The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let …
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).
Let $\{ \xi _n \} $ be a sequence of individually bounded independent random variables: \[ \xi _n = O(\varphi (n)) \] The necessary and sufficient conditions for the validity …
Let $\{ \xi _n \} $ be a sequence of individually bounded independent random variables: \[ \xi _n = O(\varphi (n)) \] The necessary and sufficient conditions for the validity of the strong law of large numbers can be expressed in terms of variances ${\bf D}\xi _n $\[ \varphi (n) = {n / {\log \log n}} \] and cannot be expressed in these terms (and, possibly, not even in terms of any finite number of moments) if\[ \varphi (n) = \left( {{n / {\log \log n}}} \right) \to \infty ,\quad n \to \infty \]In the latter case the “best” sufficient conditions are given.
The paper gives some new conditions for the strong law of large numbers (s. 1. 1. n.) to be applied to a sequence of independent symmetrical random variables (r. v.). …
The paper gives some new conditions for the strong law of large numbers (s. 1. 1. n.) to be applied to a sequence of independent symmetrical random variables (r. v.). The principal result states that the s.1.1. n. for a sequence of “adjoined” infinitely divisible r. v. implies the s. 1. 1. n. for the given sequence of r. v. This result leads to “satisfactory” sufficient conditions for s. l. l. n. In special cases some of these conditions become the necessary ones.
Let $\{ P_\alpha \}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^ * $ conjugate to a countably-Hilbert space Y) …
Let $\{ P_\alpha \}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^ * $ conjugate to a countably-Hilbert space Y) and let $\{ \chi _\alpha \}$ be the family of corresponding characteristic functionals. We investigate whether or not there exists a locally convex topology $\mathcal{T}$ with the following property: The relative compactness of $\{ {P_\alpha } \}$ is equivalent to uniform (with respect to $\alpha $) continuity of $\{ \chi _\alpha \}$. We prove that there is no such topology except for the case of the countably-Hilbert nuclear space Y.
Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y - a|^2 $, where Y is a Gaussian random element in a Hilbert space with zero …
Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y - a|^2 $, where Y is a Gaussian random element in a Hilbert space with zero mean. The estimates are sharp in the sense that starting from large enough u the ratio of upper bound to lower bound equals 8 and does not depend on any parameters of a distribution of $|Y - a|^2 $. The estimates imply two-sided bounds for probabilities ${\bf P}(|Y - a| > r)$.
Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable …
Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable Y=Q(X)is of order O(1/|t|1/k) as $|t|\to\infty$ if PX is sufficiently smooth. In addition, for every $\varepsilon \:1/k > \varepsilon > 0$ there exists a singular distribution PX such that every convolution $P^{n\star}_X$ is also singular while the characteristic function of Y is of order $O(1/|t|^{1/k-\varepsilon})$. While the characteristic function of X is small when "averaged," the characteristic function of the polynomial transformation Y of X is uniformly small.
This paper improves the inequalities for deviation of a density of the fractional part of the s‐dimensional Gaussian random vector from 1 (uniform distribution density) obtained in part I [Theory …
This paper improves the inequalities for deviation of a density of the fractional part of the s‐dimensional Gaussian random vector from 1 (uniform distribution density) obtained in part I [Theory Probab. Appl., 48 (2004), pp. 355–359].
The full availability group of trunks with an arbitrary distribution of the inter-arrival times and a negative exponential holding time distribution is considered. The possibility of evaluating the probability of …
The full availability group of trunks with an arbitrary distribution of the inter-arrival times and a negative exponential holding time distribution is considered. The possibility of evaluating the probability of loss of calls by Erlang’s formula, as a first approximation, is established under very general conditions on streams with high intensity, when not very strict requirements on the quality of service are made.
Let Y be a random variable with a completely asymmetric stable law and parameter $\alpha$. This paper proves that a probability distribution of a fractional part of the logarithm of …
Let Y be a random variable with a completely asymmetric stable law and parameter $\alpha$. This paper proves that a probability distribution of a fractional part of the logarithm of Y with respect to any base larger than 1 converges to the uniform distribution on the interval $[0,1]$ for $\alpha\to 0$. This implies that the distribution of the first significant digit of Y for small $\alpha$ can be approximately described by the Benford law.
