The Law of Iterated Logarithm for Autoregressive Processes

Autoregressive process is a basic model in engineering, insurance, and business. It is a representation of a type of random process; as such, it describes certain time-varying processes in nature, engineering, and so forth. The autoregressive model specifies that the output variable depends linearly on its own previous values. It is a special case of the more general autoregressive-moving average (ARMA)model of time series. For example, in engineering one considers a dam, with input of random amounts at random times, and a steady withdrawal of water for irrigation or power usage. This model has a Markovian representation. It is well known that stability problem is an important interest topic, which explainswhy the stochastic systems always stay in “reasonable values”: the dam does not overflow. The parameters in the process have a direct relation to the system stability, so the estimation of fixed coefficients and its dynamics behaviors are very useful in engineering. There was a rich literature which focused on the research (see Mann and Wald [1], Anderson [2], Menneteau [3], Hwang and Choi [4], Hwang and Baek [5], and Miao and Shen [6]). Recently, some attention has been directed to random coefficient autoregressive models. This way of handling the data allows for large shocks in the dynamic structure of the model and also for some flexibility in the features of the volatility of the series, which are not available in fixed coefficient autoregressive models. For simplicity, in this paper, we are concerned with some dynamics behaviors (in probability language, we call it the law of iterated logarithm) of the parameters estimation for the linear autoregressive model which is of practical importance, by a different technique from the standard method. Define the following first order autoregressive process (AR(1)) {X t , t ≥


Introduction
Autoregressive process is a basic model in engineering, insurance, and business.It is a representation of a type of random process; as such, it describes certain time-varying processes in nature, engineering, and so forth.The autoregressive model specifies that the output variable depends linearly on its own previous values.It is a special case of the more general autoregressive-moving average (ARMA) model of time series.For example, in engineering one considers a dam, with input of random amounts at random times, and a steady withdrawal of water for irrigation or power usage.This model has a Markovian representation.It is well known that stability problem is an important interest topic, which explains why the stochastic systems always stay in "reasonable values": the dam does not overflow.The parameters in the process have a direct relation to the system stability, so the estimation of fixed coefficients and its dynamics behaviors are very useful in engineering.There was a rich literature which focused on the research (see Mann and Wald [1], Anderson [2], Menneteau [3], Hwang and Choi [4], Hwang and Baek [5], and Miao and Shen [6]).Recently, some attention has been directed to random coefficient autoregressive models.This way of handling the data allows for large shocks in the dynamic structure of the model and also for some flexibility in the features of the volatility of the series, which are not available in fixed coefficient autoregressive models.
For simplicity, in this paper, we are concerned with some dynamics behaviors (in probability language, we call it the law of iterated logarithm) of the parameters estimation for the linear autoregressive model which is of practical importance, by a different technique from the standard method.Define the following first order autoregressive process (AR(1)) {  ,  ≥ 1} by where the initial value  0 is not necessarily zero and {  } is a sequence of independent and identically distributed (i.i.d.) random errors with mean zero and variance  2 .Based on the observations  1 ,  2 , . . .,   , the least squares (LS) estimators ( β() 0 , β() 1 )  of ( 0 ,  1 )  are given by ) . ( By a simple calculation, we have It is well known that, in the stable (or, in other words, asymptotically stationary) case, when      1     < 1, β() 1 is asymptotically normal under the assumption that {  } is a sequence of i.i.d.random variables with mean zero and variance 1; namely, where "⇒" denotes the convergence in distribution, although it does not hold uniformly for | 1 | < 1 (see Anderson [2] and Meyn and Tweedie [7]).In the unstable (or, in other words, unit root) case when  1 = 1, for the sequence β() 1 , we have where {() :  ∈ [0, 1]} denotes a standard Wiener process.
In this paper, we consider the convergence rates of  () 0 →  0 and  () 1 →  1 under some conditions.By constructing an -dependent random variables sequence and applying an approximation method along with a central limit theorem for random variables, we prove the LS estimators of  0 and  1 satisfy the law of iterated logarithm.Our results can be considered as an embodiment of Miao and Shen [6].
In the following statement, we have an assumption as follows: (C.1) there exists a constant  > 0 such that E exp{|    |} < +∞ for any  ≥ 1 and  ≥ 1.

Main Results
The main result is as follows.

Preliminary Lemmas
In order to prove Theorem 1, we need the following lemmas.
The following lemma is about Lévy's inequality.For completeness, we still give its proof.Lemma 3. Let {  } ≥1 be a sequence of independent random variables taking their values in R. Let   = ∑  =1   .Then for any  > 0, where m() denotes the median of r.v..