Green's Function Methods in Probability Theory by Julian Keilson

By Julian Keilson

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2). t A second example might be obtained from the theory of collective risk for an insurance operation, with Xv(t) representing the capital on hand, vt the relatively smooth input from premiums and ti the sporadic lump sum payments out - when bankruptcy is ignored. The same type of structure appears in connection with the Takacs virtual waiting-time process of queuing theory, which we will discuss in Chapters IV and V. The distribution Jt(x,t) of Xv(t) is simply related to that for X(t) by Fv(x,t) = F(x-vt,t).

4) lZ If ¢(z) is absolutely integrable over (-oo, oo) then F(x) is absolutely continuous. I where K is real. 5) -oo is then continuous for all x. Of frequent importance is the following auxiliary result (Titchmarsh, 1948). If F(x) is absolutely continuous and its density f(x) is of bounded variation in the neighbourhood of x = x', then .!. 5') e-ix' z cf,(z)dz. T -T (g) The convolution of two distributions F 1 (x) and F 2 (x) defined by F(x) = fF1 (x-x')dF2 (x') is symmetrical in F 1 and F 2 and is itself a distribution.

8) do not converge beyond V = 0. 8) converge beyond V = 0 and the convergence strip contains the real z axis in its interior. The boundaries of the convergence strip for ¢(z) are not in general the natural boundaries of the characteristic function which will have an analytic continuation beyond the boundaries of the strip. When the derivative of ¢(iV) becomes infinite as V-+ Vmax and V-+ Vmin the convergence strip of ¢(z) will be said to be complete. A plot of ¢(iV) against V in the convergence strip is shown in Fig.

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