The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation

doi:10.1016/j.spl.2022.109542

Statistics & Probability Letters

Volume 188, September 2022, 109542

https://doi.org/10.1016/j.spl.2022.109542 Get rights and content

Abstract

Let ${X_{n}; n \geq 1}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(Ω, ℋ, \hat{E})$ , $S_{n} = X_{1} + \dots + X_{n}$ . We consider the moments of ${max}_{n \geq 1} | S_{n} | / \sqrt{2 n log log n}$ . The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.

Section snippets

Introduction and basic settings

Let ${X_{n}; n \geq 1}$ be a sequence of independent and identically distributed (i.i.d) random variables on a probability space $(Ω, F, P)$ with mean zeros and finite variances, $S_{n} = X_{1} + \dots + X_{n}$ , $log x = ln max (e, x)$ and $a_{n} = \sqrt{2 n log log n}$ . Siegmund (1969) and Teicher (1971) studied the moments related to Hartman and Wintner (1941)’s law of the iterated logarithm. They obtained the sufficient and necessary conditions for the moments of the maximum of normalized partial sums ${max}_{n \geq 1} | S_{n} | / a_{n}$ to be finite. Recently, Dolera and

Main results

Let $(Ω, ℋ, \hat{E})$ be a sub-linear expectation space and ${X, X_{n}; n \geq 1}$ be a sequence random variables on it. Denote $S_{n} = \sum_{i = 1}^{n} X_{i}$ , $a_{n} = \sqrt{2 n log log n}$ .

When $X_{1}, X_{2}, \dots$ are i.i.d. random variables on a classical probability space $(Ω, F, P)$ , Siegmund (1969) and Teicher (1971) studied the moments of ${max}_{n \geq 1} | S_{n} | / a_{n}$ . Particularly, it is shown that $E [{max}_{n \geq 1} | S_{n} | / a_{n}] < \infty$ if and only if $E [X] = 0$ and $E [X^{2}] < \infty$ (see (16) of Siegmund, 1969). [1] established a reformulation of the Siegmund–Teicher inequality for $E [{max}_{n \geq 1} {| S_{n} |}^{r} / a_{n}^{r}]$ ( $r > 2)$ . In

Applications

As an application of Theorem 2.1, we show the law of the iterated logarithm for the moving average process in this section. Let ${Y_{i}; i \geq 1}$ be a sequence of i.i.d. random variables under the sub-linear expectation $\hat{E}$ . We consider the moving average process $X_{t} = \sum_{j = - \infty}^{t - 1} β_{j} Y_{t - j},$ where $B \hat{=} \sum_{j = - \infty}^{\infty} | β_{j} | < \infty, β \hat{=} \sum_{j = - \infty}^{\infty} β_{j} .$ Let $T_{n} = \sum_{t = 1}^{n} X_{t}$ . Define $Y_{t} = 0$ for $t = 0, - 1, - 2, \dots$ . Then $X_{t} = \sum_{j = - \infty}^{\infty} β_{j} Y_{t - j}, T_{n} = \sum_{j = - \infty}^{\infty} β_{j} \sum_{t = 1}^{n} Y_{t - j} .$ If ${Y_{i}}_{i = - \infty}^{\infty}$ is a bi-directional sequence of i.i.d. random variables, i.e., for any $i_{1} < i_{2} < \dots < i_{p}$ , ${Y_{i_{1}}, \dots, Y_{i_{p}}}$

Proofs

For proving the main results, we need several inequalities.

Lemma 4.1

Suppose that ${X_{1}, \dots, X_{n}}$ is a sequence of independent random variables on $(Ω, ℋ, \hat{E})$ . Set $S_{n} = \sum_{k = 1}^{n} X_{k}$ , $A_{n} (p, y) = \sum_{i = 1}^{n} \hat{E} [{(X_{i}^{+} \land y)}^{p}]$ and ${\overset{̆}{B}}_{n, y} = \sum_{i = 1}^{n} \overset{̆}{E} [{(X_{i} \land y)}^{2}]$ . Then, for all $p \geq 2$ , $x, y > 0$ , $0 < δ \leq 1$ , $\hat{V} (max_{k \leq n} \sum_{i = 1}^{k} (X_{i} - \overset{̆}{E} [X_{i}]) \geq x) \leq \hat{V} (max_{k \leq n} X_{k} > y) + 2 exp {p^{p}} {\frac{A_{n} (p, y)}{y^{p}}}^{\frac{δ x}{10 y}} + exp \{- \frac{x^{2}}{2 {\overset{̆}{B}}_{n, y} (1 + δ)}\} .$

Proof

The proof of (4.1) is the same as that of (3.2) of Zhang (2021) if we note $\hat{E} [e^{t (X_{k} \land y)}] = \overset{̆}{E} [e^{t (X_{k} \land y)}] \leq 1 + t \overset{̆}{E} [X_{k}] + \frac{e^{t y} - 1 - t y}{y^{2}} \overset{̆}{E} [{(X_{k} \land y)}^{2}], y > 0 . □$

Lemma 4.2

Suppose $X \in ℋ$ , $r > 0$ . Let $ς_{X}$ and $η_{X}$

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PengS.
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^☆: Research supported by grants from the NSF of China (Grant No. 11731012, 12031005), Ten Thousands Talents Plan of Zhejiang Province (Grant No. 2018R52042), NSF of Zhejiang Province (Grant No. LZ21A010002) and the Fundamental Research Funds for the Central Universities .

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Statistics & Probability Letters

The moments of the maximum of normalized partial sums related to laws of the iterated logarithm under the sub-linear expectation☆

Abstract

Section snippets

Introduction and basic settings

Main results

Applications

Proofs

Uniform rates of the Glivenko–Cantelli convergence and their use in approximating Bayesian inferences

Bernoulli

On the law of iterated logarithm

Amer. J. Math.

The law of the iterated logarithm for linear processes generated by a sequence of stationary independent random variables under the sub-linear expectation

Entopy

A new central limit theorem under sublinear expectations