distance in terms of characteristic functions and some of their applications”, J. Soviet Math., 27:5 (1984), 3070–3083
110.
A. Yu. Zaitsev, “Use of the concentration function for estimating the uniform distance.”, J. Soviet Math., 27:5 (1984), 3059–3070
1982
111.
T. V. Arak, A. Yu. Zaitsev, “An estimate of the rate of convergence in the second uniform limit theorem of Kolmogorov”, Sov. Math., Dokl., 26 (1982), 509–513
1981
112.
A. Yu. Zaitsev, “Some properties of $n$-fold convolutions of distributions”, Theory Probab. Appl., 26:1 (1981), 148–152
113.
A. Yu. Zaitsev, “On approximating the distributions of sums of independent terms by infinitely divisible laws in the Levy metric”, Sov. Math., Dokl., 24 (1981), 382–385
114.
A. Yu. Zaitsev, “Some estimates for the distributions of sums of independent random variables and vectors”, Theory Probab. Appl., 26:1 (1981), 188–188
115.
A. Yu. Zaitsev, “On approximating distributions of sums of independent random vectors by infinitely divisible laws”, Theory Probab. Appl., 26:3 (1981), 621–622
1984
116.
A. Yu. Zaitsev, “The estimation of proximity of distribution of sequential sums of independent identically distributed random vectors”, J. Soviet Math., 24:5 (1984), 536–539
1980
117.
A. Yu. Zaitsev, “On the approach of distributions of sums of independent nonidentically distributed random variables with accompanying laws”, Sov. Math., Dokl., 21 (1980), 732–733.
118.
A. Yu. Zaitsev, “On the approximation of distributions of sums of independent random vectors by infinitely divisible distributions”, Sov. Math., Dokl., 22 (1980), 67–69
119.
A. Yu. Zaitsev, Otsenki tochnosti approksimatsii raspredelenii summ nezavisimykh sluchainykh vektorov bezgranichno delimymi raspredeleniyami, “Diss. … kand. fiz.-matem. nauk”, Leningradskoe otdelenie Matematicheskogo instituta im. V.A.Steklova AN SSSR, Leningrad, 1980 , 129 pp.
Probability and statistics. Part 34, Zap. Nauchn. Sem. POMI, 525, ed. A. Yu. Zaitsev, M. A. Lifshits, 2023, 189 с. http://mi.mathnet.ru/book1981
Probability and statistics. Part 35, Zap. Nauchn. Sem. POMI, 526, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2023, 211 с. http://mi.mathnet.ru/book1982
Probability and statistics. Part 32, Zap. Nauchn. Sem. POMI, 510, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2022, 287 с. http://mi.mathnet.ru/book1937
Probability and statistics. Part 33, Zap. Nauchn. Sem. POMI, 515, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2022, 238 с. http://mi.mathnet.ru/book1942
Probability and statistics. Part 30, Zap. Nauchn. Sem. POMI, 501, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2021, 343 с. http://mi.mathnet.ru/book1886
Probability and statistics. Part 31, Zap. Nauchn. Sem. POMI, 505, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2021, 330 с. http://mi.mathnet.ru/book1891
Probability and statistics. Part 29, Zap. Nauchn. Sem. POMI, 495, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2020, 322 с. http://mi.mathnet.ru/book1853
Probability and statistics. Part 28, Zap. Nauchn. Sem. POMI, 486, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2019, 309 с. http://mi.mathnet.ru/book1817
Probability and statistics. Part 27, Zap. Nauchn. Sem. POMI, 474, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2018, 247 с. http://mi.mathnet.ru/book1764
Probability and statistics. Part 25, Zap. Nauchn. Sem. POMI, 457, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2017, 322 с. http://mi.mathnet.ru/book1702
Probability and statistics. Part 26, Zap. Nauchn. Sem. POMI, 466, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2017, 338 с. http://mi.mathnet.ru/book1711
Probability and statistics. Part 24, Zap. Nauchn. Sem. POMI, 454, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2016, 315 с. http://mi.mathnet.ru/book1676
Probability and statistics. Part 22, Zap. Nauchn. Sem. POMI, 441, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2015, 324 с. http://mi.mathnet.ru/book1611
Probability and statistics. Part 23, Zap. Nauchn. Sem. POMI, 442, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2015, 183 с. http://mi.mathnet.ru/book1612
Probability and statistics. Part 21, Zap. Nauchn. Sem. POMI, 431, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2014, 258 с. http://mi.mathnet.ru/book1572
Probability and statistics. Part 19, Zap. Nauchn. Sem. POMI, 412, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2013, 278 с. http://mi.mathnet.ru/book1486
Probability and statistics. Part 20, Zap. Nauchn. Sem. POMI, 420, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2013, 178 с. http://mi.mathnet.ru/book1503
Probability and statistics. Part 18, Zap. Nauchn. Sem. POMI, 408, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2012, 330 с. http://mi.mathnet.ru/book1467
Probability and statistics. Part 17, Zap. Nauchn. Sem. POMI, 396, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2011, 262 с. http://mi.mathnet.ru/book1369
Probability and statistics. Part 16, Zap. Nauchn. Sem. POMI, 384, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2010, 315 с. http://mi.mathnet.ru/book1333
Probability and statistics. Part 15, Zap. Nauchn. Sem. POMI, 368, ed. A. N. Borodin, A. Yu. Zaitsev, M. A. Lifshits, 2009, 288 с. http://mi.mathnet.ru/book1224
Problems of the theory of probability distributions. Part 13, Zap. Nauchn. Sem. POMI, 216, ed. A. Yu. Zaitsev, V. N. Sudakov, 1994, 166 с. http://mi.mathnet.ru/book918
T. V. Arak, A. Yu. Zaitsev, Uniform limit theorems for sums of independent random variables, Trudy Mat. Inst. Steklov., 174, ed. I. A. Ibragimov, 1986, 217 с. http://mi.mathnet.ru/book1163
2024
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