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		   B O S T O N   U N I V E R S I T Y
                                       
		      Computer Science Department
                                       
			  C O L L O Q U I U M
                                       
			Wednesday, May 17, 1995
                                       
				 3:00pm
		       (Coffee served at 2:45 pm)
                                       
			 Seminar Room / MCS 135
                                       

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			   Tails in Harnesses

			      Andrei Toom
	       Incarnate Word College, San Antonio, Texas

  We consider d-dimensional random processes with local interaction, whose
  components are real random variables a_s^t, which interact in a linear
  local uniform way. Here s \in Z^d and t=0,1,2,....  A symmetric random
  i.i.d. `noise' \nu is added to every component at every time step.  For
  any k\in Z_+^d define \Delta_k a_s^t as follows.  If k=(0,...,0),
  \Delta_k a_s^t=a_s^t. Then by induction: \Delta_{k+e_i} a_s^t =\Delta_k
  a_{s+e_i}^t-\Delta_k a_s^t, where e_i is the d-dimensional vector, whose
  i-th component is one and other components are zeroes.  Denote |k| the
  sum of components of k.

  For any symmetric random variable \xi {\it power decay} or {\it P-decay}
  is defined as the supremum of those r, for which the r-th absolute
  moment of \xi is finite.  Convergence a.s., in probability and in law
  when t\to\infty is examined in terms of P-decay(\nu):
    If d=1 and k=0, a_s^t diverges.
    If d=2 and k=(0,0), a_s^t also diverges (in this case logarithmically).
    In all the other cases:
    If P-decay(\nu) < (d+2)/(d+|k|), \Delta_k a_s^t diverges.
    If P-decay(\nu) > (d+2)/(d+|k|), \Delta_k a_s^t converges and
	  P-decay(\lim \Delta_k a_s^t) = P-decay(\nu). 

  For any symmetric random variable \xi {\it exponential decay} or
  {\it E-decay} is defined as 
  E-decay(\xi) = \underline{\lim}_{x\to\infty} \log_x(-\ln Prob(\xi > x)).
  Assume that E-decay(\nu) > 1.  Exclude the (divergent) cases d=1, k=0
  and d=2, k=(0,0).
  In all the other cases \Delta_k a_s^t converges when t\to\infty and:
    If d > 2, E-decay(\lim a_s^t) = \min(\hbox{E-decay}(\nu), (d+2)/2).
    If |k|=1, E-decay(\lim \Delta_k a_s^t) = \min(\hbox{E-decay}(\nu), d+2).
    If |k| \geq 2, E-decay(\lim \Delta_k a_s^t) = \hbox{E-decay}(\nu).

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For colloquium info, including directions, see http://cs-www.bu.edu/colloquium
   For more information contact Prof. Mark Crovella <crovella@bu.edu>
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