An Extension of Feller's Strong Law of Large Numbers

TL;DR

This paper extends Feller's strong law of large numbers to Banach space valued random variables by linking it with the Kolmogorov-Marcinkiewicz-Zygmund law, using recent symmetrization and probability inequality tools.

Contribution

It establishes a new almost sure convergence result in Banach spaces connecting two classical laws of large numbers, utilizing recent advanced probabilistic tools.

Findings

Proves a generalized strong law of large numbers in Banach spaces.

Links classical laws through new convergence conditions.

Utilizes recent symmetrization and probability inequalities.

Abstract

~This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller's strong law of large numbers in a Banach space setting. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of independent and identically distributed Banach space valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let $\{a_{n}; n \geq 1\}$ and $\{b_{n}; n \geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left\{b_{n}/a_{n};~ n \geq 1 \right\}$ is a nondecreasing sequence. We show that \[ \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0~~\mbox{almost surely} \] for every Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$ if $S_{n}/a_{n} \rightarrow 0$ almost surely for every symmetric Banach space valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.

Equations

$$\frac{S_n-n\mathbb{E}\left(XI\{\Vert X\Vert \le b_n\}\right)}{b_n}\to 0\text{mbox}almostsurely$$

$$b_n{n}^{-1/(1+ϵ)}\uparrow \text{mbox}and{b}_n/n\downarrow ,$$

$$b_n/n\uparrow .$$

$$\underset{n\to \infty }{\lim }\frac{S_n}{b_n}=0\text{mbox}almostsurely(a.s.)or\underset{n\to \infty }{\mathrm{limsup}}\frac{\left|S_n\right|}{b_n}=\infty \text{mbox}a.s.$$

$$\sum _{n=1}^{\infty }\mathbb{P}(|X|>b_n)<\infty \text{mbox}or=\infty .$$

$$\mathcal{C}(\mathbf{B})=\left\{(v_1,v_2,...)\in {\mathbf{B}}^{\infty }:\sum _{n=1}^{\infty }{R}_n{v}_n\text{mbox}convergesinprobability\right\}.$$

$$\mathbb{E}{\Vert \sum _{n=1}^{\infty }{R}_n{v}_n\Vert }^p\le C\sum _{n=1}^{\infty }\Vert v_n{\Vert }^p\text{mbox}forall(v_1,v_2,...)\in \mathcal{C}(\mathbf{B}).$$

$$\displaystyle{\bf(i)}\quad\mbox{The Banach space $\mathbf{B}$ is of Rademacher type $p$.}$$

$$\displaystyle{\bf(ii)}\quad\mbox{For every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. {\bf B}-valued variables},$$

$$\underset{n\to \infty }{\lim }\frac{S_n}{n^{1/p}}=0\text{mbox}a.s.ifandonlyif\mathbb{E}\Vert X{\Vert }^p<\infty \text{mbox}and\mathbb{E}X=0.$$

$$\underset{n\to \infty }{\lim }{a}_n=\infty \text{mbox}and{b}_n/a_n\uparrow .$$

$$\underset{n\to \infty }{\lim }\frac{S_n}{a_n}=0\text{mbox}a.s.whenever\sum _{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >a_n)<\infty .$$

$$\underset{n\to \infty }{\lim }\frac{S_n-{\gamma }_n}{b_n}=0\text{mbox}a.s.or\underset{n\to \infty }{\mathrm{limsup}}\frac{\Vert S_n-{\gamma }_n\Vert }{b_n}=\infty \text{mbox}a.s.$$

$$\sum _{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >b_n)<\infty \text{mbox}or=\infty .$$

$$\underset{n\to \infty }{\lim }\frac{S_n}{a_n}=\underset{n\to \infty }{\lim }\frac{S_n}{n^{1/(1+ϵ)}}=0\text{mbox}a.s.ifandonlyif\mathbb{E}|X{|}^{1+ϵ}<\infty (\text{mbox}i.e.,\sum _{n=1}^{\infty }\mathbb{P}(|X|>a_n)<\infty ).$$

