Based upon previous studies on laws of large numbers for fuzzy, random, fuzzy random and random fuzzy variables, We go further to explore weak law of large numbers(WLLN) for hybrid variables comprising fuzzy random variables and random fuzzy variables. we mainly prove Chebyshev WLLN, Poisson WLLN, Bernoulli WLLN, Markov WLLN and Khintchin WLLN for hybrid variables based on chance measure.

In this note we show how the properties of association and negative association can be combined with Stein’s method for compound Poisson approximation. Applications include k–runs in iid Bernoulli trials, an urn model with urns of limited capacity and extremes of random variables.

Abstract: For each n ≥ 1, let {Xn,j , j ≥ 1} be i.i.d. Bernoulli random variables with P{Xn,1 = 1} = pn = 1 − qn = 1 − P{Xn,1 = 0}, 0 < pn < 1. Define Wn,mn = ∑mn j=1 (Rn,j −an), where Rn,j = inf{k ≥ 0 : Xn,j+k = 0} is the number of success runs starting at j, mn is a sequences of positive integers, and an = pn/qn. We show that, under the condition mnpn → ∞, the central limit theorem Wn,mn σmn ...

and Applied Analysis 3 see 3, 4, 15, 16 . By 1.5 and 1.7 , the Witt’s formula for the q-Bernoulli numbers with weight α is given by ∫ Zp x qαdμq x β̃ α n,q , where n ∈ Z . 1.8 The q-Bernoulli polynomials with weight α are also defined by

In this paper, we study the connections between properties of the action of a countable group Γ on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of Γ on MX , where M is a measure space. In particular, we show that the action of Γ on X is amenable iff the shift Γ ↪→MX has almost invariant sets.

In this note, we prove that if G is a countable group that contains a nonabelian free subgroup then every pair of nontrivial Bernoulli shifts over G are weakly isomorphic.

We study the stability of an interconnected system of Euler−Bernoulli beam and heat equation with boundary coupling, where the boundary temperature of the heat equation is fed as the boundary moment of the Euler−Bernoulli beam and, in turn, the boundary angular velocity of the Euler−Bernoulli beam is fed into the boundary heat flux of the heat equation. We show that the spectrum of the closed-l...