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According to the Ray-Knight description, the Brownian local time in some conditional probability space is a diffusion process with respect to spatial variable. This diffusion has a local time. Thus the following definition of local time of the original Brownian local time seems to be natural: such a process is called the second-order Brownian local time. The paper studies the Laplace transform of the distribution of the Brownian local time of the second order. [ABSTRACT FROM AUTHOR]

The performance of the minimax variance (in the Huber sense) robust M-estimates of a location parameter designed in the class of symmetric distributions with bounded interquantile ranges is studied both on large and small samples. Although the general structure of these minimax M-estimates has been described by Huber (1981) long ago, they were not studied as main attention was focused on the estimates designed for the various neighborhoods of a Gaussian. In this work, the proposed estimates are compared to the sample mean, sample median, and the classical robust Huber and Hampel M-estimates under the Gaussian, contaminated Gaussian, Laplace, and Cauchy distributions. The performance of an estimate is measured by its efficiency, bias, and mean squared error. The following conclusions are made: (i) under the Gaussian, Laplace, Cauchy, and moderately contaminated Gaussian distributions, the proposed minimax M-estimates outperform the robust Huber and Hampel M-estimates with respect to asymptotic efficiency; (ii) under heavily contaminated Gaussian distributions, the Huber and Hampel M-estimates are slightly better; (iii) on small samples, these classical robust estimates also slightly outperform the proposed minimax estimates in mean squared error. [ABSTRACT FROM AUTHOR]

We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, and generalized Mittag–Leffler laws. In particular, we prove that the generalized Linnik distribution is a normal scale mixture with the generalized Mittag–Leffler mixing distribution. Based on these representations, we prove some limit theorems for a wide class of statistics constructed from samples with random sized including, e.g., random sums of independent random variables with finite variances, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is by far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution. [ABSTRACT FROM AUTHOR]

In practice, we often encounter situations where a sample size is not defined in advance and can be a random value. In the present paper, we derive second-order Chebyshev–Edgeworth and Cornish–Fisher expansions based of Student's t- and Laplace distributions and their quantiles for samples with random size of a special kind. This derivation uses a general transfer theorem, which allows us to construct asymptotic expansions for distributions of randomly normalized statistics from the distributions of the considered nonrandomly normalized statistics and of the random size of the underlying sample. Recently, interest in Cornish–Fisher expansions has increased because of study in risk management. Widespread risk measure Value at Risk (VaR) substantially depends on quantiles of the loss function, which is connected with description of investment portfolio of financial instruments. [ABSTRACT FROM AUTHOR]

We construct and justify a homogenized model of the variational inequality with the Laplace operator and a nonlinear boundary constraint on the flow on arbitrary shaped cavities generating perforation of the domain with critical values of parameters. [ABSTRACT FROM AUTHOR]

In this paper we conduct a comparative analysis of the powers of the two-sample Kolmogorov-Smirnov and Anderson-Darling tests under various alternatives using simulation. We consider two examples. In the first example the alternatives to the standard normal distribution are the distributions of the so-called contaminated normal model. We study the influence of a small contamination with a positive shift on the powers of the test. In the second example the alternatives are the logistic and the Laplace distributions, which are symmetric and differ in shape from the normal distribution having a larger kurtosis coefficient and heavier tails. [ABSTRACT FROM AUTHOR]

The paper contains an introduction to the asymptotic theory of hypothesis testing and a review of recent results of the author and his students. We consider only the asymptotic approach, for which with increasing sample size n the test size is separated from zero, and the sequences of local alternatives, for which the power is separated from one. This paper focuses on asymptotically efficient tests when testing a simple hypothesis in the case of a one-parameter family. We study the difference between the powers of the best and asymptotically efficient tests. This difference is closely related to the notion of asymptotic test deficiency. We consider the formula for the limiting deviation of the power of asymptotically optimal test from the power of the best test in the case of Laplace distribution. Due to the irregularity of the Laplace distribution, this deviation is of order n , in contrast to the usual regular families for which this order is n . We also study the Bayesian settings and the case of increasing parameter dimension. [ABSTRACT FROM AUTHOR]

In this paper, under several general assumptions, we deduce an abstract Green formula for a triple of Hilbert spaces and an (abstract) trace operator and a similar formula corresponding to sesquilinear forms. We establish existence conditions for the abstract Green formula for mixed boundary-value problems. As the main application, we deduce generalized Green formulas for the Laplace operator applied to boundary-value problems in Lipschitz domains. [ABSTRACT FROM AUTHOR]

We present some product representations for random variables with the Linnik, Mittag-Leffler, and Weibull distributions and establish the relationship between the mixing distributions in these representations. The main result is the representation of the Linnik distribution as a normal scale mixture with the Mittag-Leffler mixing distribution. As a corollary, we obtain the known representation of the Linnik distribution as a scale mixture of Laplace distributions. Another corollary of the main representation is the theorem establishing that the distributions of random sums of independent identically distributed random variables with finite variances converge to the Linnik distribution under an appropriate normalization if and only if the distribution of the random number of summands under the same normalization converges to the Mittag-Leffler distribution. [ABSTRACT FROM AUTHOR]

We develop self-adjoint extensions of the radial part of the Laplace operator for l = 1 in a special scalar product. The product arises under the passage of the standard product from ℝ to the set of functions parametrizing one of two components of the transverse vector field. Similar extensions are treated for the square of the inverse operator of the radial part in question. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

For nonlinear parabolic p-Laplace type equations we obtain estimates improving similar estimates in the known monograph of E. DiBenedetto. For the gradients of sign-variable solutions we obtain local 'entropy' estimates which are used in the study of the solvability of the Cauchy problem in the entire space with sign-variable initial data. Bibliography: 22 titles. [ABSTRACT FROM AUTHOR]

In this paper asymptotic formulas are obtained for determining the principal terms in global theorems for the distribution function of a geometric random sum. The obtained result implies that among all symmetric distributions the Bernoulli distribution has an asymptotically extreme nature under geometric summation. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional Markov diffusion process on the open half-plane as the range of values. With respect to the boundary of the half-plane, the process is represented by the normal and tangential components which are locally independent at any point of the open half-plane. The range of values is extended to the boundary by some rule representing reflection with delay. Due to this reflection, components of the process become dependent. The tangential component gets a delay as well. A relation between distributions of the initial independent and delayed dependent components is derived in terms of their Laplace images. Bibliography: 4 titles. [ABSTRACT FROM AUTHOR]