  Signals consisting of a sequence of pulses show that inherent origin of the
1/f noise is a Brownian fluctuation of the average interevent time between
subsequent pulses of the pulse sequence. In this paper we generalize the model
of interevent time to reproduce a variety of self-affine time series exhibiting
power spectral density S(f) scaling as a power of the frequency f. Furthermore,
we analyze the relation between the power-law correlations and the origin of
the power-law probability distribution of the signal intensity. We introduce a
stochastic multiplicative model for the time intervals between point events and
analyze the statistical properties of the signal analytically and numerically.
Such model system exhibits power-law spectral density S(f)~1/f**beta for
various values of beta, including beta=1/2, 1 and 3/2. Explicit expressions for
the power spectra in the low frequency limit and for the distribution density
of the interevent time are obtained. The counting statistics of the events is
analyzed analytically and numerically, as well. The specific interest of our
analysis is related with the financial markets, where long-range correlations
of price fluctuations largely depend on the number of transactions. We analyze
the spectral density and counting statistics of the number of transactions. The
model reproduces spectral properties of the real markets and explains the
mechanism of power-law distribution of trading activity. The study provides
evidence that the statistical properties of the financial markets are enclosed
in the statistics of the time interval between trades. A multiplicative point
process serves as a consistent model generating this statistics.