This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula …
This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula and some facts from the theory of representation of integers by square forms. The main attention of this part of the paper is devoted to the case of small values of s. The case of large values of s will be consider additionally.
This paper establishes estimates from above for a variance of arbitrary polynomials in binomially distributed random variables similar to the Chernoff inequality.Keywordsbinomial distributionmoment inequalitiesKrawtchouk polynomialsfactorial-power formalismcombinatorial identities
This paper establishes estimates from above for a variance of arbitrary polynomials in binomially distributed random variables similar to the Chernoff inequality.Keywordsbinomial distributionmoment inequalitiesKrawtchouk polynomialsfactorial-power formalismcombinatorial identities
This paper notes a connection among a wide class of the so-called HF-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of …
This paper notes a connection among a wide class of the so-called HF-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of examples of random variables having gamma-distribution. Let Y be a random variable having gamma-distribution with parameter $\alpha$. It is proved that the distribution of a fractional part of the logarithm of Y with respect to any base larger than 1 converges to the uniform distribution on the interval [0,1] for $\alpha$ to 0. This implies that the probability distribution of the first significant digit of Y for small $\alpha$ can be approximately described by Benford's law. The order of the approximation is illustrated by tables.
Analogues of the isoperimetric Chernoff inequality for a negative binomial distribution are obtained.Keywordsnegative binomial distributionPascal distributionmoment inequalitiesfactorial-power binomialspolynomials orthogonal with respect to a negative binomial distribution (Pascal's)
Analogues of the isoperimetric Chernoff inequality for a negative binomial distribution are obtained.Keywordsnegative binomial distributionPascal distributionmoment inequalitiesfactorial-power binomialspolynomials orthogonal with respect to a negative binomial distribution (Pascal's)
February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As …
February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As his specialization field he chose probability theory and began his studies under the supervision of Eugene Borisovich Dynkin. After graduating from the university he was recommended to graduate studies, where his advisor was Andrei Nikolaevich Kolmogorov. Other distinguished mathematicians also have had a potent effect on Zolotarev's mathematical talent. Later, he mentions more than once not only his teachers E. B. Dynkin and A. N. Kolmogorov, but also B. V. Gnedenko and Yu. V. Linnik. In his graduate studies, Vladimir begins to study the properties of stable distributions. He continues to be interested in this theme even today. At first he was dealing with the stable distributions in the scheme of summation of independent identically distributed random variables. Later, he extended the concept of a stable law to the schemes of maximum and multiplication of random variables. His studies of random variables are summarized in the monograph [1]. It has gained widespread recognition and has been translated into English. As it saw the light, the nomenclature, such as Zolotarev's theorem, Zolotarev's formula, and Zolotarev's transformation, became quite conventional. Contemporaneously with studying the properties of stable laws, Zolotarev began to work in the field of limit theorems for sums of independent random variables. His results obtained in this direction can be conventionally divided into three groups. The first group concerns refining classical theorems and convergence rate estimates. As an example, we mention the convergence rate estimate in the central limit theorem in terms of pseudomoments, which for some time remained the best estimate known. But it is even more important that the method of pseudomoments used by Zolotarev to obtain this estimate led to a new structure of convergence rate estimates in limit theorems for sums of independent random variables. His disciples have made this a powerful tool which provides us with even more sharp estimates. The principal novelty resides in that a pseudomoment allows us to recognize the contribution of every individual summand to the whole estimate. The notions of a center and a scatter (spread) introduced by Zolotarev turn out to be indispensable in establishing the weak compactness of sequences of sums of independent random variables which have no finite moments and allowed us to extend limit theorems, previously known to be valid, under quite strict moment conditions to such variables. Zolotarev's second group of results gathers together the results whose essence reduces to weakening the condition of independence of random variables so that the limit theorems remain true. The third group concerns the so-called nonclassical scheme of summation. The cornerstone of this scheme consists of breaking the habitual pattern, where an individual summand does not influence the form of the limit distribution. In the nonclassical summation theory an individual summand is allowed to play a discernible part. It is fair to say that Vladimir Zolotarev is one of the fathers of this direction in the theory of summation of random variables. He has generalized the results of his predecessors, P. Lévy and Yu. V. Linnik, who on the heuristic level of reasoning pointed to the possibility