One of our research goal is the design of algorithms capable of detecting, with some robustness guaranteed by a suitable performance criterion, random signals with unknown distributions and occurrences in additive and stationary noise. This research is motivated by numerous practical applications where the lack of prior information about the signals or most of their describing parameters is an obstacle to the use of standard likelihood ratio theory. Several results established and assessed in various signal processing applications suggest that our goal could actually be attainable in a nearby future. Part of these results concern robust and non-parametric statistical inference based on assumptions of weak sparseness for the signal. The notion of weak sparseness slightly differs from that introduced by Donoho and Johnstone. Another set of results address statistical properties of the wavelet transform of wide-sense stationary random processes and fractional brownian motions. Some of the aforementioned results have been published separately, whereas others have been submitted only recently. Our purpose is then to present these results altogether so as to examine the links between them and pinpoint possible extensions. In particular, a perspective is the design of algorithms that perform more than the detection of random sigals with unknown distributions, but actually acquire statistical knowledge about these signals to optimize their detection. Beyond the description of standard possible applications in image processing and non-parametric statistics, the presentation will also give the opportunity to initiate a discussion on the application of these results to the design of new types of cell automata.