Letu(1/Q) be an absolutely integrable function and define the random process where thet i are Poisson arrivals and thes i, are identically distributed nonnegative random variables. Under routine independence assumptions, one may then calculate a formula for the spectrum ofn(t), S n(), in terms of the probability density ofs, ps(). If any probability density ps() having the property ps() I for small is substituted into this formula, the calculated Sn() is such that Sn() 1 for small . However, this is not a spectrum of a well-defined random process; here, it is termed alimit spectrum. If a probability density having the property ps() for small , where > 0, is substituted into the formula instead, a spectrum is calculated which is indeed the spectrum of a well-defined random process. Also, if the latter ps is suitably close to the former ps, then the spectrum in the second case approximates, to an arbitrary, degree of accuracy, the limit spectrum. It is shown how one may thereby have 1/f noise with low-frequency turnover, and also strict 1/f 1– noise (the latter spectrum being integrable for > 0). Suitable examples are given. Actually, u() may be itself a random process, and the theory is developed on this basis.