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By Stieglitz J.

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We note that for sequences of independent random variables it suffices to know the one-dimensional distributions in order to develop uny multi-dimensional distribution. For the special case in which all the onedimensional distributions are equal, it is even sufficient to know one single distribution function. We shall refer to this latter case by speaking of independent und identically distributed random variables. , X, be independent and have the Same distribution with the density function The n-dimensional density function is then = exp { [i (log xi - - 2 n}'l[(2nr" .

The clairn nurnber process 45 Iheorem: rlhe "normal case" Sn(t,s)= 1 for all n, t, s applies if und only if " 1 E -- diverges (Am,m = l,2, ... are the non-negative time-independent m = 1 Am claim frequencies). ) which decreases monotonically with k to ,u(s); hence and m If 1 l/Am diverges, ~ ( s must ) be Zero, which implies that Sn(s)=1. m=n+l Conversely 1 remains bounded if l/Am converges; hence S„(T) cannot converge to 1 (which by the theorem of dominated convergence (cf. Appendix) would imply the erroneous Statement s 5 (l/A,) is bounded for all s).

56] or [32]. It is left to the reader to prove that every process with independent increments satisfies the Markov condition M and is therefore itself a Markov process. ) Exercises 1 ) Show that every process with independent increments is a Markov process. 2) Suppose that two compound Poisson processes having characteristic functions &) = e " [ ~ ( u ) - ' l resp. e"'[$'"'- 'I X are identical. What is the relation between A,p, ~ ( uand ) $(U)'! Suppose that { Y,,, Ti, ... (1tIk k! e. 2. 1. 1 the claim number process was described by an intuitive approach.

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A Theory of Color Production I by Stieglitz J.

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