To reconstruct dark energy models the redshift z eq , marking the end of radiation era and the beginning of matter-dominated era, can play a role as important as z t , the redshift at which deceleration parameter experiences a signature flip. To implement the idea we propose a variable equation of state for matter that can bring a smooth transition from radiation to matter-dominated era in a single model. A popular Λ ∝ ρ dark energy model is chosen for demonstration but found to be unacceptable. An alternative Λ ∝ ρa 3 model is proposed and found to be more close to observation.

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The sequence of random variables {Xn}n?N is said to be weighted modulus ??-statistically convergent in probability to a random variable X [16] if for any ?,? > 0, limn??1 1/T??(n) |{k ? T??(n): tk?(P(|Xk-X|? ?)) ? ?}| = 0 where ? be a modulus function and {tn}n?N be a sequence of real numbers such that limn?? inf tn > 0 and T??(n) = ? k?[?n,?n] tk ? n ? N. In this paper we study a related concept of convergence in which the value 1/ T??(n) is replaced by 1/Cn, for some sequence of real numbers {Cn}n?N such that Cn > 0 8 n ? N, lim n?1 Cn = ? and lim n?1 sup Cn T??(n)< 1 (like [30]). The results are applied to build the probability distribution for quasi-weighted modulus ??-statistical convergence in probability, quasi-weighted modulus ??-strongly Ces?ro convergence in probability, quasi-weighted modulus S??-convergence in probability and quasiweighted modulus N??-convergence in probability. If {Cn}n?N satisfying the condition lim n?1 inf Cn/T??(n) > 0, then quasi-weighted modulus ??-statistical convergence in probability and weighted modulus ??-statistical convergence in probability are equivalent except the condition lim n?1 inf Cn T??(n) = 0. So our main objective is to interpret the above exceptional condition and produce a relational behavior of above mention four convergences.

A sequence of real numbers {xn} n∈N is said to be αβ-statistically convergent of order γ (where 0 < γ ≤ 1) to a real number x [1] if for every δ > 0,where {αn} n∈N and {βn} n∈N be two sequences of positive real numbers such that {αn} n∈N and {βn} n∈N are both non-decreasing, βn ≥ αn ∀ n ∈ N, (βn −αn) → ∞ as n → ∞. In this paper we study a related concept of convergences in which the value |x k −x| is replaced by P (|X k −X| ≥ ε) and E(|X k − X| r ) repectively (Where X, X k are random variables for each k ∈ N, ε > 0, P denote the probability, E denote the expectation) and we call them αβ-statistical convergence of order γ in probability and αβ-statistical convergence of order γ in r th expectation respectively. The results are applied to build the probability distribution for αβ-strong p-Cesàro summability of order γ in probability and αβ-statistical convergence of order γ in distribution. So our main objective is to interpret a relational behavior of above mentioned four convergences. We give a condition under which a sequence of random variables will converge to a unique limit under two different (α, β) sequences and this is also use to prove that if this condition violates then the limit value of αβ-statistical convergence of order γ in probability of a sequence of random variables for two different (α, β) sequences may not be equal.Keywords: αβ-statistical convergence, αβ-statistical convergence of order γ in probability, αβ-strong p-Cesàro summability of order γ in probability, αβ-statistical convergence of order γ in r th expectation, αβ-statistical convergence of order γ in distribution. (2010) : 40A35, 40G15, 60B10. Mathematics Subject Classification

Dark energy models inspired by the cosmological holographic principle are studied in homogeneous isotropic spacetime with a general choice for the dark energy density ρ d = 3(αH 2 + βḢ). Special choices of the parameters enable us to obtain three different holographic models, including the holographic Ricci dark energy(RDE) model. Effect of interaction between dark matter and dark energy on the dynamics of those models are investigated for different popular forms of interaction. It is found that crossing of phantom divide can be avoided in RDE models for β > 0.5 irrespective of the presence of interaction. A choice of α = 1 and β = 2/3 leads to a varying Λ-like model introducing an IR cutoff length Λ −1/2 . It is concluded that among the popular choices an interaction of the form Q ∝ Hρ m suits the best in avoiding the coincidence problem in this model.