AN ACR SF340 WAS CLRED FOR THE SOUND VISUAL APCH TO RWY 34 AT HPN. FO FLYING INTERCEPTED THE ILS FINAL FOR RWY 34 AND SAW THAT THEY WERE OVERTAKING AN SMA. TRACON HAD NOT SWITCHED ACFT TO TWR; BUT THEY CHANGED ON THEIR OWN AND TOLD THE TWR THEY WERE CLBING AND TURNING TO MISS THE SMA IN FRONT OF THEM.

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AFTER A SCHEDULED PAX FLT FROM BOSTON; MA; TO WHITE PLAINS; NY; WE WERE HANDED OFF TO THE FINAL APCH CTLR (NEW YORK 126.4) FOR THE HPN ARPT. THE CTLR CLRED US TO 2000 FT MSL TO FOLLOW THE SOUTHERN CONNECTICUT SHORELINE; AND THEN ISSUED A SOUND VISUAL RWY 34 CLRNC. THE FO WAS FLYING AND HE MAINTAINED 2000 FT UNTIL INTERCEPTING THE FINAL FOR RWY 34. HE THEN FOLLOWED THE ILS FOR RWY 34. NEW YORK APCH NEVER SWITCHED US OVER TO TWR; NOR DID HE EVER GIVE US A SEQUENCE FOR THE RWY. HIS FREQ WAS COMPLETELY SATURATED. WE SERE AT THE OM WHEN WE SWITCHED TO THE TWR AND GOT AN RA ON OUR TCASII. WE WERE AT OUR APCH SPD OF 130 KIAS AND WE WERE OVERTAKING A SINGLE ENG AIRPLANE ON FINAL. WE EXECUTED A CLB AND A R TURN TO AVOID THE ACFT. WE MISSED THE ACFT BY 100 FT; HIS ALT BEING 1200 FT AND OURS 1300 FT. WE EXECUTED A R TURN TO 090 DEGS BECAUSE WE MISTOOK THE TWR'S INSTRUCTIONS TO ANOTHER ACFT. THE TWR HANDED US BACK TO NEW YORK ON 126.4 AND WE WERE RESEQUENCED BACK TO THE FINAL FOR RWY 34 (THE CTLR NEVER SAID A THING TO US). WE LANDED ON RWY 34 WITHOUT FURTHER INCIDENT. AFTER LNDG AND PARKING I CALLED THE TWR AND SPOKE WITH THE CTLR IN CHARGE AND HE INFORMED ME THAT WE WEREN'T THE FIRST ACFT THAT WASN'T HANDED OFF TO THE TWR. THIS WAS JUST ANOTHER TYPICAL DAY OF FLYING IN THE NE CORRIDOR ON A NICE NIGHT WITH A LOT OF GA ACTIVITY. I BELIEVE THAT THE FINAL CTLR WAS COMPLETELY OVERWHELMED WITH TFC AND JUST FORGOT ABOUT US.