## Green's function methods in probability theory by Julian Keilson By Julian Keilson

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T It is also superconvex. See Note 3. 4) = -av. dV The variance a: of Fv(x) will be positive (apart from the degenerate distributions U(x-v), which can never be conjugate to non-degenerate distributions). We then have the following theorem. 1. For a non-degenerate conjugate family of distributions Fy(x) the mean /1-v of Fy(x) is a continuous monotonic decreasing function of V. The range of values assumed by /1-v for the conjugate family of distributions is of interest. 5) xM = inf{x: Fc,(x) = 11.

18) A process of independent increments with X0 = 0 and g(x,t) = g~>(x) is called a Wiener-Levy process. A simple indication that sample functions of the Wiener-Levy process are continuous with probability one will be given in Section 7. The Wiener-Levy process is the only homogeneous process with this property, terms XO + vt apart. 19) is of interest in its own right since it describes a simple type of Brownian motion with positive increments. 4) with XJ(O) 0, and XD (t) is a Wiener-Levy process.

The saddlepoint approximation of Daniels for estimating A (k)(x) at large deviations and the Central Limit approximation associated with an optimum conjugate transformation are identical. The asymptotic properties of the Green distribution G(x,t) for processes in continuous time and Green probabilities g~k) and gn(t) for lattice processes may be discussed along similar lines. The structure of the convergence strip and the properties of the conjugate family of distributions associated with this strip are examined in detail.