In this article, we establish a local limit theorem for linear fields of random variables constructed from i.i.d. innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, that is, for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.

In this paper, we establish a local limit theorem for linear fields of random variables constructed from independent and identically distributed innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square-summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, i.e. for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.

In this paper, we estimate the Shannon entropy S(f ) = −E [log(f (x))] of a one-sided linear process with probability density function f (x). We employ the integral estimator S n (f ), which utilizes the standard kernel density estimator f n (x) of f (x). We show that S n (f ) converges to S(f ) almost surely and in L 2 under reasonable conditions.

In this paper, we estimate the Shannon entropy S(f)=−E[log(f(x))] of a one-sided linear process with probability density function f(x). We employ the integral estimator Sn(f), which utilizes the standard kernel density estimator fn(x) of f(x). We show that Sn(f) converges to S(f) almost surely and in Ł2 under reasonable conditions.