# Probability and Information Theory II by Alfred Rényi (auth.), M. Behara, K. Krickeberg, J. Wolfowitz By Alfred Rényi (auth.), M. Behara, K. Krickeberg, J. Wolfowitz (eds.)

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Extra resources for Probability and Information Theory II

Example text

We shall fill them up. How to define +co n -- E S ¢(o)6U(o) ? -0D Let on ~(-" C,(~) be the space of all continuous real valued functions "N with compact support, Then -O~ - 47 - +00 : defines a E ~" ¢(~)~(u(~)) ~ (~)-valued measure on ~ with support in ~ + which we denote by +W E S ¢(a)~U~,a)" -OO The following lemma shows that commuting of expectations and convolution is legitimate in our case. Lemma 5. Let for XI, X 2 i = I, 2 be be two locally compact Hausdorff spaces, let Pi a probability measure on Pi-measurable application from = 0, and F1 Ai Xi be a Pi-integrable into IR+ Xi, let such that function from Xi be an integrable continuous application from space Co(~ ) of all continuous functions on ~ be a Pi{Ui = ~ into X1 Ui ~ (~).

K 1 + ... ,N). Then 1 E~(~ 1 o . . P 1 = (1-])EU Lemma 4. For £---~ ~ compact interval of o Ol). P£ the functions converge uniformly on every "JR to (% + ~ , + ~+ (~ n) ~'~-I * (6 - n +) W-! * 6+ ) with n = E~(a), and g ~ = E~(a) is a stochastic variable on E distributed with respect to Q. - Proof. 46 By the corollary of lemma - 1 one obtains J + ~, [ ~(°J) * ~(oj_1) * ... , ~(oi+~) * ~(o i) + l

K£:! j:l g k I .... ,k£:1 j=l ~kj • {{Ul { +1 ' ' ' D 1{Ukj-I ~< +1 {{ukj > +}] £ j=1 2 Efk(J) ((~1' Ul)' .... (+k,Uk)) ~{u14 t} ... {{Uk_l-(t}. d. ~} - 43 - During The lemma respect shows the p r o o f of the last that to the same the Kj's are lemma we called independent law. P{Kj : k} : The e x p e c t a t i o n of K. J = J The (ii), function p{u Call Lemma > 3} > 0 made - PlU above. has the e x p e c t a t i o n mu > ~iEN/ = (/-1)/p{u > 3}. lemma plausible. functions 1 R N -- ~ uniformly E WN in every 1 P1 -- EU(/-1) converge 3} N i = K 1 + ...