of a new approach to limit theorems for sums of independent random variables, and developed a self-sufficient theory of summation of random variables, now referred to as nonclassical. The theory of probability metrics built by Zolotarev underlies the novel viewpoint of limit theorems of probability theory as stability theorems. Zolotarev summarizes his studies in this field in the monograph [3], which immediately became a widely used source for new investigations. In 1997, a revised and enlarged version of this monograph was published in English [4]. Along with sums of random variables, Vladimir Zolotarev deals with more general asymptotic schemes. In particular, together with his Hungarian colleague L. Szeidl, Zolotarev has made an essential contribution to the asymptotic theory of random polynomials. Their joint results are presented in the monograph [6]. Investigating the asymptotic properties of sums of independent random variables, Vladimir Zolotarev came to related studies in the theory of stochastic processes and queueing theory. He and his colleagues analyze the stability and continuity of queueing systems with the use of related concepts of the theory of summation of random variables. The quantitative characteristics of these properties suggested by Zolotarev led to a deeper understanding of these phenomena. Speaking of the results obtained by Vladimir Zolotarev, we must say that he masterfully manages to use both modern and classical technique of analysis. Zolotarev enriches the store of probability theory by a series of new analytic methods and tools; it suffices to mention the thorough study of theorems of classical analysis, harmonic analysis, and the theory of special functions. Zolotarev also clears up how to use the Mellin–Stieltjes transforms in probability theory. Modern probability theory plays an important role in many natural sciences. For example, modern statistical physics becomes infeasible without using the fundamental concepts of probability theory. Vladimir Zolotarev has given many talks on how to apply probability theory to explaining various phenomena in physics, genetics, and geology. Together with V. V. Uchaikin he wrote the monograph [5], where one finds plenty of applications of stable laws explaining a series of physical and economical phenomena. In 1956, A. N. Kolmogorov founded the journal Theory of Probability and Its Applications. From the very beginning, Vladimir Zolotarev took an active part in the operation of the journal. From the day of foundation to 1966 he was the executive secretary; from 1967 to 1990 the deputy editor-in-chief; and from 1991 to the present he is a member of the advisory board. Vladimir Zolotarev founded the series of issues Stability Problems for Stochastic Models of the Journal of Mathematical Sciences, being its editor-in-chief. He also heads the editorial board of the series of monographs Modern Probability and Statistics published by VSP/Brill (The Netherlands). Vladimir Zolotarev is the initiator and continuous leader of the international scientific seminar on stability problems of stochastic models, widely known as the Zolotarev seminar. From 1973, twenty-six sessions of this seminar have been held in various countries. These sessions take place almost every year and attract about a hundred participants from diverse countries. Vladimir Zolotarev does much to popularize science. In the brochure [2] he presented an exciting story about stable laws and their applications. At the same time he created two educational films about fundamental limit theorems of probability theory. A crowd of youthful science enthusiasts is always gathering around him. Many of his pupils have become reputable specialists in various fields of mathematics. The scientific school of Vladimir Zolotarev is highly regarded. Friends and colleagues of Zolotarev love and appreciate him because he is a man of principle, well-wishing and tenderhearted, ready to stand by and assist. Devoting heart and soul to science, he demands the same from his colleagues and students. The scientific society highly appreciates the contributions of Vladimir Zolotarev. In 1971, for a series of works on limit theorems for sums of independent random variables, the Presidium of the Academy of Sciences of the USSR awarded him the Markov prize. We wish Vladimir Zolotarev many years of good health and continued activity.
A celebration of the 75th anniversary of the publication of Foundations of the Theory of Probability by A. N. Kolmogorov.
A celebration of the 75th anniversary of the publication of Foundations of the Theory of Probability by A. N. Kolmogorov.
Details of several international conferences which were held in 2012 to commemorate the 100th anniversary of the birth of world renowned scientist and outstanding mathematician Boris Vladimirovich Gnedenko.
Details of several international conferences which were held in 2012 to commemorate the 100th anniversary of the birth of world renowned scientist and outstanding mathematician Boris Vladimirovich Gnedenko.
We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal …
We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal random variables are normalized sums of independent identically distributed summands with a “regular” distribution. Possible generalizations are considered. The estimates extend the results of previous studies, where for the distributions of the summands, the presence of either a discrete or an absolutely continuous component was required. The proofs of the bounds are based on the stochastic generalization of the I. M. Vinogradov mean value theorem, which is also obtained in the present paper.
