Universal Coding and Order Identification by Model Selection Methods by Élisabeth Gassiat

Author:Élisabeth Gassiat
Language: eng
Format: epub, pdf
ISBN: 9783319962627
Publisher: Springer International Publishing

Let be the inter-arrivals vector, and let be the element of maximizing :

Let be the set of ’s with inter-arrivals vector equal to M and with arrival time of the first 1 equal to t. Then

where is the inter-arrivals random vector:

Hence

and

The number of possible M is the number of such that , which is smaller than for some constant , by the Hardy-Ramanujan Theorem [14], so that for all integers ,

for some constant .

Moving on to the lower bound, let be a strictly positive even integer and be the set of subsets of with size . If , let be the distribution of a renewal source whose inter-arrival distribution is the uniform distribution over .

Let now W be a uniform random variable over and be the process with conditional distribution given . By Theorem 2.12

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