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**A convergence theorem with application to Bayesian inference.**
*(Spanish.
English summary)*
Zbl 0654.62031

A theorem is proved showing that, under some boundary conditions, the following hypotheses:

1. \(\{X_ n\}\) is a sequence of continuous random variables which “approaches in probability” a numerical sequence \(\{a_ n\},\)

2. \(\{Y_ n\}\) is another sequence of random variables such that, for all n, the density function of \(Y_ n\) is proportional to the product of the density of \(X_ n\) and another density not depending on n,

lead to the fact that the random sequence \(\{Y_ n\}\) also “approaches in probability” \(\{a_ n\}\). We also show some related theorems as well as applications to Bayesian inference.

1. \(\{X_ n\}\) is a sequence of continuous random variables which “approaches in probability” a numerical sequence \(\{a_ n\},\)

2. \(\{Y_ n\}\) is another sequence of random variables such that, for all n, the density function of \(Y_ n\) is proportional to the product of the density of \(X_ n\) and another density not depending on n,

lead to the fact that the random sequence \(\{Y_ n\}\) also “approaches in probability” \(\{a_ n\}\). We also show some related theorems as well as applications to Bayesian inference.

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\textit{E. G. Sánchez-Manzano}, Trab. Estad. 1, No. 1, 30--41 (1986; Zbl 0654.62031)

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### References:

[1] | BURRILL, C. W. (1972):Measure, Integration and Probability, New York, McGraw-Hill. |

[2] | De GROOT, M. H. (1970):Optional Statistical Decisions, New York, McGraw-Hill. |

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