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2 results

  • Distances on circles, toruses and spheres

  • B. W. Silverman
  • Journal:
    Journal of Applied Probability / Volume 15 / Issue 1 / March 1978
    Published online by Cambridge University Press:
    14 July 2016, pp. 136-143
    Print publication:
    March 1978

  • Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj) for some suitable function g of two arguments and independent uniformly distributed random variables Y1, Y2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.

  • Limit theorems for dissociated random variables

  • B. W. Silverman
  • Journal:
    Advances in Applied Probability / Volume 8 / Issue 4 / December 1976
    Published online by Cambridge University Press:
    01 July 2016, pp. 806-819
    Print publication:
    December 1976

  • Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj, …, Yz) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.