Convergence theory was well developed based on classical measure theory, and some applications can be found in [1-3]. As for convergence theory in fuzzy environments, information and data are usually vague or imprecise which is essentially different from the classical measure case [4-6]. Therefore, it is more reasonable to utilize quasi-probability measure, which is an important extension of probability measure [1-2] to deal with fuzziness, to study such convergence theory. Quasi-probability measure was introduced by Wang [6], which offered an efficient tool to deal with fuzzy information fusion, subjective judgement, decision making, and so forth [7-11].

Convergence concepts play an important role in classical measure theory. Some mathematics workers explored them for fuzzy (or non-additive) measures such as Liu [12-13], Wang [14], Zhang [4-5], Gianluca [15], and so on. While the measure tool is non-additive, the convergence concepts are very different from additive case. In order to investigate quasi-probability theory deeper, we will propose in the present paper some new convergence concepts on quasi-probability space, and discuss the relationships among the convergence concepts. Our work helps to build important theoretical foundations for the development of quasi-probability measure theory.

The paper is outlined as follows: Section 1 is for introduction. In Section 2, some preliminaries are given. In Section 3, we study convergence concepts of $q$-random variables sequence and, ultimately, Section 4 is for conclusions.

In this paper, let$~X~$be a nonempty set and $(X, \mathcal{F})$ be a measurable space, here $\mathcal{F}$ is a $\sigma$-algebra of $X$. If $A, ~B\in \mathcal{F}$, then the notation $A\subset B$ means that $A$ is a subset of $B$, and the complement of $A$ is denoted by $A^c$.

Definition 2.1 [6] Let $a\in(0, +\infty]$, an extended real function is called a $T$-function iff $\theta:[0, a]\rightarrow[0, +\infty]$ is continuous, strictly increasing, and such that $\theta(0)=0, ~\theta^{-1}(\{\infty\})=\emptyset$ or $\{\infty\}$, according to $a$ being finite or not.

Let $a\in(0, +\infty]$, an extended real function $\theta:[0, a]\rightarrow[0, +\infty]$ is called a regular function, if $\theta$ is continuous, strictly increasing, and $\theta(0)=0, \theta(1)=1$ [10].

Obviously, if $\theta$ is a regular function, then $\theta^{-1}$ is also a regular function.

Definition 2.2 [6] $\mu$ is called quasi-additive iff there exists a $T$-function $\theta$, whose domain of definition contains the range of $\mu$, such that the set function $\theta\circ\mu$ defined on $\cal{F}$ by $(\theta\circ\mu)(E)=\theta~[\mu(E)]~~(\forall{E}\in\cal{F}), $ is additive; $\mu$ is called a quasi-measure iff there exists a $T$-function $\theta$ such that $\theta\circ\mu$ is a classical measure on $\cal{F}$. The $T$-function $\theta$ is called the proper $T$-function of $\mu$.

Definition 2.3  If $\theta$ is a regular $T$-function of $\mu$, then $\mu$ is called a quasi-probability. The triplet $(X, \cal{F}, \mu)$ is called a quasi-probability space.

From Definition 2.3, we know probability is a quasi-probability with $\theta(x)=x$ as its $T$-function.

Example 2.1  Suppose that $X=\{1, 2, \cdots, n\}$, $\rho(X)$ is the power set of $X$. If

$ \mu(E)=(\frac{|E|}{n})^2, $

where $|E|$ is the number of those points that belong to $E$, then $\mu$ is a quasi-probability with $\theta(x)=\sqrt{x}, ~~x \in [0, 1]$ as its $T$-function [6].

Definition 2.4 Let $(X, \mathcal{F}, \mu)$ be a quasi-probability space, and $\xi=\xi(\omega), \omega\in X$, be a real set function on $X$. For any given real number $x$, if $\{\omega|~\xi(\omega)\leq{x}\}\in\mathcal{F}, $ then $\xi$ is called a quasi-random variable, denoted by $q$-random variable. The distribution function of $q$-random variable $\xi$ is defined by $F_{\mu}(x)=\mu \{\omega\in X|~ \xi(\omega)\leq x \}.$

Let $\xi$ and $\eta$ be two $q$-random variables. For any given real numbers $x, ~y$, if

$ \mu(\xi\leq{x}, \eta\leq{y})=\theta^{-1}[(\theta\circ\mu)(\xi\leq{x})\cdot(\theta\circ\mu)(\eta\leq{y})], $

then $\xi$ and $\eta$ are independent $q$-random variables [10].

