egin{abstract} hspace*{1cm} Let $X_{1}$@, $X_{2}$@,$cdots$@, $X_{n}$ be a sequence of independent indentically distributed random variables ( i@. i@. d@.) and $S_{n}=X_{1}+X_{2}+...+X_{n}$@. Denote $lambda(varepsilon)=displaystylesum_{n=1}^{infty}P(|S_{n}|geq nvarepsilon)$@. O.I. Klesov proved that if $EX_{1}=0$, $EX_{1}^{2}=sigma ^{2} eq 0$, $E|X_{1}|^{3}<infty$, then $displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$. In this thesis, it is shown that if $EX_{1}=0$, $EX_{1}^{2}=sigma ^{2} eq 0$, $E|X_{1}|^{2+delta}<infty$ for some $displaystyledeltain(frac{1}{2},1]$, then $displaystylelim_{varepsilondownarrow0}varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}})=0$. end{abstract}

egin{abstract} hspace{1cm}Let $X_{1}$, $X_{2}$, $cdots$, $X_{n}$ be a sequence of independent indentically distributed random variables ( i. i. d.) and $displaystyle S_{n}=X_{1}+X_{2} +cdots X_{n}$. Denote $displaystylelambda(varepsilon)=sum^{infty}_{n=1}P{left|S_{n} ight|geq nvarepsilon}$, the convergence rate of $displaystylelambda(varepsilon)$ is studied. O.I. Klesov proved that if $E|X_{1}|^{3}$ exists, then $displaystyle varepsilon^{frac{3}{2}}(lambda(varepsilon)-frac{sigma^{2}}{varepsilon^{2}}) ightarrow 0$. In this thesis, we show that if $E|X_{1}|^{2+delta}<infty$ for some $displaystyle deltain(frac{sqrt{7}-1}{3},1]$, the result of O.I. Klesov still holds. end{abstract}

The viscoelastic and creep properties of salt create challenges in the design of salt mines. Salt undergoes steady state creep for a long period of time, and the time of failure is not easily predicted. Developing functions for creep behavior is important in predicting the deformation of salt pillars. Through literature reviews, it was found that there are many relationships to determine the deformation rate of salt specimens through constitutive models. Mine panels have also been modeled to understand the stress and deformational behavior of the pillars. The purpose of this was project was to develop a relationship that determines the convergence rate from knowing the pillar width to pillar height ratio and thickness of the salt strata immediately above and below the mine. The third power law was adopted in the modeling of salt pillars, which is applicable to low stresses of less than 10 MPa that is typical of salt mine conditions. The finite difference software, FLAC3D was used for the simulations of salt pillar models. A square pillar was modeled using four pillar width to pillar height ratios from 1.5 to 4.6. In mining practices, the pillar width to pillar height ratios are designed to be 1.0 to 5.0. Three sets of pillar dimensions were used for each pillar width to pillar height ratio, this was done to determine whether different room and pillar dimensions for each pillar width to pillar height ratio resulted in different convergence rates. Eight salt thicknesses of 0 m to 26 m were modeled for each set of pillar dimensions, which was sufficient to determine the effect of salt thickness on convergence rate. From the modeled results, general trends among the various pillar width to pillar height ratios were observed. The convergence rate increased as the pillar width to pillar height ratio decreased. In addition, an exponential relationship was found between the convergence rate and the pillar width to pillar height ratio. There was a strong correlation between convergence values calculated from the developed function and the modeled values for the power law exponent of three. The developed expression can be used to estimate the convergence rate due to pillar compression and room convergence.

