By Shaked M., Singpurwalla N. D.

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**Extra resources for A Bayesian approach for quantile and response probability estimation with applications to reliability**

**Sample text**

3.

Thus the continuous-time representation of the random walk is a step function, which coincides with the random walk at integer arguments. However, the step function is not the only possible continuous-time representation of the random walk. , by performing a linear interpolation. 1) 18 1. 12. Possible initial segments of the two continuous-time stochastic processes constructed from one realization of an uncentered random walk with uniform steps for the case n = 10. The step-function representation appears on the left, while the linear-interpolation representation appears on the right.

16) follows from Donsker’s theorem. 13), B is standard Brownian motion and the function space D is endowed with an appropriate topology. 3. Even though Brownian motion has a relatively simple characterization, it is a special stochastic process. For example, it has the self-similarity property observed in the plots (without limit). , the random vector (c−1/2 B(ct1 ), . . , c−1/2 B(ctk )) has a distribution in Rk that is independent of c for any positive integer k and any k time points ti , 1 ≤ i ≤ k, with 0 < t1 < · · · < tk ≤ 1.