THEORY OF MULTIVARIATE STATISTICS
THEOREM 4 13 LET THEN CAUCHY SEQUENCES IN FIRST, LET US DESCRIBE SOME PROPERTIES OF RANDOM VARIABLES FROM DEFINITION 4 14 LET AND LET BE A SEQUENCE SUCH THAT E|?K??|2?0 THEN WE SAY THAT THIS SEQUENCE CONVERGES TO ? IN (OR ?K?? AS K?+? IN OR ?=LIM ?K IN REMEMBER THAT IS THE SET OF CLASSES OF P-EQUIVALENT RANDOM VARIABLES FROM THEOREM 4 17 LET AND LET BE SUCH THAT AS K?+? THEN IS A CAUCHY SEQUENCE IN WHERE THEORY OF MULTIVARIATE STATISTICS THIS SEQUENCE CONVERGES IN AND ITS LIMIT DEPENDS ONLY ON F AND DOES NOT DEPEND ON THE CHOICE OF THE APPROXIMATING SEQUENCE (IN THE SENSE THAT ALL POSSIBLE MODIFICATIONS OF THE LIMIT ARE P-EQUIVALENT) DEFINITION 4 18 THE LIMIT OF I(FK) IN FROM THE THEOREM ABOVE IS SAID TO BE THE ITO INTEGRAL (STOCHASTIC INTEGRAL) THEOREM 4 19 LET THEN (I) (II) (III) PROOF FOLLOWS FROM THE PROPERTIES FOR APPROXIMATING FUNCTIONS FROM THEOREM 4 20 LET THEN (I) (II) (III) PROOF FOLLOWS AGAIN FROM THE PROPERTIES FOR APPROXIMATING FUNCTIONS FROM DEFINITION 4 21 A MODIFICATION OF A PROCESS ?(T, ?) IS ANY PROCESS ??(T, ?) SUCH THAT ?=(T, ?))=??(T, ?) FOR A E T, ? 2 IN FACT, PROCESSES ARE MEASURABLE AS MAPPINGS ?:[0, T]×P?R WITH RESPECT TO THE COMPLETION OF THE ?-ALGEBRA GENERATED BY ALL MAPPINGS ?0:[0, T]×P?R SUCH THAT THEOREM 4 22 LET T>0 BE FIXED, AND LET THEN THE PROCESS IS PATHWISE CONTINUOUS IN (MORE PRECISELY, THERE EXISTS A MODIFICATION OF THE PROCESS THAT IS CONTINUOUS IN A S (I E , WITH PROBABILITY 1, OR FOR A E (ALMOST EVERY) ?) THEORY OF MULTIVARIATE STATISTICS