Maximal Inequalities for Dependent Random Variables

Abstract

Maximal inequalities play a crucial role in many probabilistic limit theorem; for instance, the law of large numbers, the law of the iterated logarithm, the martingale limit theorem and the central limit theorem. Let X ₁, X ₂, … be random variables with partial sums S _k  = X ₁ + ⋯ + X _k . Then a maximal inequality gives conditions ensuring that the maximal partial sum M _n  = max_{1 ≤ i ≤ n}  S _i is of the same order as the last sum S _n . In the literature there exist large number of maximal inequalities if X ₁, X ₂, … are independent but much fewer for dependent random variables. In this paper, I shall focus on random variables X ₁, X ₂, … having some weak dependence properties; such as positive and negative In-correlation, mixing conditions and weak martingale conditions.

Mathematics Subject Classification (2010). Primary 60E15; Secondary 60F05

Appendix

In this appendix, I shall give a purely analytic solution to a certain recursive, functional inequality which is closely linked to the Rademacher-Menchoff inequalities of Sect. 2. But first let me prove the following simple lemma.

Lemma A.1

Let $(g_{i,j})_{(i,j)\in \boldsymbol{\Delta }_{0}}$ be a triangular schemes of non-negative numbers. Let $(i,j) \in \boldsymbol{ \Delta }_{2}$ be a given pair and let a > 0 and h ≥ 0 be given numbers satisfying

$$\displaystyle{g_{i,i} \vee (a\,g_{j,j}) \leq h\ \text{ and }\ \max _{i<k<k}\,(g_{i,k} + g_{k,j}) \leq (1 + \tfrac{1} {a})\,h\,.}$$

Then we have $\min \limits _{i<k\leq j}(g_{i,k-1} \vee (a\,g_{k,j})) \leq h$ .

Proof

I shall split the proof in three cases:

Case 1::

g _i, j−1 ≤ h. Since a g _j, j  ≤ h, we have g _i, j−1 ∨ (a g _j, j ) ≤ h.

Case 2::

g _i, i+1 ≤ h < g _i, j−1. Then there exists an integer i < k < j such that g _i, k−1 ≤ h ≤ g _i, k and since $h + g_{k.j} \leq g_{i,k} + g_{k,j} \leq (1 + \frac{1} {a})\,h$, we have g _i, k−1 ∨ (a g _k, j ) ≤ h.

Case 3::

g _i, i+1 > h. Since j ≥ 2, we have i < i + 1 < j and so we have $h + g_{i+1,j} \leq g_{i,i+1} + g_{i+1,j} \leq (1 + \frac{1} {a})\,h$ and since g _i, i  ≤ h, we have g _i, i ∨ (a g _i+1, j ) ≤ h.

Since the three cases exhaust all possibilities, we see that there exists an integer i < k ≤ j such that g _i, k−1 ∨ (a g _k, j ) ≤ h. □ 

Proposition A.2

Let $p,q \in \mathbb{R}_{+}$ be given numbers and let $D \subseteq \mathbb{R}_{+}$ be a non-empty set such that pt ∈ D and qt ∈ D for all t ∈ D. Let $\Gamma: \mathbb{R}_{+}^{2} \rightarrow \mathbb{R}_{+}$ , be an increasing homogeneous function and set $\Gamma (x,\infty ) = \Gamma (\infty,x) = \infty$ for all x ∈ [0,∞]. Let A _i,j ,B _i,j ,V _i,j : D → [0,∞] be given functions for $(i,j) \in \boldsymbol{ \Delta }_{0}$ and let $h: \mathbb{N}_{0} \rightarrow \mathbb{R}_{+}$ and $\xi: \mathbb{N}_{0} \rightarrow \mathbb{N}_{0}$ be increasing functions such that ξ(0) = 0 and

• (a) $A_{i,j}(t) \leq \Gamma (A_{k,j}(\,pt) + A_{i,k-1}(qt),B_{i,k}(t))\quad \forall (i,k,j,t) \in \nabla \times D\,,$

• (b) $A_{i,i}(t) \leq h(0)\,V _{i,i}(t)\ \text{ and }\ V _{i,j}(t) \leq V _{i,j+1}(t) <\infty \quad \,\forall \,(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D\,,$

• (c) $B_{i,j}(t) \leq h(\xi (\,j - i))\,V _{i,j}(t)\quad \forall \,(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D\,.$

Then we have A _i,j (t) < ∞ and B _i,j (t) < ∞ for all $(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D$ . Let s ≥ 0 be a given number satisfying $h(0) \leq \Gamma (s\,h(0),h(1))$ and let us define

$$\displaystyle{ \Upsilon _{s} =\{ (i,j,t) \in \boldsymbol{ \Delta }_{1} \times D\mid h_{s}^{\Gamma }(\xi (\,j - i))\,V _{ i,j}(t) <A_{i,j}(t)\}\,. }$$

Then $h_{s}^{\Gamma }$ is increasing and if

• (d) $\min \limits _{k\in D_{i,j}^{\xi }}\,(V _{i,k-1}(qt) + V _{k,j}(\,pt)) \leq s\,V _{i,j}(t)\quad \forall \,(i,j,t) \in \Upsilon _{r}\,,$

then we have $A_{i,j}(t) \leq h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$ for all $(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D$ .