Theorem 2.1 Let $\mu$ be a quasi-probability on $\mathcal{F}$. Then there exists a regular $T$-function $\theta, $ such that $\theta\circ\mu$ is a probability on $\mathcal{F}$ [4].

Theorem 2.2 If $\mu$ is a quasi-probability, then $\mu$ is continuous and $\mu(\emptyset)=0$ [4].

Theorem 2.3 Let $\mu$ be a quasi-probability on $\mathcal{F}$, $A, B\in\mathcal{F}$, then we have

(1) if $A\subset{B}$, then $\mu(A)<\mu(B)$;

(2) if $\mu(A)=0$, then $\mu(A^c)=1;$

(3) $\mu(A\bigcup B)\leq \theta^{-1}[(\theta\circ\mu)(A)+(\theta\circ\mu)(B)].$

Proof (1) Since $A\subset{B}$, and there exists a $T$-function $\theta$ such that $~\theta\circ\mu$ is a probability, we have $(\theta\circ\mu)(A)<(\theta\circ\mu)(B)$, $\theta$ is continuous, strictly increasing, it is clear that $~\mu(A)<\mu(B)$.

(2) Since $~\theta\circ\mu$ is a probability, one can have

$ 1=(\theta\circ\mu)(X)=(\theta\circ\mu)(A\bigcup A^c)=(\theta\circ\mu)(A)+(\theta\circ\mu)( A^c), $

which implies that

$ (\theta\circ\mu)( A^c)=1-(\theta\circ\mu)(A)=1-\theta(\mu(A))=1-\theta(0)=1, $

namely, $\theta(\mu(A^c))=1.$ It follows from the regularity of $\theta$ that $\mu(A^c)=1.$

(3) $~\theta\circ\mu$ is a probability, we have

$ (\theta\circ\mu)(A\bigcup B)\leq (\theta\circ\mu)(A)+(\theta\circ\mu)(B), $$ $

that is,

$ \mu(A\bigcup B)\leq \theta^{-1}[(\theta\circ\mu)(A)+(\theta\circ\mu)(B)]. $

In the section, we introduce some new convergence concepts such as convergence almost surely, convergence in distribution, fundamental convergence almost everywhere, fundamental convergence in quasi-probability, etc., and we will investigate the relationships among the convergence concepts.

Definition 3.1 [4] Suppose that $\xi$, $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If

$ \mu\{{\lim\limits_{n\rightarrow\infty}\xi_n=\xi}\}=1, $

then we say that $\{\xi_n\}$ converges almost surely to $\xi.$ Denoted by

$ \lim\limits_{n\rightarrow\infty}\xi_n=\xi \qquad (\mu-\mbox{a.s.}). $

Definition 3.2 Suppose that $\xi$, $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If there exists $E\in \mathcal{F}$ with $\mu(E)=0$ such that $\{\xi_n\}$ converges to $\xi$ on $E^c, $ then we say $\{\xi_n\}$ converges to $\xi$ almost everywhere. Denoted by

$ \xi_n\rightarrow \xi\qquad(\mbox{a.e.}). $

Definition 3.3 Suppose that $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If there exists $E\in \mathcal{F}$ with $\mu(E)=0$ such that for any $x\in E^c, $

$ \lim\limits_{n, ~m\rightarrow\infty}(\xi_n(x)-\xi_m(x))=0, $

then we say $\{\xi_n\}$ is fundamentally convergent almost everywhere.

