grows approximately as ". Taking the difference between and its approximation and then dividing by the next term in the expansion, we arrive at a more refined statement about : Here one can say that the difference between the function and its approximation grows approximately as . The idea is that dividing the function by appropriate normalizing functions, and looking at the limiting behavior of the result, can tell us much about the limiting behavior of the original function itself. Informally, something along these lines happens when the sum, , of independent identically distributed random variables, , is studied in classical probability theory. If each has finite mean , then by the law of large numbers, . If in addition each has finite variance , then by the central limit theorem, where is distributed as . This provides values of the first two constants in the informal expansion In the case where the do not have finite mean or variance, convergence of the shifted and rescaled sum can also occur with different centering and scaling factors: or informally Distributions which can arise in this way are called stable. Clearly, the normal distribution is stable, but there are also other stable distributions, such as the Cauchy distribution, for which the mean or variance are not defined. The scaling factor may be proportional to , for any ; it may also be multiplied by a slowly varying function of . The law of the iterated logarithm specifies what is happening "in between" the law of large numbers and the central limit theorem. Specifically it says that the normalizing function , intermediate in size between of the law of large numbers and of the central limit theorem, provides a non-trivial limiting behavior. Alternative statements of the theorem Density functions The density of the sum of two or more independent variables is the convolution of their densities (if these densities exist). Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound. These theorems require stronger hypotheses than the forms of the central limit theorem given above. Theorems of this type are often called local limit theorems. See Petrov for a particular local limit theorem for sums of independent and identically distributed random variables.