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| PEOPLE |
| Zaitsev Andrei Yurevich |
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| Senior Researcher |
| Doctor of physico-mathematical sciences (1989) |
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.
ISTINA