Definition 3.4 [4] Suppose that $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ is a sequence of $q$-random variables. If there exists a $q$-random variable $\xi$, such that $\forall\varepsilon>0, $

$ \lim\limits_{n\rightarrow\infty}\mu\{\mid\xi_n-\xi\mid \geq \varepsilon\}=0, $

namely,

$ \lim\limits_{n\rightarrow\infty}\mu\{\mid\xi_n-\xi\mid < \varepsilon\}=1, $

then we say that $\{\xi_n\}$ converges in quasi-probability to $\xi$. Denoted by

$ \xi_n\rightarrow \xi \qquad (\mu). $

Definition 3.5 Suppose that $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables. If for any given $\varepsilon >0$,

$ \lim\limits_{n, ~m\rightarrow\infty}\mu(|\xi_n-\xi_m|\geq \varepsilon)=0, $

then we say $\{\xi_n\}$ fundamentally converges in quasi-probability.

Definition 3.6 Suppose that $F_{\mu}(x), ~F_{\mu}^1(x)~F_{\mu}^2(x)\cdots$ are the distribution functions of $q$-random variables $\xi$, $\xi_1, ~\xi_2, ~\cdots, $ respectively. The sequence $\{\xi_n\}$ is said to be convergent in distribution to $\xi$ if

$ \lim\limits_{n\rightarrow\infty}F_{\mu}^n(x)=F_{\mu}(x) $

at any continuity point of $F_{\mu}(x).$

Theorem 3.1 If $\{\xi_n\}$ converges in quasi-probability to $\xi, $ then $\{\xi_n\}$ fundamentally converges in quasi-probability.

Proof Suppose that $\{\xi_n\}$ converges in quasi-probability to $\xi, $ then for any given $\varepsilon>0$, we have

$ \lim\limits_{n\rightarrow\infty}\mu\{\mid\xi_n-\xi\mid \geq \frac{\varepsilon}{2}\}=0. $

According to [2],

$ \{|\xi_n-\xi_m|\geq \varepsilon\}\subset \{|\xi_n-\xi|\geq \frac{\varepsilon}{2}\}\bigcup \{|\xi_m-\xi|\geq \frac{\varepsilon}{2}\}. $

It follows from Theorem 2.3 that

$ \begin{eqnarray*}&&\mu\{|\xi_n-\xi_m| \geq \varepsilon\}\leq \mu[\{|\xi_n-\xi|\geq \frac{\varepsilon}{2}\}\bigcup \{|\xi_m-\xi|\geq \frac{\varepsilon}{2}\}]\\ &\leq& \theta^{-1}[(\theta\circ \mu)\{|\xi_n-\xi|\geq \frac{\varepsilon}{2}\}+(\theta\circ \mu)\{|\xi_m-\xi|\geq \frac{\varepsilon}{2}\}].\end{eqnarray*} $

And $\theta, ~~\theta^{-1}$ are strictly increasing and continuous,

$ \begin{eqnarray*}\lim\limits_{n, ~m\rightarrow\infty}\mu\{|\xi_n-\xi_m|\geq \varepsilon\}&\leq& \lim\limits_{n, ~m\rightarrow\infty} \theta^{-1}[(\theta\circ \mu)\{|\xi_n-\xi|\geq \frac{\varepsilon}{2}\}+(\theta\circ \mu)\{|\xi_m-\xi|\geq \frac{\varepsilon}{2}\}]\\ &=&\theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty} \mu\{|\xi_n-\xi|\geq \frac{\varepsilon}{2}\})+\theta(\lim\limits_{m\rightarrow\infty} \mu\{|\xi_m-\xi|\geq \frac{\varepsilon}{2}\})]\\ &=&\theta^{-1}[\theta(0)+\theta(0)]=\theta^{-1}(0)=0.\end{eqnarray*} $

This means that $\{\xi_n\}$ fundamentally converges in quasi-probability.

