Aspects of bivariate CDF iteration
Soetarto Sastrosoewignjo · 1981
Bivariate iteration may be viewed as generating a sequence of bivariate functions, resulting from certain successive operations. Stochastically, this occurs in the following two situations: The first situation concerns a sequence of two-valued bivariate functions, a natural example of which may be found in bivariate branching processes. The second situation, considered in this research, concerns a sequence of single-valued bivariate functions, referred to here as bivariate CDF iterations of the form F(,n)(x,y) = (lamda)('(n))(G(x),H(y),F(x,y)) = (lamda)(G(,n-1)(x),H(,n-1)(y),F(,n-1)(x,y)), where G(,n)(x) and H(,n)(y), the marginals of F(,n)(x,y), can be expressed, respectively, in terms of suitable functions (phi)(,1) and (phi)(,2) as G(,n)(x) = (phi)(,1)(G(,n-1)(x)) and H(,n)(y) = (phi)(,2)(H(,n-1)(y)). A natural example of this type of CDF iteration occurs in terms of the joint CDF's of the values of a certain pair of correlated games of perfect information, with iid terminal payoffs having the common cdf F; this is referred to as bivariate "maximin" or "minimax" CDF iteration. An analogous operation, bivariate "maximum" or "minimum" iteration, occurs also in bivariate extremes, on geometric subsequences of the integers. Given the marginal limit laws, the issues of weak convergence and asymptotic independence are studied for both bivariate maximin or minimax CDF iteration and bivariate extremes, with the latter restricted to the case of essentially bounded iid bivariate random vectors;In both cases, the upper Frechet bound (i.e., UFB) appears to play a dominant role in establishing stochastic dependence. In the first case, the UFB provides essentially the only instance of dependence, while, in the latter case, any mixture involving the UFB provides such an instance. In the first case, asymptotic independence holds, essentially, unless the UFB is achieved at a certain critical point, while, in the latter case, asymptotic independence is established for the case when F(x,y) (LESSTHEQ) G(x)H(y) at (x,y) values near a certain critical point.