The viscoelastic and creep properties of salt create challenges in the design of salt mines. Salt undergoes steady state creep for a long period of time, and the time of failure is not easily predicted. Developing functions for creep behavior is important in predicting the deformation of salt pillars. Through literature reviews, it was found that there are many relationships to determine the deformation rate of salt specimens through constitutive models. Mine panels have also been modeled to understand the stress and deformational behavior of the pillars. The purpose of this was project was to develop a relationship that determines the convergence rate from knowing the pillar width to pillar height ratio and thickness of the salt strata immediately above and below the mine. The third power law was adopted in the modeling of salt pillars, which is applicable to low stresses of less than 10 MPa that is typical of salt mine conditions. The finite difference software, FLAC3D was used for the simulations of salt pillar models. A square pillar was modeled using four pillar width to pillar height ratios from 1.5 to 4.6. In mining practices, the pillar width to pillar height ratios are designed to be 1.0 to 5.0. Three sets of pillar dimensions were used for each pillar width to pillar height ratio, this was done to determine whether different room and pillar dimensions for each pillar width to pillar height ratio resulted in different convergence rates. Eight salt thicknesses of 0 m to 26 m were modeled for each set of pillar dimensions, which was sufficient to determine the effect of salt thickness on convergence rate. From the modeled results, general trends among the various pillar width to pillar height ratios were observed. The convergence rate increased as the pillar width to pillar height ratio decreased. In addition, an exponential relationship was found between the convergence rate and the pillar width to pillar height ratio. There was a strong correlation between convergence values calculated from the developed function and the modeled values for the power law exponent of three. The developed expression can be used to estimate the convergence rate due to pillar compression and room convergence.

Theory of extreme values is very important and it’s use of range is very wide. A lot of research occurrence usually is described with a few measurements (for example, testing pollution of atmosphere, usually consider all superior limits of pollution concentration, not only one war gas maximum concentration) that’s why are analyzing multidimensional extreme values. Herein work we analyze convergence rate of the multidimensional extreme values in the transfer limit theorem. We solve this problem using particular distributions. In this work we will give nonuniform estimate of convergence rate of multidimensional extreme values in the transfer limit theorem.

Herein work is analyzing convergence rate in transfer theorems for extreme values of independent identically distributed random variables. Analyzing various distributions is god nonuniform estimate of convergence rate in transfer theorems. Transfer theorem of density of minima have been proved. Analyzing convergence rate in transfer theorems of density for extreme. Computation was developed using SAS and Mathcad.

This thesis poses and solves a distribution free learning problem with consensus that arises in the study of estimation and control strategies for distributed sensor networks. Each node i for i = 1, . . . , n of the sensor network collects independent and identically distributed local measurements {z i} := {z i j}j∈N := {(x i j , yi j )}j∈N ⊆ X × Y := Z that are generated by the probability measure ρ i on Z. Each node i for i = 1, . . . , n of the network constructs a sequence of estimates {f i k }k∈N from its local measurements {z i} and from information functionals whose values are exchanged with other nodes as specified by the communication graph G for the network. The optimal estimate of the distribution free learning problem with consensus is cast as a saddle point problem which characterizes the consensus-constrained optimal estimate. This thesis introduces a two stage learning dynamic wherein local estimation is carried out via local least square approximations based on wavelet constructions and information exchange is associated with the Lagrange multipliers of the saddle point problem. Rates of convergence for the two stage learning dynamic are derived based on certain recent probabilistic bounds derived for wavelet approximation of regressor functions. / Master of Science

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In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski’s method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an (r + 1)-dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic (r + 1)-dimensional function, we show that by choosing Besov balls of mixed smoothness, we can avoid the ”curse of dimensionality” and, hence, obtain higher than usual convergence rates when r is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy two-dimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a two-dimensional function. By studying the two-dimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as two-dimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with long-range dependent Gaussian errors. We do not limit our consideration to a specific type of long-range dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the iii smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels.

Estimating the integrated volatility of high frequency realized prices is an important issue in microstructure literature. Bandi and Russell (2006) derived the optimal-sampling frequency, and Zhang et al. (2005) proposed a "two-scales estimator" to solve the problem. In this study, we propose a new estimator based on a signal to noise ratio statistic with convergence rate of Op (n^(−1/ 4) ). The method is applicable to both constant and stochastic volatility models and modi¡Âes the Op (n^(−1/ 6) ) convergence rate of Zhang et al. (2005). The proposed estimator is shown to be asymptotic e¡Ócient as the maximum likelihood estimate for the constant volatility case. Furthermore, unbiased estimators of the two elements, the variance of the microstructure noise and the fourth moment of the realized log returns, are also proposed to facilitate the estimation of integrated volatility. The asymptotic prop- erties and e&#x00AE;ectiveness of the proposed estimators are investigated both theoretically and via simulation study.