$$b_n/a_n=b_n{n}^{-1/(1+ϵ)}\uparrow ,$$

$$\underset{n\to \infty }{\lim }\frac{S_n-{\gamma }_n}{b_n}=0\text{mbox}a.s.or\underset{n\to \infty }{\mathrm{limsup}}\frac{\left|S_n-{\gamma }_n\right|}{b_n}=\infty \text{mbox}a.s.$$

$$\sum _{n=1}^{\infty }\mathbb{P}(|X|>b_n)<\infty \text{mbox}or=\infty ,$$

$$\underset{n\to \infty }{\lim }\frac{S_n-{\gamma }_n}{b_n}=0\text{mbox}a.s.ifandonlyif\sum _{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >b_n)<\infty ,$$

$$\underset{n\to \infty }{\mathrm{limsup}}\frac{\Vert S_n-{\gamma }_n\Vert }{b_n}=\infty \text{mbox}a.s.ifandonlyif\sum _{n=1}^{\infty }\mathbb{P}(\Vert X\Vert >b_n)=\infty .$$

$$0<\underset{n\to \infty }{\mathrm{limsup}}\frac{\Vert S_n-{\gamma }_n\Vert }{b_n}<\infty \text{mbox}a.s.$$

$$\underset{n\to \infty }{\lim }{a}_n=\infty \text{mbox}and{n}^{1/p}/a_n\uparrow .$$

$$b_n/n^{1/p}\uparrow .$$

$$\sum _{n=1}^{\infty }\mathbb{P}\left(\Vert X\Vert >b_n\right)<\infty$$

$$\underset{n\to \infty }{\lim }\frac{{\sum }_{i=1}^n\left(X_i-\mathbb{E}\left(XI\{\Vert X\Vert \le b_i\}\right)\right)}{b_n}=0\text{mbox}a.s.$$

$$\underset{n\to \infty }{\lim }\frac{{\sum }_{i=1}^n{X}_i}{b_n}=0\text{mbox}a.s.$$

$$0<b_n\uparrow ,b_n\sum _{i=n}^{\infty }\frac{1}{b_i}=O(n),\text{mbox}and\sum _{n=1}^{\infty }\mathbb{P}\left(|X|>b_n\right)<\infty .$$

$$I(1)=\left\{i;b_i\le 2\right\}\text{mbox}andI(m)=\left\{i;2^{m-1}<b_i\le 2^m\right\},m\ge 2.$$

$$\{1,2,3,...,n,...\}=\bigcup _{m=1}^{\infty }I(m).$$

$$k_1<k_2<...<k_n<...,m_1<m_2<...<m_n<...,$$

Full text

An Extension of Feller’s Strong Law of Large Numbers

**Deli Li1, Han-Ying Liang*2,Corresponding author: Han-Ying Liang (Telephone: 86-21-65983242, FAX: 86-21-65983242), and Andrew Rosalsky3

1Department of Mathematical Sciences, Lakehead University,

Thunder Bay, Ontario, Canada

2Department of Mathematics, Tongji University,

Shanghai, China

3Department of Statistics, University of Florida,

Gainesville, Florida, USA

Abstract  This paper presents a general result that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let $\{X,X_{n};n\geq 1\}$ be a sequence of independent and identically distributed Banach space valued random variables and set $S_{n}=\sum_{i=1}^{n}X_{i},~{}n\geq 1$. Let $\{a_{n};n\geq 1\}$ and $\{b_{n};n\geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n\rightarrow\infty}a_{n}=\infty$ and $\left\{b_{n}/a_{n};~{}n\geq 1\right\}$ is a nondecreasing sequence. We show that

[TABLE]

for every Banach space valued random variable $X$ with $\sum_{n=1}^{\infty}\mathbb{P}(\|X\|>b_{n})<\infty$ if $S_{n}/a_{n}\rightarrow 0$ almost surely for every symmetric Banach space valued random variable $X$ with $\sum_{n=1}^{\infty}\mathbb{P}(\|X\|>a_{n})<\infty$. To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.

