IEEE Transactions on Information Theory, Jul 1, 1978
An erroneow method for maximizing the projected divergence between two Gaussian multivariate hypo... more An erroneow method for maximizing the projected divergence between two Gaussian multivariate hypotheses appeared in a recent paper. The correct solution is given.
We revisit processes generated by iterated random functions driven by a stationary and ergodic se... more We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as stochastic approximation and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.
Starting in 2007, the MFO publishes a preprint series which mainly contains research results rela... more Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1-200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website www.mfo.de as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by email to or , respectively. The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX-XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation.
We study piecewise linear density estimators from the L 1 point of view: the frequency polygons i... more We study piecewise linear density estimators from the L 1 point of view: the frequency polygons investigated by SCOTT (1985) and JONES et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L 1 strongly consistent. We derive large deviation inequalities. For twice dierentiable densities with compact support their expected L 1 error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.
IEEE Transactions on Information Theory, Nov 1, 2005
We consider the rate of convergence of the expected distortion redundancy of empirically optimal ... more We consider the rate of convergence of the expected distortion redundancy of empirically optimal vector quantizers. Earlier results show that the mean-squared distortion of an empirically optimal quantizer designed from independent and identically distributed (i.i.d.) source samples converges uniformly to the optimum at a rate of (1), and that this rate is sharp in the minimax sense. We prove that for any fixed distribution supported on a given finite set the convergence rate is (1) (faster than the minimax lower bound), where the corresponding constant depends on the source distribution. For more general source distributions we provide conditions implying a little bit worse (log) rate of convergence. Although these conditions, in general, are hard to verify, we show that sources with continuous densities satisfying certain regularity properties (similar to the ones of Pollard that were used to prove a central limit theorem for the code points of the empirically optimal quantizers) are included in the scope of this result. In particular, scalar distributions with strictly log-concave densities with bounded support (such as the truncated Gaussian distribution) satisfy these conditions.
Multi-type inhomogeneous Galton-Watson process with immigration is investigated, where the offspr... more Multi-type inhomogeneous Galton-Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.
In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determ... more In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representation of the distribution function of the possible semistable limits. In the representation, each term corresponds to a given maximum, in particular this result explains that the semistable behavior is caused by the typical values of the maximum.
ABSTRACT This paper discusses some metrics of TCP based on real measurements over the Internet. W... more ABSTRACT This paper discusses some metrics of TCP based on real measurements over the Internet. We present algorithms to measure the congestion window related metrics and use these metrics to study the stationary behavior of TCP. Our statistical analysis shows that the distribution of the congestion window process in a stable period has a bell-like shape and can be approximated by a normal distribution
We investigate the performance of the constantly rebalanced portfolios, when the random vectors o... more We investigate the performance of the constantly rebalanced portfolios, when the random vectors of the market process {Xi} are independent, and each of them distributed as (X (1) , X (2) ,. .. , X (d) , 1), d ≥ 1, where X (1) , X (2) ,. .. , X (d) are nonnegative iid random variables. Under general conditions we show that the optimal strategy is the uniform: (1/d,. .. , 1/d, 0), at least for d large enough. In case of St. Petersburg components we compute the average growth rate and the optimal strategy for d = 1, 2. In order to make the problem non-trivial, a commission factor is introduced and tuned to result in zero growth rate on any individual St. Petersburg components. One of the interesting observations made is that a combination of two components of zero growth can result in a strictly positive growth. For d ≥ 3 we prove that the uniform strategy is the best, and we obtain tight asymptotic results for the growth rate.
We provide exact asymptotics for the tail probabilities P{S n > x} and P{S n − X * n > x} as x → ... more We provide exact asymptotics for the tail probabilities P{S n > x} and P{S n − X * n > x} as x → ∞, for fix n, where S n and X * n is the partial sum and partial maximum of i.i.d. St. Petersburg random variables. We show that while the order of the tail of the sum S n is x −1 , the order of the tail of the trimmed sum S n − X * n is x −2. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also provide an infinite series representation of the distribution function of the limiting distribution of the trimmed sum, and analyze its tail behavior.
Springer proceedings in mathematics & statistics, 2015
We discuss Chernoff-type large deviation results for χ 2 divergence errors on partitions. In cont... more We discuss Chernoff-type large deviation results for χ 2 divergence errors on partitions. In contrast to the total variation and the I-divergence, the χ 2-divergence has an unconventional large deviation rate. In this paper we extend the result of Quine and Robinson in Ann. Stat. 13:727–742, 1985 from uniform distribution to arbitrary distribution.
