One SCI\'ice mathematics bas rendered the 'Et moi, .... si j'avait su comment en revcnir. je n'y serais point aile: human race. It bas put common sc:nsc back where it belongs, on the topmost shelf next Jules Verne to the dusty canister labelled 'discarded n- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Hcavisidc Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non­ linearities abound. Similarly. all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. :; 'One service logic has rendered com­ puter science .. :; 'One service category theory has rendered mathematics .. :. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

0. Preliminary Informationh.- 0.1 Probability Space.- 0.2 Random Functions and Processes.- 0.3 Conditional Probabilities.- 0.4 Independence.- 1. Sums of Independent Random Variables.- 1.1 Main Inequalities.- 1.2 Renewal Scheme.- 1.3 Random Walks. Recurrence.- 1.4 Distribution of Ladder Functions.- 2. General Processes with Independent Increments (Random Measures).- 2.1 Nonnegative Random Measures with Independent Values (r.m.i.v.).- 2.2 Random Measures with Alternating Signs.- 2.3 Stochastic Integrals and Countably Additive r.m.i.v.- 2.4 Random Linear Functional and Generalized Functions.- 3. Processes with Independent Increments. General Properties.- 3.1 Decomposition of a Process. Properties of Sample Functions.- 3.2 Stochastically Continuous Processes.- 3.3 Properties of Sample Functions.- 3.4 Locally Homogeneous Processes with Independent Increments.- 4. Homogeneous Processes.- 4.1 General Properties.- 4.2 Additive Functionals.- 4.3 Composed Poisson Process.- 4.4 Homogeneous Processes in R.- 5. Multiplicative Processes.- 5.1 Definition and General Properties.- 5.2 Multiplicative Processes in Abelian Groups.- 5.3 Stochastic Semigroups of Linear Operators in Rd.- Notes.- References.