Limit theorem for maximal segmental score for random sequences of random length. (English) Zbl 1199.60168

Prakasa Rao, B. L. S.; Sreehari, M.

Teor. Jmovirn. Mat. Stat. 76, 138-141 (2007) and Theory Probab. Math. Stat. 76, 155-158 (2008).

Let \(\{X_n\}\) be a sequence of independent and identically distributed (i.i.d.) random variables satisfying the following assumptions: \(P(X_1>0)> 0\), \(E(X_1)<0\), the random variable \(X_1\) is bounded; i.e., \(P(| X_1| < c) = 1\) for some constant \(c > 0\). Let \(S_0=0, S_n =\sum_{k=1}^nX_k,n\geq1\), and \(Z_n=\max_{1\leq i\leq j\leq n}(S_j-S_i)\). D. L. Iglehart [Ann. Math. Stat. 43, 627–635 (1972; Zbl 0238.60072)] proved the following theorem.
Theorem 1. If \(\{X_n\}\) is a sequence of non-lattice i.i.d. random variables satisfying the conditions given above, then \[ P\{Z_n-\theta\log{n}\leq x\}\to G(x)=\exp\{-ke^{-x/\theta}\} \] for every \(x\in\mathbb R\), where \(\theta\) and \(k\) are positive constants depending on the distribution of \(X_1\).
S. Karlin and A. Dembo [Adv. Appl. Probab. 24, No. 1, 113–140 (1992; Zbl 0767.60017)] extended this result to the lattice case. In this paper the authors obtain a random version of Theorem 1.
Theorem 2. Suppose \(\{X_n\}\) is a sequence of non-lattice i.i.d. random variables satisfying the conditions stated above. Then \[ P\{Z_{N_n}-\theta\log{N_n}\leq x\}\to G(x) \] as \(n\to\infty\) for every \(x\in\mathbb R\) and \[ P\{Z_{N_n}-\theta\log{k_n}\leq x\}\to \int_0^{\infty}G(x-\theta\log{u})dP(N\leq u), \] as \(n\to\infty\). Here \(\{N_n\}\) is a sequence of positive integer-valued random variables satisfying the condition \(N_n/k_n\to N\) in probability for some sequence of integers \(\{k_n\}\), \(0<k_n\to\infty\), \(N\) is a positive random variable.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

60G50

Sums of independent random variables; random walks

Keywords:

maximum segmental score; limit theorem; random sequence; random length

Citations:

Zbl 0238.60072; Zbl 0767.60017