We discuss Chernoff-type large deviation properties of the Hellinger distance on partitions. If H... more We discuss Chernoff-type large deviation properties of the Hellinger distance on partitions. If H n denotes the Hellinger distance of the empirical distribution and the distribution restricted to a partition then for small \( > 0,P\left\{ {H_n > } \right\} \approx e^{ - n\left( {^2 + o\left( 1 \right)} \right)} , \) where n is the sample size.
Springer proceedings in mathematics & statistics, 2014
For Gaussian process, we present an open problem whether or not there is a data driven predictor ... more For Gaussian process, we present an open problem whether or not there is a data driven predictor of the conditional expectation of the current value given the past such that the difference between the predictor and the conditional expectation tends to zero almost surely for all stationary, ergodic, Gaussian process. We show some related negative and positive findings. 1 Open problem Let {Y n } ∞ −∞ be a stationary, ergodic, mean zero Gaussian process. The predictor is a sequence of functions g = {g i } ∞ i=1. It is an open problem whether it is possible to learn the best predictor from the past data in a strongly consistent way, i.e., whether there exists a prediction rule g such that lim n→∞
IEEE Transactions on Information Theory, Jul 1, 1975
signals in Section V make use of Price's theorem and a cross correlation property for separable r... more signals in Section V make use of Price's theorem and a cross correlation property for separable random processes. Generalizations of results due to Richardson (absolutely integrable signals) and Prosser (periodic signals) have been obtained, and it is suggested that the constructive methods used for deterministic signals may be used to obtain results for signals depending on two or more arguments.
The strong universal pointwise consistency of some modified versions of the standard regression f... more The strong universal pointwise consistency of some modified versions of the standard regression function estimates of partitioning, kernel, and nearest neighbor type is shown.
IEEE Transactions on Information Theory, May 1, 1992
A general theorem is proved showing how to ohtain a constant-weight binary cyclic code from a p-a... more A general theorem is proved showing how to ohtain a constant-weight binary cyclic code from a p-ary linear cyclic code, where p is a prime,. by using a representation of Cl;(p) as cyclic shifts of a binary p-tuple. Based on this theorem, constructions are given for four classes of binary constant-weight codes. The first two classes are shown to achieve the Johnson upper bound on minimum distance asymptotically for long block lengths. The other two classes are shown similarly to asymptotically meet the low-rate Plotkin upper bound on minimum distance. A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. A simple method is given for selecting virtually the maximum number of cyclically distinct codewords with full cyclic order from Reed-Solomon codes and from Berlekamp-Justesen maximum-distance-separable codes. Two correspondingly optimum classes of constant-weight cyclically permutable codes are constructed by appropriate selection of codewords from the first two classes of binary constant-weight codes. It is shown that cyclically permutable codes provide a natural solution to the problem of constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback.
For the tree algorithm introduced by Capetanakis (1979) and Tsybakov and Mihailov (1978) let LN d... more For the tree algorithm introduced by Capetanakis (1979) and Tsybakov and Mihailov (1978) let LN denote the expected collision resolution time given the collision multiplicity N. If L(z) stands for the Poisson transform of LN , then we show that LN − L(N) 1.29 • 10 −4 cos(2π log 2 N + 0.698).
ABSTRACT A composite hypothesis testing procedure, originally introduced in [5], is examined for ... more ABSTRACT A composite hypothesis testing procedure, originally introduced in [5], is examined for robustness in the binary case. A density-free uniform exponential bound for the error probability is derived which tightens the bound of [5], it is shown that this procedure is equivalent to a hard-limited likelihood-ratio test, and asymptotic and nonasymptotic robustness is discussed.
IEEE Transactions on Information Theory, Jul 1, 1978
An erroneow method for maximizing the projected divergence between two Gaussian multivariate hypo... more An erroneow method for maximizing the projected divergence between two Gaussian multivariate hypotheses appeared in a recent paper. The correct solution is given.
We revisit processes generated by iterated random functions driven by a stationary and ergodic se... more We revisit processes generated by iterated random functions driven by a stationary and ergodic sequence. Such a process is called strongly stable if a random initialization exists, for which the process is stationary and ergodic, and for any other initialization the difference of the two processes converges to zero almost surely. Under some mild conditions on the corresponding recursive map, without any condition on the driving sequence we show the strong stability of iterations. Several applications are surveyed such as stochastic approximation and queuing. Furthermore, new results are deduced for Langevin-type iterations with dependent noise and for multitype branching processes.
