Research | Open | Published:
Probability inequality concerning a chirp-type signal
Journal of Inequalities and Applicationsvolume 2013, Article number: 131 (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.
In  Whittle (1952) considered the problem of estimating the parameters of a sine wave:
where ’s are the observations and ’s are independent, identically distributed random variables with mean zero and finite(unknown) variance. Whittle’s solution to the problem of estimating the parameters and the proof of consistency of parameters of the above sine wave used arguments which are not mathematically rigorous. In 1973, a rigorous solution to Whittle’s problem was given by Walker . In his proof of consistency of parameters, he used the following result:
where is a linear process.
In this paper, we extend the above result to obtain
We came across this problem while attempting to establish the consistency of the parameters of the model
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
Definition 1 Let be a sequence of random variables. We say that if for given , there exists M such that
Let be a sequence of nonzero real numbers. We say that if .
Lemma 1 Let be a sequence of random variables, and let and be two sequence of nonzero real numbers. Then
Lemma 2 Let be a sequence of random variables, let be a sequence of real numbers, and let q be a positive real number. Then
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 4 Let be a sequence of random variables, and let β be a positive real number. 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.
Lemma 7 Let ; , be complex numbers. Then
Proof The is the sum of the terms in the table:
We can add by summing along the diagonals. So, the sum is equal to
Lemma 8 Let ; be complex numbers. Then
Proof Since , by using in Lemma 7, then
3 Main result
Theorem 1 Let be a sequence of independent random variables such that , and . Then
Proof Letting in Lemma 8, we see that
Using the triangle inequality, we obtain
Fix s and consider . By Lemma 8 with , we obtain
and it follows from the triangle inequality that
Substituting this in equation (2), we obtain
It follows that
Taking the expectation of both sides of the above inequality, we obtain
We claim that
We note that since
we can ignore the term provided, and we show the other term is .
Taking in Schwarz’s inequality yields . Using this on the right-hand side of the above inequality, we get
Using Schwarz’s inequality again on the first term on the right, we get
Consider . Since , is the unique largest subscript, the ’s are independent and , we have
Now consider the term . Since , and the ’s are independent,
Similarly, since , , and the ’s are independent,
It follows by virtue of Lemma 6 that
Using Lemma 3, we obtain
which completes the proof of the theorem. □
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The author declares that she has no competing interests.