Keywords  Feller’s strong law of large numbers $\cdot$ Kolmogorov-Marcinkiewicz-Zygmund strong law of large numbers $\cdot$ Rademacher type $p$ Banach space $\cdot$ Sums of independent random variables

Mathematics Subject Classification (2000): 60F15 $\cdot$ 60B12 $\cdot$ 60G50

Running Head: On Feller’s Strong Law of Large Numbers

1 Introduction and the main result

We begin with stating Feller’s (1946) strong law of large numbers (SLLN) as follows.

Theorem A. (Feller’s SLLN. Theorems 1 and 2 of Feller (1946)). Let $\{X,X_{n};n\geq 1\}$ be a sequence of independent and identically distributed (i.i.d.) real-valued random variables, and let $S_{n}=\sum_{i=1}^{n}$, $n\geq 1$. Let $\{b_{n};n\geq 1\}$ be an increasing sequence of positive real numbers. Suppose that one of the following two sets of conditions holds:

(i)*  For some $0<\delta<1$, $\mathbb{E}|X|^{1+\delta}=\infty$, $\mathbb{E}X=0$, and there exists an $\epsilon$ with $0\leq\epsilon<1$ such that*

[TABLE]

(ii)*  $\mathbb{E}|X|=\infty$ and*

[TABLE]

Then we have

[TABLE]

according as

[TABLE]

Feller’s SLLN is a remarkable limit theorem concerning sums of i.i.d. random variables. From the internet, one can find that Feller’s SLLN has received more than 150 citations where many of them have been received within the most recent 5 years.

This paper presents a general result in a Banach space setting that allows for establishing a link between the Kolmogorov-Marcinkiewicz-Zygmund SLLN and Feller’s SLLN.

For stating our main result, we introduce some notation as follows. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and let $(\mathbf{B},\|\cdot\|)$ be a real separable Banach space equipped with its Borel $\sigma$-algebra $\mathcal{B}$ ($=$ the $\sigma$-algebra generated by the class of open subsets of $\mathbf{B}$ determined by $\|\cdot\|$). A B-valued random variable $X$ is defined as a measurable function from $(\Omega,\mathcal{F})$ into $(\mathbf{B},\mathcal{B})$. Let $\{X,X_{n};~{}n\geq 1\}$ be a sequence of i.i.d. B-valued random variables and put $S_{n}=\sum_{i=1}^{n}X_{i}$, $n\geq 1$. Let $\{R,R_{n};~{}n\geq 1\}$ be a Rademacher sequence; that is, $\{R_{n};~{}n\geq 1\}$ is a sequence of i.i.d. random variables with $\mathbb{P}\left(R=1\right)=\mathbb{P}\left(R=-1\right)=1/2$. Let $\mathbf{B}^{\infty}=\mathbf{B}\times\mathbf{B}\times\mathbf{B}\times\cdots$ and define

[TABLE]

Let $1\leq p\leq 2$. Then $\mathbf{B}$ is said to be of Rademacher type $p$ if there exists a constant $0<C<\infty$ such that

[TABLE]

The following remarkable theorem, which is due to de Acosta (1981), provides a characterization of Rademacher type $p$ Banach spaces.

Theorem B. (de Acosta (1981)). Let $1\leq p<2$. Then the following two statements are equivalent:

[TABLE]

[TABLE]

The main result of this paper is the following theorem.

Theorem 1.1**.**

Let $(\mathbf{B},\|\cdot\|)$ be a real separable Banach space. Let $\{a_{n};n\geq 1\}$ and $\{b_{n};n\geq 1\}$ be increasing sequences of positive real numbers such that

[TABLE]

Suppose that for every symmetric sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables,

[TABLE]

Then, for every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables, we have that

[TABLE]

according as

[TABLE]

Here and below $\gamma_{n}=n\mathbb{E}\left(XI\{\|X\|\leq b_{n}\}\right)$, $n\geq 1$.

