Inequalities for the rth Absolute Moment of a Sum of Random Variables, 1 r 2

Let X₁, X₂, ⋯, Xₙ be a sequence of random variables (r.v.'s) and put Sₘ = ∑ᵐν = 1 X_ν, 1 m n. It is well-known that {equation*}{(1)}E|S_n|^r nr - 1 ∑^nν = 1 E|X_ν|^r r > 1,{equation*} E|Sₙ|ʳ ∑ⁿν = 1 E|X_|nu|ʳ, r 1. However, if the r.v.'s satisfy the relations {equation*}{2}E(Xm + 1 S_m) = 0 a.s. 1 m n - 1,{equation*} it is possible to improve the first inequality considerably. The case $r > 2$ with independent r.v.'s will be treated elsewhere by one of the authors, von Bahr. If $r = 2$, we have, under (2), {equation*}{3}ES^2_n = ∑^nν = 1 EX^2_ν.{equation*} In the case 1 r 2, we will show that under (2) {equation*}{(4)}E|S_n|^r C(r, n) ∑^nν = 1e|X_ν|^r,{equation*} where $C(r, n)$ is a bounded function of r and n. In Theorem 2 we show that (4) is true with $C(r, n) = 2$. If the distribution of each Xm + 1 conditioned by Sₘ is symmetric about zero, one can put $C(r, n) = 1$ (Theorem 1). Further, if the r.v.'s satisfy the following conditions {equation*}{(5)}E(X_i Rₘᵢ) = 0a.s. 1 i m + 1 n,{equation*} where Rₘᵢ = ∑m + 1ν = 1, ν ≠ i X_ν it is possible to put C(r, n) = 2 - n⁻¹. The conditions (2) and (5) are fulfilled if the r.v.'s are independent and have zero means. In this case, however, it is possible to make $C(r, n)$ dependent on r, so that C(r, n) → 1 as r → 2. It is possible to show by an example, that (4) is not generally true with $C(r, n) = 1$ even in this case. If 1 r < s 2 and E|X_ν|ˢ < ∞, 1 ν n, it is generally better not to use (4) directly, but to use it together with E|Sₙ|ʳ (E|Sₙ|ˢ)r/s, so that E|Sₙ|ʳ (C(s, n) ∑ⁿν = 1 E|X_ν|ˢ)r/s. The case $r < 1$ is by (1) trivial.