Lemma 3.1 [2] Suppose that $\xi_n, ~\xi\in \mathcal{F}, $ and for any given $\varepsilon_k>0, ~\lim\limits_{n\rightarrow\infty}\varepsilon_k=0, $ then we have

(1) $\{\xi_n\rightarrow \xi\}=\bigcap\limits_{\varepsilon>0}\bigcup\limits_{m=1}^\infty\bigcap\limits_{n=m}^\infty \{|\xi_{n}-\xi|\geq \varepsilon\}=\bigcap\limits_{k=1}^\infty\bigcup\limits_{m=1}^\infty\bigcap\limits_{n=m}^\infty \{|\xi_n-\xi|< \varepsilon_k\};$

(2) $\{|\xi_n-\xi_m|\rightarrow 0\} =\bigcap\limits_{\varepsilon>0}^\infty\bigcup\limits_{n=1}^\infty\bigcap\limits_{v=1}^\infty \{|\xi_{n+v}-\xi_n|< \varepsilon\} =\bigcap\limits_{k=1}^\infty\bigcup\limits_{n=1}^\infty\bigcap\limits_{v=1}^\infty \{|\xi_{n+v}-\xi_n|< \varepsilon_k\}.$

Theorem 3.2 Suppose that $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables, then

$ \xi_n\rightarrow \xi \qquad (\mbox{a.e.}) $

if and only if

$ \mu(\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}=0, \quad \forall\varepsilon>0 $

if and only if

$ \lim\limits_{n\rightarrow\infty}\mu(\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\})=0, \quad \forall\varepsilon>0. $

Proof If $\xi_n\rightarrow \xi~~\mbox{a.e.}, $ then $\forall\varepsilon>0$. According to Lemma 3.1,

$ \mu(\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}\leq \mu(\bigcup\limits_{\varepsilon>0}\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}=\mu(\{\xi_n\rightarrow \xi\}^c)=0. $

On the other hand, if

$ \mu(\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}=0, \quad \forall\varepsilon>0, $

then for any given $\varepsilon_k>0, ~\lim\limits_{k\rightarrow\infty}\varepsilon_k=0, $ it follows from Theorems 2.1 and 2.3 that

$ \begin{eqnarray*}&&\mu(\{\xi_n\rightarrow \xi\}^c)=\mu(\bigcup\limits_{k=1}^\infty\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon_k\})\\ &=&\theta^{-1}[(\theta\circ\mu) (\bigcup\limits_{k=1}^\infty\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon_k\})]\\ &\leq& \theta^{-1}[\sum\limits_{k=1}^\infty(\theta\circ\mu) (\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon_k\})]\\ &=&\theta^{-1}[\sum\limits_{k=1}^\infty\theta(\mu (\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon_k\}))]=\theta^{-1}(0)=0, \end{eqnarray*} $

that is $\xi_n\rightarrow \xi~\mbox{a.e.}.$ And since

$ \bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\} $

is decreasing for $n$, it follows from the continuity of $\mu$ that

$ \mu(\bigcap\limits_{n=1}^\infty\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}=\lim\limits_{n\rightarrow\infty}\mu(\bigcup\limits_{v=1}^\infty \{|\xi_{n+v}-\xi|\geq \varepsilon\}). $

Now the theorem is proved.

Theorem 3.3 Suppose that $\xi$, $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If $\{\xi_n\}$ converges almost surely to $\xi$, then $\{\xi_n\}$ converges in quasi-probability to $\xi$.

Example 3.1 [4] Suppose that $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are independent $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If

$ \mu\{\xi_n=\frac{1}{n}\}=1-\frac{1}{n}, ~~\mu\{\xi_n=n+1\}=\frac{1}{n}, ~~ n=1, 2, \cdots, $

then $\{\xi_n\}$ converges in quasi-probability to $0$. However, $\{\xi_n\}$ does not converges almost surely to $0$.

Theorem 3.3 shows that convergence almost surely implies convergence in quasi-probability. Example 3.1 shows that convergence in quasi-probability does not imply convergence almost surely. But for independent $q$-random series, convergence almost surely is equivalent to convergence in quasi-probability.

Theorem 3.4 [4] If $\{\xi_n\}$ is a sequence of independent $q$-random variables, then $~\sum\limits_{n=1}^\infty\xi_n$ converges almost surely if and only if $~\sum\limits_{n=1}^\infty\xi_n$ converges in quasi-probability.