Starting in 2007, the MFO publishes a preprint series which mainly contains research results rela... more Starting in 2007, the MFO publishes a preprint series which mainly contains research results related to a longer stay in Oberwolfach. In particular, this concerns the Research in Pairs-Programme (RiP) and the Oberwolfach-Leibniz-Fellows (OWLF), but this can also include an Oberwolfach Lecture, for example. A preprint can have a size from 1-200 pages, and the MFO will publish it on its website as well as by hard copy. Every RiP group or Oberwolfach-Leibniz-Fellow may receive on request 30 free hard copies (DIN A4, black and white copy) by surface mail. Of course, the full copy right is left to the authors. The MFO only needs the right to publish it on its website www.mfo.de as a documentation of the research work done at the MFO, which you are accepting by sending us your file. In case of interest, please send a pdf file of your preprint by email to or , respectively. The file should be sent to the MFO within 12 months after your stay as RiP or OWLF at the MFO. There are no requirements for the format of the preprint, except that the introduction should contain a short appreciation and that the paper size (respectively format) should be DIN A4, "letter" or "article". On the front page of the hard copies, which contains the logo of the MFO, title and authors, we shall add a running number (20XX-XX). We cordially invite the researchers within the RiP or OWLF programme to make use of this offer and would like to thank you in advance for your cooperation.
We study piecewise linear density estimators from the L 1 point of view: the frequency polygons i... more We study piecewise linear density estimators from the L 1 point of view: the frequency polygons investigated by SCOTT (1985) and JONES et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L 1 strongly consistent. We derive large deviation inequalities. For twice dierentiable densities with compact support their expected L 1 error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.
IEEE Transactions on Information Theory, Nov 1, 2005
We consider the rate of convergence of the expected distortion redundancy of empirically optimal ... more We consider the rate of convergence of the expected distortion redundancy of empirically optimal vector quantizers. Earlier results show that the mean-squared distortion of an empirically optimal quantizer designed from independent and identically distributed (i.i.d.) source samples converges uniformly to the optimum at a rate of (1), and that this rate is sharp in the minimax sense. We prove that for any fixed distribution supported on a given finite set the convergence rate is (1) (faster than the minimax lower bound), where the corresponding constant depends on the source distribution. For more general source distributions we provide conditions implying a little bit worse (log) rate of convergence. Although these conditions, in general, are hard to verify, we show that sources with continuous densities satisfying certain regularity properties (similar to the ones of Pollard that were used to prove a central limit theorem for the code points of the empirically optimal quantizers) are included in the scope of this result. In particular, scalar distributions with strictly log-concave densities with bounded support (such as the truncated Gaussian distribution) satisfy these conditions.
Multi-type inhomogeneous Galton-Watson process with immigration is investigated, where the offspr... more Multi-type inhomogeneous Galton-Watson process with immigration is investigated, where the offspring mean matrix slowly converges to a critical mean matrix. Under general conditions we obtain limit distribution for the process, where the coordinates of the limit vector are not necessarily independent.
In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determ... more In this paper, we revisit the classical results on the generalized St. Petersburg sums. We determine the limit distribution of the St. Petersburg sum conditioning on its maximum, and we analyze how the limit depends on the value of the maximum. As an application, we obtain an infinite sum representation of the distribution function of the possible semistable limits. In the representation, each term corresponds to a given maximum, in particular this result explains that the semistable behavior is caused by the typical values of the maximum.
ABSTRACT This paper discusses some metrics of TCP based on real measurements over the Internet. W... more ABSTRACT This paper discusses some metrics of TCP based on real measurements over the Internet. We present algorithms to measure the congestion window related metrics and use these metrics to study the stationary behavior of TCP. Our statistical analysis shows that the distribution of the congestion window process in a stable period has a bell-like shape and can be approximated by a normal distribution
We investigate the performance of the constantly rebalanced portfolios, when the random vectors o... more We investigate the performance of the constantly rebalanced portfolios, when the random vectors of the market process {Xi} are independent, and each of them distributed as (X (1) , X (2) ,. .. , X (d) , 1), d ≥ 1, where X (1) , X (2) ,. .. , X (d) are nonnegative iid random variables. Under general conditions we show that the optimal strategy is the uniform: (1/d,. .. , 1/d, 0), at least for d large enough. In case of St. Petersburg components we compute the average growth rate and the optimal strategy for d = 1, 2. In order to make the problem non-trivial, a commission factor is introduced and tuned to result in zero growth rate on any individual St. Petersburg components. One of the interesting observations made is that a combination of two components of zero growth can result in a strictly positive growth. For d ≥ 3 we prove that the uniform strategy is the best, and we obtain tight asymptotic results for the growth rate.
