Probability inequality concerning a chirp-type signal
© Perera; licensee Springer 2013
Received: 21 August 2012
Accepted: 11 March 2013
Published: 27 March 2013
Probability inequalities of random variables play important roles, especially in the theory of limiting theorems of sums of independent random variables. In this note, we give a proof of the following:
Let be a sequence of independent random variables such that , and . Then
It is shown that this result will be useful in estimating the parameters of a chirp-type statistical model and in establishing the consistency of estimators.
where is a linear process.
where ’s and ’s are as mentioned above. Models of this type are referred to as ‘chirp’ models [3, 4] and , and they have drawn the attention of many researchers [6, 7] and . Although we tried to find our main result given in (1) in the literature, we were unable to find one. We make use of the following basic results to establish our main result.
2 Basic results
Let be a sequence of nonzero real numbers. We say that if .
Definition 2 Let be a sequence of random variables. We say that if in probability. (So, if for each , we have .)
Lemma 3 Let be a sequence of random variables, and let be a sequence of real numbers with as . If , then .
Lemma 5 Let be a sequence of random variables, and let β be any real number. If and , then .
Lemma 6 Let μ be any real number. Let be a sequence of random variables such that . Then .
We now establish two lemmas that we need to prove the main result.
3 Main result
we can ignore the term provided, and we show the other term is .
which completes the proof of the theorem. □
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