Remark 1.1**.**

We now can see how Feller’s SLLN can be easily derived from the Kolmogorov-Marcinkiewicz-Zygmund SLLN and Theorem 1.1 above. For given $\epsilon$ with $0\leq\epsilon<1$, write $a_{n}=n^{1/(1+\epsilon)},~{}n\geq 1$. The celebrated Kolmogorov-Marcinkiewicz-Zygmund SLLN (see Kolmogoroff (1930) for $\epsilon=0$ and Marcinkiewicz and Zygmund (1937) for $0<\epsilon<1$) asserts that, for every symmetric sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. real-valued (i.e., ${\bf B}=\mathbb{R}$) random variables

[TABLE]

Then by Theorem 1.1, for every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. real-valued random variables and every increasing sequence $\{b_{n};n\geq 1\}$ of positive real numbers with

[TABLE]

we have

[TABLE]

according as

[TABLE]

where $\gamma_{n}=n\mathbb{E}\left(XI\{|X|\leq b_{n}\}\right)$, $n\geq 1$. That is, Feller’s SLLN follows from the Kolmogorov-Marcinkiewicz-Zygmund SLLN and Theorem 1.1 above.

Remark 1.2**.**

Under the assumptions of Theorem 1.1, it follows from the conclusion of Theorem 1.1 that, for every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables,

[TABLE]

[TABLE]

Hence under the assumptions of Theorem 1.1, there does not exist a sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables such that

[TABLE]

Also, combining Theorem 1.1 and Theorem B above, we immediately obtain the following two remarks.

Remark 1.3**.**

Let $1\leq p<2$ and let $\{a_{n};n\geq 1\}$ be an increasing sequences of positive real numbers such that

[TABLE]

Let $(\mathbf{B},\|\cdot\|)$ be a real separable Banach space such that (1.2) holds for every symmetric sequence $\{X,X_{n};n\geq 1\}$ of i.i.d. B-valued random variables. Then the Banach space ${\bf B}$ is of Rademacher type $p$.

Remark 1.4**.**

Let $1\leq p<2$ and let $\{b_{n};n\geq 1\}$ be a sequence of positive real numbers such that

[TABLE]

If $\mathbf{B}$ is of Rademacher type $p$, then for every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables, (1.3) and (1.4) are equivalent.

Remark 1.5**.**

Remark 1.4 should be compared with Theorem 4 of Adler, Rosalsky, and Taylor (1989) and with (in the unweighted case) the key lemma (Lemma 6) of that article. In Lemma 6 (wherein $1\leq p\leq 2$) and Theorem 4 (wherein $1<p\leq 2$) of Adler, Rosalsky, and Taylor (1989), for a sequence $\left\{X,X_{n};~{}n\geq 1\right\}$ of i.i.d. random variables in a real separable Rademacher type $p$ Banach space and a sequence of positive constants $b_{n}\uparrow\infty$, conditions are provided under which

[TABLE]

ensures that

[TABLE]

and

[TABLE]

The proof of Theorem 1.1 will be provided in Section 2. To establish Theorem 1.1, we invoke two tools (obtained recently by Li, Liang, and Rosalsky (2017a, b)): a symmetrization procedure for the SLLN and a probability inequality which is a comparison theorem for sums of independent ${\bf B}$-valued random variables.

We close this section by remarking that a version of Feller’s SLLN was obtained by Martikainen and Petrov (1980) for a sequence of identically distributed real-valued random variables $\left\{X,X_{n};~{}n\geq 1\right\}$ without any independence conditions being imposed on the summands; the result holds irrespective of the joint distributions of the summands. Specifically, in Theorem 2 of Martikainen and Petrov (1980) it is shown that

[TABLE]

if

[TABLE]

2 Proof of Theorem 1.1

Throughout this section, $\{a_{n};n\geq 1\}$ and $\{b_{n};n\geq 1\}$ are increasing sequences of positive real numbers satisfying (1.1). Write

[TABLE]

It follows from (1.1) that $0<b_{n}\uparrow\infty$ and

[TABLE]

Note that the $I(m),~{}m\geq 1$ are mutually exclusive sets. Thus there exist positive integers $k_{n},m_{n},~{}n\geq 1$ such that

[TABLE]

[TABLE]

To prove Theorem 1.1, we use the following four preliminary lemmas.

