Studia Scientiarium Mathematicarum Hungarica 34. (1998)
1-3. szám - Kiesel R.-Stadtmüller U.: Erdős-Rényi-Shepp laws for dependent random variables
254 R. KIESEL and U. STADTMULLER is a convex, continuous function. For {Zn} the large deviation principle holds true with the good rate function I, that is, (1.2) limsupn-1 log P (Zn G F) ^ — inf I(x) n->oo x£F for each closed set F QR, and (1.3) liminf n~l log P (Zn G G) ^ — inf I(x) n-y oo x eG for each open set G ^ K. Lemma. Let {Xn} be a stationary f-mixing sequence of random variables, such that |2fi| ^ C < oo, E(Xi) = 0 but E(A^) > 0 and (1.1) holds. Then we have with xq := sup{x ^ 0; I(x) = 0}, x\ sup{x > 0; I(x) < oo} and A\= sup {I(x)} (xvhere sup{/(.)} := — oo} that 0<x<xi 0 (i) L(.) is convex and hence continuous on R and L(A) CA, A ^ 0; (ii) O^xq^xi^C; (iii) If .To <x\, I ■ [to,ti) —> [0, Ä) is continuous and strictly increasing and hence I*~ : [0, A) —> [to, xi) exists. CONVENTION. To obtain our main result in a closed form we define I<~(x)::=xi if x ^ max{0, A}. Using (1.2) resp. (1.3) for F — [a, oo) resp. G = (a, oo) with any a G (xo,x\) we obtain by the strict monotonicity of I that for any sufficiently small e > 0 and n sufficiently large (1.4) P(Zn ^ a) ^ exp{-n/(a - e)} and (1.5) P(Zn > a) ^exp{—nl(a + e)}. We consider in the sequel the following random variables Sk+b„ ~ “Sfc(1.6) Vn := max 0<k<n—bn with Sn := X\ 4------b Xn and bn := [clog n] for c > 0. We can now state our main result, in which A := sup {I(x)} as above. 0<i<ii THEOREM. Let {xn} be a stationary, <f>-mixing sequence of random variables, such that |Xi| ^ C < oo, E(X[) = 0 but E(Ai^) > 0 and (1.1) holds. Then we have for any c > 0 lim Vn = I*~ (1/c) a.s.. n—*oo