Many Heads in a Short Block

P. Deheuvels, K. Grill, P. Erdõs and P. Révész
Additional contact information
P. Deheuvels: Université Paris VI
K. Grill: Technical University
P. Erdõs: Mathematical Institute
P. Révész: Mathematical Institute Budapest and Technical University

A chapter in Mathematical Statistics and Probability Theory, 1987, pp 53-67 from Springer

Abstract: Abstract Let X 1.X 2.... be a sequence of i.i.d.r.v.’s with P(X 1 = +1) = P(X 1, = -1) = 1/2. Further let S o = 0. S n = X 1 + X 2 +... + X n (n = 1,2....) and I(N. K) = max0≤ n≤N-K (S n - K - S n ) (K = 1.2.....N: N = 1.2....). Consider a sequence {Kv} of positive integers and investigate the properties of the maximal increments I(N.K N ). This problem was studied by many authors in case of different {K N }’s. In the present paper we intend to summarize the results and prove a few new theorems. We are especially interested in the case K v = log N + o(log N). In section 1 we introduce a few notations and concepts and recall the known results in the case K N ≤ c log N. In section 2 a key-inequality will be proved. The main results are presented in section 3. Section 4 gives a survey of the case log N ≪ K N ≤ N

Date: 1987
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DOI: 10.1007/978-94-009-3963-9_6

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