We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal …
We obtain upper bounds for the absolute values of the characteristic functions of the $k$th powers of asymptotically normal random variables. Estimates are proved for the case when asymptotically normal random variables are normalized sums of independent identically distributed summands with a “regular” distribution. Possible generalizations are considered. The estimates extend the results of previous studies, where for the distributions of the summands, the presence of either a discrete or an absolutely continuous component was required. The proofs of the bounds are based on the stochastic generalization of the I. M. Vinogradov mean value theorem, which is also obtained in the present paper.
Details of several international conferences which were held in 2012 to commemorate the 100th anniversary of the birth of world renowned scientist and outstanding mathematician Boris Vladimirovich Gnedenko.
Details of several international conferences which were held in 2012 to commemorate the 100th anniversary of the birth of world renowned scientist and outstanding mathematician Boris Vladimirovich Gnedenko.
A celebration of the 75th anniversary of the publication of Foundations of the Theory of Probability by A. N. Kolmogorov.
A celebration of the 75th anniversary of the publication of Foundations of the Theory of Probability by A. N. Kolmogorov.
February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As …
February 27, 2007, was the 75th birthday of the eminent mathematician, Professor Vladimir Mikhailovich Zolotarev. In 1949, Vladimir entered the faculty of mechanics and mathematics of Moscow State University. As his specialization field he chose probability theory and began his studies under the supervision of Eugene Borisovich Dynkin. After graduating from the university he was recommended to graduate studies, where his advisor was Andrei Nikolaevich Kolmogorov. Other distinguished mathematicians also have had a potent effect on Zolotarev's mathematical talent. Later, he mentions more than once not only his teachers E. B. Dynkin and A. N. Kolmogorov, but also B. V. Gnedenko and Yu. V. Linnik. In his graduate studies, Vladimir begins to study the properties of stable distributions. He continues to be interested in this theme even today. At first he was dealing with the stable distributions in the scheme of summation of independent identically distributed random variables. Later, he extended the concept of a stable law to the schemes of maximum and multiplication of random variables. His studies of random variables are summarized in the monograph [1]. It has gained widespread recognition and has been translated into English. As it saw the light, the nomenclature, such as Zolotarev's theorem, Zolotarev's formula, and Zolotarev's transformation, became quite conventional. Contemporaneously with studying the properties of stable laws, Zolotarev began to work in the field of limit theorems for sums of independent random variables. His results obtained in this direction can be conventionally divided into three groups. The first group concerns refining classical theorems and convergence rate estimates. As an example, we mention the convergence rate estimate in the central limit theorem in terms of pseudomoments, which for some time remained the best estimate known. But it is even more important that the method of pseudomoments used by Zolotarev to obtain this estimate led to a new structure of convergence rate estimates in limit theorems for sums of independent random variables. His disciples have made this a powerful tool which provides us with even more sharp estimates. The principal novelty resides in that a pseudomoment allows us to recognize the contribution of every individual summand to the whole estimate. The notions of a center and a scatter (spread) introduced by Zolotarev turn out to be indispensable in establishing the weak compactness of sequences of sums of independent random variables which have no finite moments and allowed us to extend limit theorems, previously known to be valid, under quite strict moment conditions to such variables. Zolotarev's second group of results gathers together the results whose essence reduces to weakening the condition of independence of random variables so that the limit theorems remain true. The third group concerns the so-called nonclassical scheme of summation. The cornerstone of this scheme consists of breaking the habitual pattern, where an individual summand does not influence the form of the limit distribution. In the nonclassical summation theory an individual summand is allowed to play a discernible part. It is fair to say that Vladimir Zolotarev is one of the fathers of this direction in the theory of summation of random variables. He has generalized the results of his predecessors, P. Lévy and Yu. V. Linnik, who on the heuristic level of reasoning pointed to the possibility of a new approach to limit theorems for sums of independent random variables, and developed a self-sufficient theory of summation of random variables, now referred to as nonclassical. The theory of probability metrics built by Zolotarev underlies the novel viewpoint of limit theorems of probability theory as stability theorems. Zolotarev summarizes his studies in