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Extra info for An Introduction to Statistical Signal Processing last edition

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Such limits might not exist, or they might exist and not converge to the same thing for different repetitions of the same experiment. Even when the limits do exist there is no guarantee they will behave as intuition would suggest when one tries to do calculus with probabilities, that is, to compute probabilities of complicated events from those of simple related events. Attempts to get around these problems uniformly failed and probability was not put on a rigorous basis until the axiomatic approach was completed by Kolmogorov.

In fact, the notations are interchangeable; we could denote waveforms as {x(t); t ∈ } or as {xt ; t ∈ }. The notation using subscripts for sequences and parentheses for waveforms is the most common, and we will usually stick to it. Yet another notation for discrete time signals is x[n], a common notation in the digital signal processing literature. It is worth remembering that vectors, sequences, and waveforms are all just indexed collections of numbers; the only difference is the index set: finite for vectors, countably infinite for sequences, and continuous for waveforms.

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 → ∞. By a similar argument, if we are told that each set in a decreasing sequence Fn is an event, then the limit must be an event, since it is an intersection of a countable number of events. Thus, for example, if we are told that all finite intervals of the form (a, b) are events, then the points of singleton sets must also be events, since a point {a} is the limit of the decreasing sequence of sets (a − 1/n, a + 1/n).

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