• Journal of the Korean Institute of Intelligent Systems
• /
• v.13 no.6
• /
• pp.754-758
• /
• 2003

The present paper establishes a strong law of large numbers for sums of tight fuzzy random variables as a generalization of SLLN for sums of independent and identically distributed fuzzy random variables.

• Communications of the Korean Mathematical Society
• /
• v.22 no.3
• /
• pp.441-452
• /
• 2007

The exact distributions of X + Y, XY and X/(X + Y) are derived when X and Y are independent generalized Planck random variables.

• Proceedings of the Korean Institute of Intelligent Systems Conference
• /
• 2004.04a
• /
• pp.385-388
• /
• 2004

The present paper establish the improved version of central limit theorem for sums of level-continuous fuzzy random variables as a generalization of central limit theorem for sums of independent and identically distributed random sets.

• Communications of the Korean Mathematical Society
• /
• v.18 no.2
• /
• pp.375-383
• /
• 2003

Under some conditions on an array of rowwise independent random variables, Hu et at. (1998) obtained a complete convergence result for law of large numbers with rate {a$\_$n/, n $\geq$ 1} which is bounded away from zero. We investigate the general situation for rate {a$\_$n/, n $\geq$ 1) under similar conditions.

• Bulletin of the Korean Mathematical Society
• /
• v.33 no.3
• /
• pp.419-425
• /
• 1996

Let ${X, X_n, n \geq 1}$ be a sequence of independent and identically distributed(i.i.d) random variables with EX = 0 and $E$\mid$X$\mid$^p < \infty$ for some $p \geq 1$. Let ${a_{ni}, 1 \leq i \leq n, n \geq 1}$ be a triangular arrary of constants. The almost sure(a.s) convergence of weighted sums $\sum_{i=1}^{n} a_{ni}X_i$ can be founded in Choi and Sung[1], Chow[2], Chow and Lai[3], Li et al. [4], Stout[6], Sung[8], Teicher[9], and Thrum[10].

• Journal of the Korean Mathematical Society
• /
• v.32 no.2
• /
• pp.363-370
• /
• 1995

A famous result of Chow and Robbins [8] asserts that if ${X_n, n \geq 1}$ are independent and identically distributed (i.i.d.) random variables with $E$\mid$X_1$\mid$ = \infty$, then for each sequence of constants ${M_n, n \geq 1}$ either $$ (1) lim inf_{n\to\infty} $\mid$\frac{M_n}{\sum_{j=1}^{n}X_j}$\mid$ = 0 almost certainly (a.c.) $$ or $$ (2) lim sup_{n\to\infty}$\mid$\frac{M_n}{\sum_{j=1}^{n}X_j}$\mid$ = \infty a.c. $$ and thus $P{lim_{n\to\infty} \sum_{j=1}^{n}X_j/M_n = 1} = 0$. Note that both (1) and (2) may indeed prevail.

• Journal of the Korean Institute of Intelligent Systems
• /
• v.15 no.3
• /
• pp.337-342
• /
• 2005

The present paper establishes the improved version of central limit theorem for sums of level-continuous fuzzy set-valued random variables as a generalization of central limit theorem for sums of independent and identically distributed set-valued random variables.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.4
• /
• pp.617-626
• /
• 2009

Let {$Y_i$,-$\infty$ < i < $\infty$} be a doubly infinite sequence of i.i.d. random variables with E|$Y_1$| < $\infty$, {$a_{ni}$,-$\infty$ < i < $\infty$ n $\geq$ 1} an array of real numbers. Under some conditions on {$a_{ni}$}, we obtain necessary and sufficient conditions for $\sum\;_{n=1}^{\infty}\frac{1}{n}P(|\sum\;_{i=-\infty}^{\infty}a_{ni}(Y_i-EY_i)|$>$n{\epsilon})$<{\infty}$. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.

• Bulletin of the Korean Mathematical Society
• /
• v.48 no.5
• /
• pp.923-938
• /
• 2011

Some properties for negatively orthant dependent sequence are discussed. Some strong limit results for the weighted sums are obtained, which generalize the corresponding results for independent sequence and negatively associated sequence. At last, exponential inequalities for negatively orthant dependent sequence are presented.

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.467-476
• /
• 2007

A complete convergence theorem for arrays of rowwise independent random variables was proved by Sung, Volodin, and Hu [14]. In this paper, we extend this theorem to the Banach space without any geometric assumptions on the underlying Banach space. Our theorem also improves some known results from the literature.