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POISSON APPROXIMATIONS FOR 2-DIMENSIONAL PATTERNS

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

^{1} *Department of Statistics, The University of Manitoba,*

Winnipeg, Manitoba, Canada R3T 2N2

^{2} *Department 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* \to \infty. 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.

**Source**
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