Proof

By (b) and (c), we have A _i, i (t) < ∞ for all $(i,t) \in \mathbb{N}_{0} \times D$ and B _i, j (t) < ∞ for all $(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D$. Let n ≥ 0 be a given integer such that A _i, i+n (t) < ∞ for all $(i,t) \in \mathbb{N}_{0} \times D$ and let $(i,t) \in \mathbb{N}_{0} \times D$ be given. Since pt ∈ D and qt ∈ D, we have A _i, i+n (qt) + A _{i+n+1, i+n+1}( pt) < ∞. Hence, by (a), we see that A _i, i+n+1(t) < ∞ and so by induction, we have A _i, j (t) < ∞ for all $(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D$.

Suppose that $h(0) \leq \Gamma (s\,h(0),h(1))$. By (2.6), we have $h_{s}^{\Gamma }(0) = h(0) \leq \Gamma (s\,h(0),h(1)) = h_{r}^{\Gamma }(1)$. Let n ≥ 1 be a given integer such that $h_{s}^{\Gamma }(n - 1) \leq h_{s}^{\Gamma }(n)$. By (2.6), we have $h_{s}^{\Gamma }(n + 1) = \Gamma (s\,h_{s}^{\Gamma }(n),h(n + 1))$ and since $\Gamma$ and h are increasing we have $h_{s}^{\Gamma }(n + 1) \geq \Gamma (s\,h_{s}^{\Gamma }(n - 1)\,,h(n)) = h_{s}^{\Gamma }(n)$. So by induction, we see that $h_{s}^{\Gamma }$ is increasing.

Suppose in addition that (d) holds. Since ξ(0) = 0, we have $h(0) = h_{s}^{\Gamma }(\xi (0))$ and so by (b) we have $A_{i,j}(t) \leq h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$ for all $(i,j,t) \in \boldsymbol{ \Delta }^{0} \times D$. Let n ≥ 0 be a given integer such that $A_{i,j}(t) \leq f_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$ for all $(i,j,t) \in \boldsymbol{ \Delta }^{n} \times D$. Let $(i,j,t) \in \boldsymbol{ \Delta }^{n+1} \times D$ be given and let me show that $A_{i,j}(t) \leq h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$.

If j − i ≤ n, this follows from the induction hypothesis and if $A_{i,j}(t) \leq h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$, this holds trivially. So suppose that j − i = n + 1 and $h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t) <A_{i,j}(t)$ and set ν = ξ(n + 1). Since j − i = n + 1 ≥ 1, we have $(i,j,t) \in \Upsilon _{s}$. Recall that min _{k ∈ ∅} a _k  = ∞ and V _i, j (t) < ∞. So by (d) there exists k ∈ D _i, j ^ξ such that V _i, k−1(qt) + V _k, j ( pt) ≤ s V _i, j (t).

Since k ∈ D _i, j ^ξ, we have ξ(j − k) ∨ξ(k − i − 1) ≤ ν − 1 and since $h_{s}^{\Gamma }$ is increasing, we have $h_{s}^{\Gamma }(\xi (\,j - k)) \leq h_{s}^{\Gamma }(\nu -1)$ and $h_{s}^{\Gamma }(\xi (k - i - 1)) \leq h_{s}^{\Gamma }(\nu -1)$. Since h and ξ are increasing and k − i ≤ n + 1, we have h(ξ(k − i)) ≤ h(ν) and since i < k ≤ j and n + 1 = j − i, we have $(i,k - 1) \in \boldsymbol{ \Delta }^{n}$ and $(k,j) \in \boldsymbol{ \Delta }^{n}$. Since t ∈ D, we have pt, qt ∈ D and so by induction hypothesis, we have $A_{i,k-1}(qt) \leq h_{s}^{\Gamma }(\nu -1))\,V _{i,k-1}(qt)$ and $A_{k,j}(\,pt) \leq h_{s}^{\Gamma }(\nu -1)\,V _{k,j}(\,pt)$. Since V _i, k−1(qt) + V _k, j ( pt)) ≤ s V _i, j (t), we have

$$\displaystyle{A_{i,k-1}(qt) + A_{k,j}(\,pt) \leq s\,h_{s}^{\Gamma }(\nu -1)\,V _{ i,j}(t)\,.}$$

By (c), we have B _i, k (t) ≤ h(ξ(k − i)) V _i, k (t) and since ξ and h are increasing and k − i ≤ j − i = n + 1, we have h(ξ(k − i)) ≤ h(ν). So we have B _i, k (t) ≤ h(ν) V _i, k (t) and by (a) and homogeneity and monotonicity of $\Gamma$ we have

$$\displaystyle\begin{array}{rcl} A_{i,j}(t)& \leq & \Gamma (A_{i,k-1}(qt) + A_{k,j}(\,pt),B_{i,k}(t)) {}\\ & \leq & \Gamma (s\,h_{s}^{\Gamma }(\nu -1),\,h(\nu ))\,V _{ i,j}(t) = h_{s}^{\Gamma }(\nu )\,V _{ i,j}(t)\,. {}\\ \end{array}$$

Hence, by induction we see that $A_{i,j}(t) \leq h_{s}^{\Gamma }(\xi (\,j - i))\,V _{i,j}(t)$ for all $(i,j,t) \in \boldsymbol{ \Delta }_{0} \times D$. □