POISSON APPROXIMATIONS FOR 2-DIMENSIONAL PATTERNS

JAMES C. FU^{1} AND MARKOS V. KOUTRAS^{2}

1Department of Statistics, The University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2

2Department of Mathematics, Section of Statistics and O.R., University of Athens, Panepistemiopolis, Athens 157 10, Greece

(Received November 5, 1992; revised March 18, 1993)

Abstract. Let X = (X_{ij})_{n×n} be a random matrix whose elements are independent Bernoulli random variables, taking the values 0 and 1 with probability q_{ij} and p_{ij} (p_{ij} + q_{ij} = 1) respectively. Upper and lower bounds for the probabilities of m non-overlapping occurrences of a square submatrix with all its elements being equal to 1, are obtained. Some Poisson convergence theorems are established for n → ∞. Numerical results indicate that the proposed bounds perform very well, even for moderate and small values of n.

Key words and phrases: Random matrix, Bernoulli random variables, Poisson approximation, patterns, consecutive-k-out-of-n:F system.