###### Q2. What is the name of the paper?

During the past few decades, linear combinations of random variables have been extensively studied in statistics, operations research, reliability theory, actuarial science and other fields.

###### Q3. What is the usual order of the stochastic order?

Since the usual stochastic order is closed under convolution, the authors only need to proveφ(X1 − p2) + φ(X2 − p1) ≥st φ(X1 − p1) + φ(X2 − p2). (4.1)Under assumptions (A1) and (A2), (4.1) holds due to Corollary 3.8.

###### Q4. What is the support of the first author?

The first author is supported by the NNSF of China (No. 11401558), the Fundamental Research Funds for the Central Universities (No. WK2040160010) and China Postdoctoral12Science Foundation (No. 2014M561823).

###### Q5. What is the capital allocation strategy?

If the joint density function of X1, . . . , Xn satisfies assumptions (A1) and (A2) of Section 3, and if p∗ = (p∗1, . . . , p ∗ n) is the solution to the best capital allocation strategy, then, the authors have p∗1 ≤ p∗2 ≤ . . . ≤ p∗n .

###### Q6. What is the way to allocate the capital?

As defined in Xu and Hu (2012), the loss functionL(p) = n∑ i=1 φ(Xi − pi), p ∈ A = {p ∈ <n+ : p1 + . . .+ pn = p}is a reasonable criterion to set the capital amount pi to Xi, where φ is convex.

###### Q7. What is the proof of the following theorem?

An (1998) remarked that if X has a log-concave density, then its density has at most an exponential tail, i.e.,f(x) = O (exp(−λx)) , λ > 0, x→∞.9

###### Q8. What is the main result of the lemma?

In this section, the authors study stochastic comparisons of weighted sums of the form ∑ni=1 φ(Xi, ai)where X1, . . . , Xn are random variables with joint density function f(x).

###### Q9. what is the g if x is concave?

Then g is log-concave on <m.5Lemma 3.2 (Pan et al., 2013)If g : <2 → <+ is log-concave and −g is AI, i.e.g(x(2), x(1)) ≥ g(x(1), x(2)) for all (x1, x2) ∈ <2, (3.1)then(x1, x2) m (y1, y2) =⇒ g(x(1), x(2)) ≥ g(y(1), y(2)).

###### Q10. What is the significance of the notions of dependence?

They proved that characterizations of these notions are related to properties of arrangement increasing (AI) functions (to be defined in Section 2).

###### Q11. What is the main problem of the paper?

In this paper the authors further study the problem of stochastic comparisons of linear combinations of AI random variables not only for increasing convex ordering, but also for the usual stochastic ordering.

###### Q12. what is the a(1) b(2)?

it is sufficient to proveh (ϕ (x,a12)) + h (ϕ (x,a21))− h (ϕ (x,b12))− h (ϕ (x,b21)) ≥ 0. (3.6)Since φ is convex and supermodular, for a(1) ≤ b(1) ≤ b(2) ≤ a(2) and x1 ≤ x2, the authors haveϕ(x,a12)− ϕ(x,b12) = φ(x1, a(1)) + φ(x2, a(2))− φ(x1, b(1))− φ(x2, b(2)) = [ φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2)) ] + [ φ(x2, a(2))− φ(x1, a(2)) + φ(x1, b(2))− φ(x2, b(2)) ] ≥ 0,ϕ(x,a12)− ϕ(x,b21) = φ(x1, a(1)) + φ(x2, a(2))− φ(x2, b(1))− φ(x1, b(2)) = [ φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2)) ] + [ φ(x2, a(2))− φ(x1, a(2)) + φ(x1, b(1))− φ(x2, b(1)) ] ≥ 0andϕ(x,a12) + ϕ(x,a21)− ϕ(x,b12)− ϕ(x,b21)= φ(x1, a(1)) + φ(x1, a(2))− φ(x1, b(1))− φ(x1, b(2))+φ(x2, a(2)) + φ(x2, a(1))− φ(x2, b(2))− φ(x2, b(1)) ≥ 0.Thus, for any increasing convex function h, if ϕ(x,a21) ≥ ϕ(x,b21), then h(ϕ(x,a21)) ≥ h(ϕ(x,b21)) and h(ϕ(x,a12)) ≥ h(ϕ(x,b12)), which implies (3.6).