Liangzhen LEI (雷良貞)School of Mathematical Science,Capital Normal University，Beijing 100048，China E-mail : leiliangzhen@cnu.edu.cn

Yutao MA(馬宇韜)+School of Mathematical Sciences & Laboratory of Mathematics and Complea Systems of Ministry of Education,，Beijing Normal University，Beijing 100875，China E-mail: mayt@bnu.edu.cn

1 Introduction

For γ >0, we get another large deviation.

The rest of this paper is organized as follows: Section 2 is devoted to the proof of Theorem 1.1 and we prove Theorem 1.2 in Section 3.Appendix focuses on some lemmas.

2 Proof of Theorem 1.1

For simplicity, set Xi= pλi/p1for 1 ≤i ≤n.We now give the joint density function of(X1,··· ,Xn).

2.1 The Verification of (2.1)

The limit (2.1) is verified.

2.2 The Verification of (2.2)

For any x ∈(0,1), on the set {X(n)≤x} we know

since X(n)≥0.The proof of the limit (2.2) is complete.

2.3 The Verification of (2.3)

If x<1,the limit(2.3)holds automatically,since Jσ(x)=-∞,and Jσ(x)=0 if and only if x=1.By Theorem 2 in[9], X(n)converges almost surely to 1, which completes the case x=1.

Now we suppose that x>1.Since G is open, we choose r,a,b such that 1

3 Proof of Theorem 1.2

As for Theorem 1.1, we just verify Theorem 1.2 for σ > 0, and it is enough to take the limit when σ = 0.For the large deviation principle of X(n), we need to prove the weak large deviation principle of X(n)and the exponential tightness.As in [11], we are going to check the following three limits:

for weak large deviation and

for the exponential tightness.

Based on this theorem, following the standard argument, as in [1] and[11], we are going to check the limits (3.1)–(3.4).

3.1 The Verification of (3.3)

For all x

By the large deviation principle for Ln, we have that

This immediately implies (3.3).

3.2 The Verification of (3.4)

Recall that u2

3.3 The Verification of (3.1)

Starting from (3.4), it remains to prove that

on the condition that 0 ≤xn≤M and that supp (Ln-1) ?[0,M] for n large enough under the condition A.By the large deviation principle for Ln-1, Lemma A.2 and the boundedness of φM, the second term in the above bound can be neglected.Therefore, we have that

3.4 The Verification of (3.2)

Here, for the last equality, we use the continuity of

The verifications(3.1),(3.2),(3.3)and(3.4)are all done now and the proof of the large deviation is complete.

Conflict of InterestThe authors declare no conflict of interest.

Appendix