Let f ( n ) f(n) be an arbitrary arithmetical function and let A N {A_N} and B N {B_N} be sequences of real numbers with 0 > B N → + ∞ 0 > {B_N} \to + \infty with N N . We give a sufficient condition for ( f ( n ) − A N ) / B N (f(n) - {A_N})/{B_N} to have a limiting distribution. The case when f ( n ) f(n) is defined by f ( n ) = Σ g ( d ) f(n) = \Sigma g(d) , where the summation is over all divisors d d of n n and g ( d ) g(d) is any given arithmetical function, is discussed in more detail. A concrete example is given as an application of our result, in which example f ( n ) f(n) is neither additive nor multiplicative. Our method of proof is to approximate f ( n ) f(n) by a suitably chosen additive function, as proposed in [4], and then to apply general theorems available for additive functions.