CAPT OF MLG ACFT TOOK EVASIVE ACTION ON A FINAL VISUAL APCH WHEN ANOTHER ACFT OVERSHOT THE TURN FROM BASE LEG TO FINAL ON A PARALLEL (ALMOST) RWY INTERSECTING FLT PATHS.

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WE WERE AT APPROX 2000 FT ON A VISUAL APCH TO RWY 25L AT LAX; MONITORING THE ILS; ON COURSE AND ON GS. I MANUALLY WAS HAND FLYING THE APCH. I RECEIVED A TCASII RA COMMAND TO CLB. I INITIATED A CLB AND TOLD THE FO TO TELL THE TWR THAT WE WERE CLBING IN RESPONSE TO A TCASII COMMAND. HE SO INFORMED THE TWR AND THEY REPLIED THAT OUR TFC WAS ON A VISUAL APCH TO RWY 24. I STOPPED THE CLB MOMENTARILY THINKING THEY MUST HAVE THE TFC IN SIGHT. AT THAT TIME; MY FO SAID 'I HAVE HIM AND HE'S OVERSHOOTING BADLY AND HDG FOR US.' I ACQUIRED THE ACFT VISUALLY AND INITIATED A CLB AND BANKED AWAY FROM HIM. HE SLID UNDERNEATH US WITH 0 HORIZ CLRNC AND APPROX 300 FT OF VERT CLRNC. THE FO INFORMED THE TWR OF HIS POS AND THE TWR RESPONDED THAT THEY WERE NOT IN RADIO CONTACT WITH THE ACFT. HE REAPPEARED TO OUR R AND BELOW US AS HE TURNED BACK TOWARDS RWY 24. I REALIGNED MY ACFT WITH THE RWY AND MADE A NORMAL LNDG. THE TCASII PROBABLY SAVED OUR LIVES. THE DANGER IN THIS SITUATION THOUGH WAS THE TWR TELLING US THAT THE ACFT THE TCASII HAD ORDERED US TO CLB WAS ON APCH TO 24. THAT MADE ME THINK THAT THE TCASII HAD MADE A MISTAKE AND THE TWR KNEW THE POS OF THE OTHER ACFT. IT CAUSED ME TO SLOW OR STOP MY CLB GREATLY REDUCED OUR VERT CLRNC WHEN THE OTHER ACFT CONTINUED TO INTERCEPT OUR APCH PATH.