Theorem 3.5 Suppose that $\xi$, $\xi_1, ~\xi_2, ~\cdots, ~\xi_n, ~\cdots$ are $q$-random variables defined on the quasi-probability space $(X, \mathcal{F}, \mu)$. If $\{\xi_n\}$ converges in quasi-probability to $\xi$, then $\{\xi_n\}$ converges in distribution to $\xi.$

Proof Assume that $F_{\mu}^n(x), F_{\mu}(x)~$ are the distribution functions of $\xi_n, \xi, $ respectively. Let $x, y, z$ be the given continuity points of the distribution function $F_{\mu}(x).$

On the one hand, for any $y<x, $ we have

$ \{\xi\leq y\}=\{\xi_n\leq x, ~\xi\leq y\}\bigcup \{\xi_n> x, ~\xi\leq y\}\subset \{\xi_n\leq x\}\bigcup \{|\xi_n-\xi|\geq x- y\}. $

It follows from Theorem 2.3 that

$ \begin{eqnarray*}&&\mu\{\xi\leq y\}\leq \mu[\{\xi_n\leq x\}\bigcup \{|\xi_n-\xi|\geq x- y\}]\\ &\leq& \theta^{-1}[(\theta\circ\mu)\{\xi_n\leq x\}+(\theta\circ\mu)\{|\xi_n-\xi|\geq x-y\}].\end{eqnarray*} $

Since $\{\xi_n\}$ converges in quasi-probability to $\xi, $ and $\theta, ~~\theta^{-1}$ are continuous, we have

$ \begin{eqnarray*}&&\mu\{\xi\leq y\}\leq \theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi_n\leq x\})+\theta(\lim\limits_{n\rightarrow\infty}\mu\{|\xi_n-\xi|\geq x- y\})]\\ &=&\theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi_n\leq x\})+\theta(0)]=\theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi_n\leq x\})], \end{eqnarray*} $

which implies that

$ F_{\mu}(y)\leq \lim\limits_{n\rightarrow\infty} F_{\mu}^n(x) $

for any $y<x.$

Let $y\rightarrow x$, we obtain

$ F_{\mu}(x)\leq \lim\limits_{n\rightarrow\infty} F_{\mu}^n(x). $

On the other hand, for any $z>x, $ we have

$ \{\xi_n\leq x\}=\{\xi_n\leq x, ~\xi\leq z\}\bigcup \{\xi_n\leq x, ~\xi>z\}\subset \{\xi\leq z\}\bigcup \{|\xi_n-\xi|\geq z-x\}. $

Since $\mu\{|\xi_n-\xi|\geq z-x\}\rightarrow 0$ as $n\rightarrow\infty, $ and $\theta, ~~\theta^{-1}$ are continuous, we get

$ \begin{eqnarray*}&&\lim\limits_{n\rightarrow\infty}\mu\{\xi_n\leq x\}\leq \theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi\leq z\})+\theta(\lim\limits_{n\rightarrow\infty}\mu\{|\xi_n-\xi|\geq z- x\})]\\ &=&\theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi\leq z\})+\theta(0)]=\theta^{-1}[\theta(\lim\limits_{n\rightarrow\infty}\mu\{\xi\leq z\})]=\mu\{\xi\leq z\}.\end{eqnarray*} $

It means that

$ \lim\limits_{n\rightarrow\infty} F_{\mu}^n(x)\leq F_{\mu}(z) $

for any $z>x.$ Let $z\rightarrow x$, we get

$ \lim\limits_{n\rightarrow\infty} F_{\mu}^n(x)\leq F_{\mu}(x). $

Finally, one can see that

$ \lim\limits_{n\rightarrow\infty}F_{\mu}^n(x)=F_{\mu}(x), $

that is to say $\{\xi_n\}$ converges in distribution to $\xi.$

According to Theorems 3.3 and 3.5, we conclude that convergence almost surely implies convergence in quasi-probability; convergence in quasi-probability implies convergence in distribution.

This paper proposed some new convergence concepts for quasi-random variables. Firstly, the properties of quasi-probability measure were further discussed. Secondly, the concepts of convergence in quasi-probability, convergence almost surely, convergence in distribution and convergence almost everywhere were introduced on quasi-probability space. Finally, the relationships among the convergence concepts were investigated in detail. All investigations helped to lay important theoretical foundations for the systematic and comprehensive development of quasi-probability measure theory.