• Communications for Statistical Applications and Methods
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• v.20 no.3
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• pp.193-197
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• 2013

Almost sure central limit theorems are established for a stationary bootstrap sample mean of strong mixing processes. Both weak and strong consistencies are obtained.

• Communications of the Korean Mathematical Society
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• v.29 no.2
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• pp.351-358
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• 2014

In this paper, under some suitable integrability and smoothness conditions on f, we establish the central limit theorems for $$\sum_{k{\leq}N}k^{-1}f(S_k/{\sigma}\sqrt{k})$$, where $S_k$ is the partial sums of strictly stationary mixing random variables with $EX_1=0$ and ${\sigma}^2=EX^2_1+2\sum_{k=1}^{\infty}EX_1X_{1+k}$. We also establish an almost sure limit behaviors of the above sums.

• Journal of applied mathematics & informatics
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• v.26 no.1_2
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• pp.417-425
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• 2008

For stationary m-dimensional martingale difference sequences we prove the random functional central limit theorems and propose an almost sure consistent estimator for the limiting covariance matrix.

• Journal of the Korean Mathematical Society
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• v.59 no.6
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• pp.1185-1201
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• 2022

Feller introduced an unfair-fair-game in his famous book [3]. In this game, at each trial, player will win 2^k yuan with probability p_k = 1/2^kk(k + 1), k ∈ ℕ, and zero yuan with probability p₀ = 1 - Σ^∞_k=1 p_k. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed "fair", Feller [2] proved that when the total trial number n is large enough, player will loss n yuan with its probability approximate 1. So it's an "unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take 2^k = m to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.

• Communications for Statistical Applications and Methods
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• v.19 no.5
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• pp.647-654
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• 2012

We discuss the strong convergence for weighted sums of linearly negative quadrant dependent(LNQD) random variables under suitable conditions and the central limit theorem for weighted sums of an LNQD case is also considered. In addition, we derive some corollaries in LNQD setting.