Lemma 2.1**.**

(Lemma 3.1 of Li, Liang, and Rosalsky (2017 b))*  There exist two continuous and increasing functions $\varphi(\cdot)$ and $\psi(\cdot)$ defined on $[0,\infty)$ such that*

[TABLE]

[TABLE]

Lemma 2.2**.**

Let $\{V_{n};~{}n\geq 1\}$ be a sequence of independent and symmetric B-valued random variables. Set $k_{0}=0$. Then the following two statements hold.

(i)*   If*

[TABLE]

then

[TABLE]

(ii)**

[TABLE]

if and only if

[TABLE]

Proof  We first prove Part (i). Clearly, (2.3) implies that

[TABLE]

Since the $\sum_{i=k_{n-1}+1}^{k_{n}}V_{i},~{}n\geq 1$ are independent B-valued random variables, (2.4) follows from (2.7) and the Borel-Cantelli lemma.

We now establish Part (ii). From the proof of Part (i), we only need to show that (2.5) follows from (2.6). Since $\{V_{n};~{}n\geq 1\}$ is a sequence of independent and symmetric B-valued random variables, by the remarkable Lévy inequality in a Banach space setting (see, e.g., see Proposition 2.3 of Ledoux and Talagrand (1991)), we have that for every $n\geq 1$,

[TABLE]

Thus it follows from (2.6) that

[TABLE]

which ensures that

[TABLE]

Now by the Toeplitz lemma, we conclude from (2.8) that

[TABLE]

i.e., (2.5) holds.  $\Box$

The following probability inequality is due to Li, Liang, and Rosalsky (2017 b) and is a comparison theorem for sums of independent $\mathbf{B}$-valued random variables.

Lemma 2.3**.**

(Theorem 1.1 (ii) of Li, Liang, and Rosalsky (2017 b))*  Let $\varphi(\cdot)$ and $\psi(\cdot)$ be two continuous and increasing functions defined on $[0,\infty)$ with $\varphi(0)=\psi(0)=0$ and satisfying (2.1). If $\{V_{n};~{}n\geq 1\}$ is a sequence of independent and symmetric B-valued random variables, then for every $n\geq 1$ and all $t\geq 0$,*

[TABLE]

The following symmetrization procedure for the SLLN for independent ${\bf B}$-valued random variables is due to Li, Liang, and Rosalsky (2017 a).

Lemma 2.4**.**

(Corollary 1.3 of Li, Liang, and Rosalsky (2017 a))* Let $\{X,X_{n};~{}n\geq 1\}$ be a sequence of i.i.d. B-valued random variables. Let $\{X_{n}^{\prime};~{}n\geq 1\}$ be an independent copy of $\{X_{n};~{}n\geq 1\}$. Write $S_{n}=\sum_{i=1}^{n}X_{i}$, $S_{n}^{\prime}=\sum_{i=1}^{n}X_{i}^{\prime}$, $n\geq 1$. Let $\{b_{n};n\geq 1\}$ be an increasing sequence of positive real numbers such that $\lim_{n\rightarrow\infty}b_{n}=\infty$. Then*

[TABLE]

if and only if

[TABLE]

With the preliminaries accounted for, Theorem 1.1 may be proved.

Proof of Theorem 1.1  To establish this theorem, it suffices to show that, for every sequence $\{X,X_{n};~{}n\geq 1\}$ of i.i.d. B-valued random variables, the following three statements are equivalent:

[TABLE]

[TABLE]

[TABLE]

The three statements (2.9)-(2.11) are equivalent if we can show that (2.9) and (2.11) are equivalent and (2.9) and (2.10) are equivalent.

For establishing the implication “(2.9) $\Rightarrow$ (2.11)”, let $\{X,X_{n};~{}n\geq 1\}$ be a sequence of i.i.d. B-valued random variables satisfying (2.9). It follows from (2.9) that

[TABLE]

which implies that

[TABLE]

By the Borel-Cantelli lemma, (2.12) is equivalent to

[TABLE]

Note that $\left\{\|X^{\prime}\|\leq b_{n}/2,\|X\|>b_{n}\right\}\subseteq\left\{\|X-X^{\prime}\|>b_{n}/2\right\}$ and $\displaystyle\lim_{n\rightarrow\infty}\mathbb{P}\left(\|X^{\prime}\|\leq b_{n}/2\right)=1$. We thus have that, for all large $n$,

[TABLE]

which, together with (2.13), implies (2.11).

