Some inequalities for characteristic functions

Abstract

In a recent paper, Matysiak and Szablowski [V. Matysiak, P.J. Szablowski, Theory Probab. Appl. 45 (2001) 711-713] posed an interesting conjecture about a lower bound of real-valued characteristic functions. Under a suitable moment condition on distributions, we prove the conjecture to be true. The unified approach proposed here enables us to obtain new inequalities for characteristic functions. We also show by example that the improvement in the bounds is significant if more information about the distribution is available.

Some inequalities for characteristic functions. II

“…. , y m together satisfy conditions (8)-(10), we can show, proceeding along the same lines as in the proof of Lemma 3 in Hu and Lin [3], that the set of symmetric sums s 1 , s 2 , . .…”

“…Actually, in the previous paper we obtained the following two general results for lower and upper bounds of the real part of characteristic functions (Hu and Lin [3], . For convenience, denote by supp(X) the support of X and by |supp(X)| the number of support points of X. Theorem 1.…”

“…This together with δ 0 = g 2m (0)/g 2m+1 (0) and the above inequality for h completes the proof. [3]). This implies that the set {y i } m i=1 is the same as in Lemma 4(b).…”

Some inequalities for Laplace transforms

“…Let X be a symmetric random variable with α 6 < ∞ and characteristic function f . Then there exists a constant δ = δ(α 2 , α 4 , α 6 ) > 0 such that the following inequality holds (if the support of X contains at least four points): f (t) p 1 cos(y 1 t) + p 2 cos(y 2 t) for |t| δ,…”

“…Moreover, by solving a differential inequality, we are able to find the range of argument for which inequality (1) holds under a stronger moment condition E(|X| 9 ) < ∞. Actually, in two previous papers we have obtained some general results for lower and upper bounds of the real part of characteristic functions (Hu and Lin [5,6]). …”