  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.
