Course in Probability Theory

242 1 CENTRAL LIMIT THEOREM AND ITS RAMIFICATIONSan asymptotic evaluation ofI- F,(x,)1 - 4)(x,)as x„ -a oc more rapidly than indicated above . This requires a different typeof approximation by means of "bilateral Laplace transforms", which will notbe discussed here .EXERCISES1 . If F and G are d .f .'s with finite first moments, then~DOIF(x) - G(x)I dx < oo .[HINT : Use Exercise 18 of Sec . 3 .2 .]2 . If f and g are ch .f .'s such that f (t) = g(t) for Itl < T, then00TIF(x)-G(x)Idx< ./ - TThis is due to Esseen (Acta Math, 77(1944)) .*3 . There exists a universal constant A 1 > 0 such that for any sequenceof independent, identically distributed integer-valued r .v .'s {X j } with mean 0and variance 1, we havesup IF ., (x) - (D (x) I ?1/2'where F t? is the d .f . of (E~,1 X j )/~ . [HINT : Use Exercise 24 of Sec . 6 .4 .]4. Prove that for every x > 0 :00X e-x2/2 < e-y2 /2 d < l e -x2 /21 -+- x2 I - - x7 .5 Law of the iterated logarithmThe law of the iterated logarithm is a crowning achievement in classical probabilitytheory . It had its origin in attempts to perfect Borel's theorem on normalnumbers (Theorem 5 .1 .3) . In its simplest but basic form, this asserts : if N„ (w)denotes the number of occurrences of the digit 1 in the first n places of thebinary (dyadic) expansion of the real number w in [0, 1], then N n (w) n/2for almost every w in Borel measure . What can one say about the deviationN, (a)) - n/2? The order bounds O(n(l/21+E) E > 0; O((n log n) 1 / 2 ) ( cf.