A local limit theorem for random walks conditioned to stay positive
A local limit theorem for random walks conditioned to stay positive
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条件保持为正的随机游走的局部极限定理
DOI：
10.1007/s00440-005-0444-5
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发表时间：
2004
影响因子：
2
通讯作者：
F. Caravenna
中科院分区：
文献类型：
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作者：
F. Caravenna
我们考虑一个实随机游动S_n=X_1+…+X_n被吸引(无中心)于正态规律：这意味着对于适当的正规化序列A，我们有弱收敛S_n/An⇒ϕ(X)Dx，ϕ(X)是标准正态密度。Gnedenko的局部极限定理和Stone的局部极限定理分别在晶格和非晶格的情况下提供了这种收敛的局部精化。现在设表示事件(s1>；0，…，sn>；0)，并令sn+表示随机变量sn，条件是：已知sn+/an↠ϕ+(X)dx，其中ϕ+(X)：=xexp(−x2/2)1(x≥0)。我们在本文中建立的弱收敛的局部极限定理等价于Gnedenko和Stone的局部极限定理。我们还考虑了当X1具有绝对连续律时的特殊情况：在这种情况下，与经典情况类似，在标准的附加假设下，Sn+/An的密度向ϕ+(X)的一致收敛成立。最后，我们讨论了我们的主要结果在阶梯变量过程的联合更新测度的渐近行为中的应用。与LLT的经典证明不同，我们不使用特征函数：我们的技术更多地取自随机游动的所谓涨落理论。
We consider a real random walk Sn=X1+...+Xn attracted (without centering) to the normal law: this means that for a suitable norming sequence an we have the weak convergence Sn/an⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S1>0,...,Sn>0) and let Sn+ denote the random variable Sn conditioned on : it is known that Sn+/an ↠ ϕ+(x) dx, where ϕ+(x):=x exp (−x2/2)1(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X1 has an absolutely continuous law: in this case the uniform convergence of the density of Sn+/an towards ϕ+(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.