By Daniel Dugue, E. Lukacs, V. K. Rohatgi

**Read Online or Download Analytical Methods in Probability Theory. Proc. conf. Oberwolfach, 1980 PDF**

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**Extra resources for Analytical Methods in Probability Theory. Proc. conf. Oberwolfach, 1980**

**Example text**

For this choice of Y and Z, (16) yie]ds +¢o4~o (18) 0 = f f [G(y+z) - G(y)G(z)]dP(Y

The advantage of the method of characteristic functions is that it is a promising avenue for extending a characterization theorem to a stability theorem. Namely, if instead of independence, we assume only almost independence in a well defined sense, then the deviation of the joint characteristic function from the corresponding product (which would have been obtained under independence) can be estimated. On the other hand, there are stand- ard methods in probability theory to estimate the difference of two distribution functions when an estimate is available for the difference of their characteristic functions.

Therefore P(X~Xr: n < x+ Ax) ~ r(7)Fr-l(x)[l-F(x)]n-r[F(x+Ax)-F(x)] the error being 0[(Ax)2]. If we divide by Ax and let Ax ÷ 0 we get an exact formu- la for fr:n(X), which is indeed the same as (4). This same method yields the following result. tinuous.