Algebraic Moments, Elementary Exponential Inequalities

Abstract

In this chapter, we start by giving upper bounds for algebraic moments of partial sums from a strongly mixing sequence. These inequalities are similar to Rosenthal’s inequalities (Rosenthal, Israel J Math, 8:273–303, 1970) concerning moments of sums of independent random variables.

Exercises

(1) Let $(X_{i})_{i\in {\mathrm{Z\!\!\;\!\! Z}}}$ be a sequence of centered real-valued random variables with finite fourth moments, and let $(\alpha _n)_{n\ge 0}$ be defined by (2.1).

(a) Let $i \le j \le k \le l$ be natural integers. Prove that

$$ |\mathrm{I\! E}(X_i X_j X_k X_l)| \le 2\int _0^1 {1\! {\mathrm{I}}}_{u< \alpha _{j-i} } {1\! {\mathrm{I}}}_{u< \alpha _{l-k} } Q_i (u) Q_j (u) Q_k (u) Q_l (u) du . \qquad {(1)} $$

(b) Prove that

$$ \mathrm{I\! E}(S_n^4) \le 12 \sum _{1\le i\le j \le k\le l \le n} |\mathrm{I\! E}(X_i X_j X_k X_l)| (1 + {1\! {\mathrm{I}}}_{j<k}) . $$

(c) Prove that

$$ \mathrm{I\! E}(S_n^4) \le 24 \sum _{j=1}^n \sum _{k=1}^n \int _0^1 [\alpha ^{-1} (u) \wedge n]^2 Q_j^2 (u) Q_k^2 (u) du . \qquad {(2)} $$

(d) Suppose now that $\Vert X_k \Vert _\infty \le 1$ for any k in [1, n]. Derive from the above inequalities that

$$ \mathrm{I\! E}(S_n^4) \le 24 n^2 \sum _{m=0}^{n-1} (2m+1) \alpha _m . \qquad {(3)} $$

Compare (3) with (2.13) and (2.11).

(2) Let $(S_n)_{n \ge 0}$ be a martingale sequence in $L^p$ for some $p>2$ and $X_n = S_n - S_{n-1}$. Either use Inequality (2.3) in Pinelis (1994) or adapt the proof of Theorem 2.5 to prove the inequality (4) below, given in Rio (2009):

$$ \Vert S_n \Vert _p^2 \le \Vert S_0 \Vert _p^2 + (p-1) \sum _{k=1}^n \Vert X_k \Vert _p ^2 . \qquad {(4)} $$