We consider asymptotic behavior of a random point measure generated by a finite number of points sequentially allocated in a finite domain of d -dimensional space. The probability distribution of a point depends on all previously allocated points. We consider a special case when this dependence vanishes as the domain is saturated by points. The law of large numbers and the central limit theorem are proved as the number of points goes to infinity. These measures belong to the class of random point measures generated by the spatial processes arising in random sequential packing and deposition problems. The typical example is when m points are sequentially allocated in a unit cube. Each point is uniformly distributed in the cube and is accepted with probability depending on configuration of previously accepted points in the ball of radius 1/m around the point. So, the interaction radius is inversely proportional to the number of points. It is not the case in our model where the interaction radius is a fixed constant regardless of the number of points.