We now prove “(2.11) $\Rightarrow$ (2.9)”. Let $\{X,X_{n};~{}n\geq 1\}$ be a sequence of i.i.d. B-valued random variables satisfying (2.11). Since $\{a_{n};n\geq 1\}$ and $\{b_{n};n\geq 1\}$ are increasing sequences of positive real numbers satisfying (1.1), by Lemma 2.1, there exist two continuous and increasing functions $\varphi(\cdot)$ and $\psi(\cdot)$ defined on $[0,\infty)$ such that both (2.1) and (2.2) hold. Write

[TABLE]

and

[TABLE]

Then $\{\tilde{X},\tilde{X}_{n};~{}n\geq 1\}$ is a sequence of i.i.d. symmetric B-valued random variables such that

[TABLE]

and $\{Y,Y_{n};~{}n\geq 1\}$ is a sequence of of i.i.d. symmetric B-valued random variables such that

[TABLE]

Hence, by Lemma 2.2 (i), we conclude from (2.15) and (1.2) that

[TABLE]

By Lemma 2.3, we have that, for every $n\geq 1$,

[TABLE]

It thus follows from (2.15) and (2.16) that

[TABLE]

By Lemma 2.2 (ii), (2.17) is equivalent to

[TABLE]

Hence

[TABLE]

By Lemma 2.4, (2.9) follows.

The implication “(2.9) $\Rightarrow$ (2.10)” is obvious.

We now establish the implication “(2.10) $\Rightarrow$ (2.9)”. It follows from (2.10) that

[TABLE]

which implies that

[TABLE]

By the Borel-Cantelli lemma, (2.18) is equivalent to: for some constant $0<\lambda<\infty$,

[TABLE]

That is, (2.11) holds with $X$ replaced by symmetric random variable $(X-X^{\prime})/\lambda$. Since (2.9) and (2.11) are equivalent, we conclude that

[TABLE]

Thus

[TABLE]

which, by Lemma 2.4, implies (2.9). The proof of Theorem 1.1 is therefore complete.  $\Box$

**Acknowledgments

**

The research of Deli Li was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (grant #: RGPIN-2014-05428)and the research of Han-Ying Liang was partially supported by the National Natural Science Foundation of China (grant #: 11271286).

References

Adler, A., Rosalsky, A., Taylor, R. L.: Strong laws of large numbers for weighted sums of random elements in normed linear spaces. Internat. J. Math. & Math. Sci. 12, 507-529 (1989). 2. 2.

de Acosta, A.: Inequalities for B-valued random vectors with applications to the law of large numbers. Ann. Probab. 9, 157-161 (1981). 3. 3.

Feller, W.: A limit theoerm for random variables with infinite moments. Amer. J. Math. 68, 257-262 (1946). 4. 4.

Kolmogoroff, A.: Sur la loi forte des grands nombres. C. R. Acad. Sci. Paris Sér. Math. 191 (1930), 910-912. 5. 5.

Ledoux, M., Talagrand, M.: Probability in Banach Spaces: Isoperimetry and Processes. Springer-Verlag, Berlin (1991). 6. 6.

Li, D., Liang, H.-Y., Rosalsky, A.: A note on symmetrization procedures for the laws of large numbers. Statist. Probab. Lett. 121 (2017 a), 136-142. 7. 7.

Li, D., Liang, H.-Y., Rosalsky, A.: A probability inequality for sums of independent Banach space valued random variables. arXiv:1703.07868, (2017 b), 10 pages. 8. 8.

Marcinkiewicz, J., Zygmund, A.: Sur les fonctions indépendantes. Fund. Math. 29 (1937), 60-90. 9. 9.

Martikainen, A. I., Petrov, V. V.: On a theorem of Feller. Teor. Veroyatnost. i Primenen. 25 (1980),  194-197 (in Russian). English translation in Theory Probab. Appl. 25 (1980), 191-193.