this field in the monograph [3], which immediately became a widely used source for new investigations. In 1997, a revised and enlarged version of this monograph was published in English [4]. Along with sums of random variables, Vladimir Zolotarev deals with more general asymptotic schemes. In particular, together with his Hungarian colleague L. Szeidl, Zolotarev has made an essential contribution to the asymptotic theory of random polynomials. Their joint results are presented in the monograph [6]. Investigating the asymptotic properties of sums of independent random variables, Vladimir Zolotarev came to related studies in the theory of stochastic processes and queueing theory. He and his colleagues analyze the stability and continuity of queueing systems with the use of related concepts of the theory of summation of random variables. The quantitative characteristics of these properties suggested by Zolotarev led to a deeper understanding of these phenomena. Speaking of the results obtained by Vladimir Zolotarev, we must say that he masterfully manages to use both modern and classical technique of analysis. Zolotarev enriches the store of probability theory by a series of new analytic methods and tools; it suffices to mention the thorough study of theorems of classical analysis, harmonic analysis, and the theory of special functions. Zolotarev also clears up how to use the Mellin–Stieltjes transforms in probability theory. Modern probability theory plays an important role in many natural sciences. For example, modern statistical physics becomes infeasible without using the fundamental concepts of probability theory. Vladimir Zolotarev has given many talks on how to apply probability theory to explaining various phenomena in physics, genetics, and geology. Together with V. V. Uchaikin he wrote the monograph [5], where one finds plenty of applications of stable laws explaining a series of physical and economical phenomena. In 1956, A. N. Kolmogorov founded the journal Theory of Probability and Its Applications. From the very beginning, Vladimir Zolotarev took an active part in the operation of the journal. From the day of foundation to 1966 he was the executive secretary; from 1967 to 1990 the deputy editor-in-chief; and from 1991 to the present he is a member of the advisory board. Vladimir Zolotarev founded the series of issues Stability Problems for Stochastic Models of the Journal of Mathematical Sciences, being its editor-in-chief. He also heads the editorial board of the series of monographs Modern Probability and Statistics published by VSP/Brill (The Netherlands). Vladimir Zolotarev is the initiator and continuous leader of the international scientific seminar on stability problems of stochastic models, widely known as the Zolotarev seminar. From 1973, twenty-six sessions of this seminar have been held in various countries. These sessions take place almost every year and attract about a hundred participants from diverse countries. Vladimir Zolotarev does much to popularize science. In the brochure [2] he presented an exciting story about stable laws and their applications. At the same time he created two educational films about fundamental limit theorems of probability theory. A crowd of youthful science enthusiasts is always gathering around him. Many of his pupils have become reputable specialists in various fields of mathematics. The scientific school of Vladimir Zolotarev is highly regarded. Friends and colleagues of Zolotarev love and appreciate him because he is a man of principle, well-wishing and tenderhearted, ready to stand by and assist. Devoting heart and soul to science, he demands the same from his colleagues and students. The scientific society highly appreciates the contributions of Vladimir Zolotarev. In 1971, for a series of works on limit theorems for sums of independent random variables, the Presidium of the Academy of Sciences of the USSR awarded him the Markov prize. We wish Vladimir Zolotarev many years of good health and continued activity.
This paper notes a connection among a wide class of the so-called HF-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of …
This paper notes a connection among a wide class of the so-called HF-random variables, approximately uniform distributions, and Benford's law. This connection is considered in detail with the help of examples of random variables having gamma-distribution. Let Y be a random variable having gamma-distribution with parameter $\alpha$. It is proved that the distribution of a fractional part of the logarithm of Y with respect to any base larger than 1 converges to the uniform distribution on the interval [0,1] for $\alpha$ to 0. This implies that the probability distribution of the first significant digit of Y for small $\alpha$ can be approximately described by Benford's law. The order of the approximation is illustrated by tables.
This paper improves the inequalities for deviation of a density of the fractional part of the s‐dimensional Gaussian random vector from 1 (uniform distribution density) obtained in part I [Theory …
This paper improves the inequalities for deviation of a density of the fractional part of the s‐dimensional Gaussian random vector from 1 (uniform distribution density) obtained in part I [Theory Probab. Appl., 48 (2004), pp. 355–359].