We provide exact asymptotics for the tail probabilities P{S n > x} and P{S n − X * n > x} as x → ... more We provide exact asymptotics for the tail probabilities P{S n > x} and P{S n − X * n > x} as x → ∞, for fix n, where S n and X * n is the partial sum and partial maximum of i.i.d. St. Petersburg random variables. We show that while the order of the tail of the sum S n is x −1 , the order of the tail of the trimmed sum S n − X * n is x −2. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also provide an infinite series representation of the distribution function of the limiting distribution of the trimmed sum, and analyze its tail behavior.
Springer proceedings in mathematics & statistics, 2015
We discuss Chernoff-type large deviation results for χ 2 divergence errors on partitions. In cont... more We discuss Chernoff-type large deviation results for χ 2 divergence errors on partitions. In contrast to the total variation and the I-divergence, the χ 2-divergence has an unconventional large deviation rate. In this paper we extend the result of Quine and Robinson in Ann. Stat. 13:727–742, 1985 from uniform distribution to arbitrary distribution.
We discuss Chernoff-type large deviation properties of the Hellinger distance on partitions. If H... more We discuss Chernoff-type large deviation properties of the Hellinger distance on partitions. If H n denotes the Hellinger distance of the empirical distribution and the distribution restricted to a partition then for small \( > 0,P\left\{ {H_n > } \right\} \approx e^{ - n\left( {^2 + o\left( 1 \right)} \right)} , \) where n is the sample size.
Springer proceedings in mathematics & statistics, 2014
For Gaussian process, we present an open problem whether or not there is a data driven predictor ... more For Gaussian process, we present an open problem whether or not there is a data driven predictor of the conditional expectation of the current value given the past such that the difference between the predictor and the conditional expectation tends to zero almost surely for all stationary, ergodic, Gaussian process. We show some related negative and positive findings. 1 Open problem Let {Y n } ∞ −∞ be a stationary, ergodic, mean zero Gaussian process. The predictor is a sequence of functions g = {g i } ∞ i=1. It is an open problem whether it is possible to learn the best predictor from the past data in a strongly consistent way, i.e., whether there exists a prediction rule g such that lim n→∞
IEEE Transactions on Information Theory, Jul 1, 1975
signals in Section V make use of Price's theorem and a cross correlation property for separable r... more signals in Section V make use of Price's theorem and a cross correlation property for separable random processes. Generalizations of results due to Richardson (absolutely integrable signals) and Prosser (periodic signals) have been obtained, and it is suggested that the constructive methods used for deterministic signals may be used to obtain results for signals depending on two or more arguments.
The strong universal pointwise consistency of some modified versions of the standard regression f... more The strong universal pointwise consistency of some modified versions of the standard regression function estimates of partitioning, kernel, and nearest neighbor type is shown.
IEEE Transactions on Information Theory, May 1, 1992
A general theorem is proved showing how to ohtain a constant-weight binary cyclic code from a p-a... more A general theorem is proved showing how to ohtain a constant-weight binary cyclic code from a p-ary linear cyclic code, where p is a prime,. by using a representation of Cl;(p) as cyclic shifts of a binary p-tuple. Based on this theorem, constructions are given for four classes of binary constant-weight codes. The first two classes are shown to achieve the Johnson upper bound on minimum distance asymptotically for long block lengths. The other two classes are shown similarly to asymptotically meet the low-rate Plotkin upper bound on minimum distance. A cyclically permutable code is a binary code whose codewords are cyclically distinct and have full cyclic order. A simple method is given for selecting virtually the maximum number of cyclically distinct codewords with full cyclic order from Reed-Solomon codes and from Berlekamp-Justesen maximum-distance-separable codes. Two correspondingly optimum classes of constant-weight cyclically permutable codes are constructed by appropriate selection of codewords from the first two classes of binary constant-weight codes. It is shown that cyclically permutable codes provide a natural solution to the problem of constructing protocol-sequence sets for the M-active-out-of-T users collision channel without feedback.
For the tree algorithm introduced by Capetanakis (1979) and Tsybakov and Mihailov (1978) let LN d... more For the tree algorithm introduced by Capetanakis (1979) and Tsybakov and Mihailov (1978) let LN denote the expected collision resolution time given the collision multiplicity N. If L(z) stands for the Poisson transform of LN , then we show that LN − L(N) 1.29 • 10 −4 cos(2π log 2 N + 0.698).
ABSTRACT A composite hypothesis testing procedure, originally introduced in [5], is examined for ... more ABSTRACT A composite hypothesis testing procedure, originally introduced in [5], is examined for robustness in the binary case. A density-free uniform exponential bound for the error probability is derived which tightens the bound of [5], it is shown that this procedure is equivalent to a hard-limited likelihood-ratio test, and asymptotic and nonasymptotic robustness is discussed.
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