## Central limit theorems for weakly dependent stochastic processes

Studenteropgave: Kandidatspeciale og HD afgangsprojekt

- Ege Rubak

4. semester, Matematik, Kandidat (Kandidatuddannelse)

This thesis investigates the performance of a wireless communication system using a so-called equalizer. The specific environment in which the wireless communication system is used is called the communication channel. Since wireless systems usually are mobile the channel is a random quantity changing according to the dynamic environment. The channel distorts the transmitted signals, and it is necessary to compensate for this distortion in the design of the system. One way of reducing the effect of the distortion is to use an equalizer in the receiver, which attempts to reverse the effect of the channel and thereby restore the original signal. As a measure of performance of the system the minimum means square error (MMSE) is used. The MMSE value is determined for each channel making it a random quantity. It is interesting to describe the distribution of the MMSE for a given channel model, since it then would be possible to give confidence bands on the performance of the system. The MMSE depends on the bandwidth B used by the communication system. In modern ultra wide band (UWB) systems very large bandwidths are used making asymptotic results for increasing bandwidth interesting. The problem of an asymptotic characterization of the MMSE is the starting point of this thesis.

First a mathematical model for the communication system is presented. In this model two different equalizers are introduced: The infinite and the finite equalizer, which give rise to two different MMSEs denoted respectively MMSE_inf and MMSE_N. The communication channel can be described via the channel response or equivalently the Fourier transform of this response called the transfer function H. Due to the random nature of the channel this is a stochastic process, and it turns out that this process determines MMSE_inf uniquely via an integral expression.

A brief simulation study indicates a central limit theorem (CLT) holds for MMSE_inf, and the rest of the thesis is devoted to the investigation of this. As a background for this investigation a wide range of topics within the field of stochastic processes is presented. Especially a detailed proof of a CLT for continuous time stochastic processes is given. This establishes that under suitable conditions the normalized integral of a stochastic process is asymptotically Gaussian, when the integration limit grows to infinity. For this to hold a key assumption is weak dependence of the stochastic process. The concept of weak dependence is defined via mixing properties, and this topic is also studied. In this classical central limit theory the process is not allowed to depend on the integration limit, which the process H does. To deal with this problem it is necessary to consider the limiting behavior of H as B grows. A powerful tool for studying asymptotics of stochastic processes is weak convergence in metric spaces, and a brief introduction to this topic is provided.

Finally the properties of MMSE_inf, MMSE_N, and H are studied, and a weak limit of H is determined. Furthermore it is proven that under suitable conditions a CLT holds for MMSE_inf, and finally it is proven that MMSE_N approximates MMSE_inf when N tends to infinity, which is used to give a condition ensuring the CLT behavior is inherited by MMSE_N.

First a mathematical model for the communication system is presented. In this model two different equalizers are introduced: The infinite and the finite equalizer, which give rise to two different MMSEs denoted respectively MMSE_inf and MMSE_N. The communication channel can be described via the channel response or equivalently the Fourier transform of this response called the transfer function H. Due to the random nature of the channel this is a stochastic process, and it turns out that this process determines MMSE_inf uniquely via an integral expression.

A brief simulation study indicates a central limit theorem (CLT) holds for MMSE_inf, and the rest of the thesis is devoted to the investigation of this. As a background for this investigation a wide range of topics within the field of stochastic processes is presented. Especially a detailed proof of a CLT for continuous time stochastic processes is given. This establishes that under suitable conditions the normalized integral of a stochastic process is asymptotically Gaussian, when the integration limit grows to infinity. For this to hold a key assumption is weak dependence of the stochastic process. The concept of weak dependence is defined via mixing properties, and this topic is also studied. In this classical central limit theory the process is not allowed to depend on the integration limit, which the process H does. To deal with this problem it is necessary to consider the limiting behavior of H as B grows. A powerful tool for studying asymptotics of stochastic processes is weak convergence in metric spaces, and a brief introduction to this topic is provided.

Finally the properties of MMSE_inf, MMSE_N, and H are studied, and a weak limit of H is determined. Furthermore it is proven that under suitable conditions a CLT holds for MMSE_inf, and finally it is proven that MMSE_N approximates MMSE_inf when N tends to infinity, which is used to give a condition ensuring the CLT behavior is inherited by MMSE_N.

Sprog | Engelsk |
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Udgivelsesdato | jul. 2007 |