Analogues of the isoperimetric Chernoff inequality for a negative binomial distribution are obtained.Keywordsnegative binomial distributionPascal distributionmoment inequalitiesfactorial-power binomialspolynomials orthogonal with respect to a negative binomial distribution (Pascal's)
Analogues of the isoperimetric Chernoff inequality for a negative binomial distribution are obtained.Keywordsnegative binomial distributionPascal distributionmoment inequalitiesfactorial-power binomialspolynomials orthogonal with respect to a negative binomial distribution (Pascal's)
Let Y be a random variable with a completely asymmetric stable law and parameter $\alpha$. This paper proves that a probability distribution of a fractional part of the logarithm of …
Let Y be a random variable with a completely asymmetric stable law and parameter $\alpha$. This paper proves that a probability distribution of a fractional part of the logarithm of Y with respect to any base larger than 1 converges to the uniform distribution on the interval $[0,1]$ for $\alpha\to 0$. This implies that the distribution of the first significant digit of Y for small $\alpha$ can be approximately described by the Benford law.
This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula …
This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula and some facts from the theory of representation of integers by square forms. The main attention of this part of the paper is devoted to the case of small values of s. The case of large values of s will be consider additionally.
This paper establishes estimates from above for a variance of arbitrary polynomials in binomially distributed random variables similar to the Chernoff inequality.Keywordsbinomial distributionmoment inequalitiesKrawtchouk polynomialsfactorial-power formalismcombinatorial identities
This paper establishes estimates from above for a variance of arbitrary polynomials in binomially distributed random variables similar to the Chernoff inequality.Keywordsbinomial distributionmoment inequalitiesKrawtchouk polynomialsfactorial-power formalismcombinatorial identities
Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable …
Let X be a random variable with probability distribution PX concentrated on [-1,1]$ and let Q(x) be a polynomial of degree $k\ge 2$. The characteristic function of a random variable Y=Q(X)is of order O(1/|t|1/k) as $|t|\to\infty$ if PX is sufficiently smooth. In addition, for every $\varepsilon \:1/k > \varepsilon > 0$ there exists a singular distribution PX such that every convolution $P^{n\star}_X$ is also singular while the characteristic function of Y is of order $O(1/|t|^{1/k-\varepsilon})$. While the characteristic function of X is small when "averaged," the characteristic function of the polynomial transformation Y of X is uniformly small.
Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y - a|^2 $, where Y is a Gaussian random element in a Hilbert space with zero …
Two-sided bounds are constructed for a density function $p(u,a)$ of a random variable $|Y - a|^2 $, where Y is a Gaussian random element in a Hilbert space with zero mean. The estimates are sharp in the sense that starting from large enough u the ratio of upper bound to lower bound equals 8 and does not depend on any parameters of a distribution of $|Y - a|^2 $. The estimates imply two-sided bounds for probabilities ${\bf P}(|Y - a| > r)$.
The full availability group of trunks with an arbitrary distribution of the inter-arrival times and a negative exponential holding time distribution is considered. The possibility of evaluating the probability of …
The full availability group of trunks with an arbitrary distribution of the inter-arrival times and a negative exponential holding time distribution is considered. The possibility of evaluating the probability of loss of calls by Erlang’s formula, as a first approximation, is established under very general conditions on streams with high intensity, when not very strict requirements on the quality of service are made.
Let $\{ P_\alpha \}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^ * $ conjugate to a countably-Hilbert space Y) …
Let $\{ P_\alpha \}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^ * $ conjugate to a countably-Hilbert space Y) and let $\{ \chi _\alpha \}$ be the family of corresponding characteristic functionals. We investigate whether or not there exists a locally convex topology $\mathcal{T}$ with the following property: The relative compactness of $\{ {P_\alpha } \}$ is equivalent to uniform (with respect to $\alpha $) continuity of $\{ \chi _\alpha \}$. We prove that there is no such topology except for the case of the countably-Hilbert nuclear space Y.
Let $\{ \xi _n \} $ be a sequence of individually bounded independent random variables: \[ \xi _n = O(\varphi (n)) \] The necessary and sufficient conditions for the validity …
Let $\{ \xi _n \} $ be a sequence of individually bounded independent random variables: \[ \xi _n = O(\varphi (n)) \] The necessary and sufficient conditions for the validity of the strong law of large numbers can be expressed in terms of variances ${\bf D}\xi _n $\[ \varphi (n) = {n / {\log \log n}} \] and cannot be expressed in these terms (and, possibly, not even in terms of any finite number of moments) if\[ \varphi (n) = \left( {{n / {\log \log n}}} \right) \to \infty ,\quad n \to \infty \]In the latter case the “best” sufficient conditions are given.
