An Introduction to Statistical Signal Processing new edition by Robert Gray

By Robert Gray

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These considerations motivate the following general definition. Given a sample space Ω (such as the real line ) and an arbitrary class G of subsets of Ω — usually the class of all open intervals of the form (a, b) when Ω = — define σ(G), the sigma-field generated by the class G, to be the smallest sigma-field containing all of the sets in G, where by “smallest” we mean that if F is any sigma-field and it contains G, then it contains σ(G). ) For example, as noted before, we might require that a sigma-field of the real line contain all intervals; then it would also have to contain at least all complements of intervals and all countable unions and intersections of intervals and all countable complements, unions, and intersections of these results, ad infinitum.

Of sets in the sigma-field. Since the limit of the sequence is defined as a union and since the union of a countable number of events must be an event, then the limit must be an event. For example, if we are told that the sets [1, 2 − 1/n) are all events, then the limit [1, 2) must also be an event. If we are told that all finite intervals of the form (a, b), where a and b are finite, are events, then the semi-infinite interval (−∞, b) must also be an event, since it is the limit of the sequence of sets (−n, b) and n → ∞.

26) is true and Gn is an arbitrary sequence of events, then define the increasing sequence n Fn = Gi . 19), since ∞ ∞ Fn = lim Fn ∈ F . Gi = i=1 n=1 n→∞ Examples As we have noted, for a given sample space the selection of an event space is not unique; it depends on the events to which it is desired to assign probabilities and also on analytical limitations on the ability to assign probabilities. We begin with two examples that represent the extremes of event spaces — one possessing the minimum quantity of sets and the other possessing the maximum.

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