The paper gives some new conditions for the strong law of large numbers (s. 1. 1. n.) to be applied to a sequence of independent symmetrical random variables (r. v.). …
The paper gives some new conditions for the strong law of large numbers (s. 1. 1. n.) to be applied to a sequence of independent symmetrical random variables (r. v.). The principal result states that the s.1.1. n. for a sequence of “adjoined” infinitely divisible r. v. implies the s. 1. 1. n. for the given sequence of r. v. This result leads to “satisfactory” sufficient conditions for s. l. l. n. In special cases some of these conditions become the necessary ones.
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let …
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).
Book 22 in the Princeton Mathematical Series. Originally published in 1960. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished …
Book 22 in the Princeton Mathematical Series. Originally published in 1960. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried …
When this book was written, methods of algebraic topology had caused revolutions in the world of pure algebra. To clarify the advances that had been made, Cartan and Eilenberg tried to unify the fields and to construct the framework of a fully fledged theory. The invasion of algebra had occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. This book presents a single homology (and also cohomology) theory that embodies all three; a large number of results is thus established in a general framework. Subsequently, each of the three theories is singled out by a suitable specialization, and its specific properties are studied. The starting point is the notion of a module over a ring. The primary operations are the tensor product of two modules and the groups of all homomorphisms of one module into another. From these, "higher order" derived of operations are obtained, which enjoy all the properties usually attributed to homology theories. This leads in a natural way to the study of "functors" and of their "derived functors." This mathematical masterpiece will appeal to all mathematicians working in algebraic topology.
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a …
This famous book was the first treatise on Lie groups in which a modern point of view was adopted systematically, namely, that a continuous group can be regarded as a global object. To develop this idea to its fullest extent, Chevalley incorporated a broad range of topics, such as the covering spaces of topological spaces, analytic manifolds, integration of complete systems of differential equations on a manifold, and the calculus of exterior differential forms. The book opens with a short description of the classical groups: unitary groups, orthogonal groups, symplectic groups, etc. These special groups are then used to illustrate the general properties of Lie groups, which are considered later. The general notion of a Lie group is defined and correlated with the algebraic notion of a Lie algebra; the subgroups, factor groups, and homomorphisms of Lie groups are studied by making use of the Lie algebra. The last chapter is concerned with the theory of compact groups, culminating in Peter-Weyl's theorem on the existence of representations. Given a compact group, it is shown how one can construct algebraically the corresponding Lie group with complex parameters which appears in the form of a certain algebraic variety (associated algebraic group). This construction is intimately related to the proof of the generalization given by Tannaka of Pontrjagin's duality theorem for Abelian groups. The continued importance of Lie groups in mathematics and theoretical physics make this an indispensable volume for researchers in both fields.
These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over …
These lectures, delivered by Professor Mumford at Harvard in 1963-1964, are devoted to a study of properties of families of algebraic curves, on a non-singular projective algebraic curve defined over an algebraically closed field of arbitrary characteristic. The methods and techniques of Grothendieck, which have so changed the character of algebraic geometry in recent years, are used systematically throughout. Thus the classical material is presented from a new viewpoint.
The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a …
The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected.
Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving …
Einstein's General Theory of Relativity leads to two remarkable predictions: first, that the ultimate destiny of many massive stars is to undergo gravitational collapse and to disappear from view, leaving behind a 'black hole' in space; and secondly, that there will exist singularities in space-time itself. These singularities are places where space-time begins or ends, and the presently known laws of physics break down. They will occur inside black holes, and in the past are what might be construed as the beginning of the universe. To show how these predictions arise, the authors discuss the General Theory of Relativity in the large. Starting with a precise formulation of the theory and an account of the necessary background of differential geometry, the significance of space-time curvature is discussed and the global properties of a number of exact solutions of Einstein's field equations are examined. The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions. A discussion of the Cauchy problem for General Relativity is also included in this 1973 book.
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let …
The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re $ be the c.s.m.s. and v a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of L-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]).