SmileBASIC, 6161 58 bytes
@L
PUSH B$,RB$=B$+R()IF LEN(B$)>9THENI>8THEN R=RND(9)?B$[R];:B$[R]="
GOTO@LI=I+1GOTO@L
Each character of the infinite list is added to the end of the buffer. When the buffer length is greater than 911, a random character is printed and removed.
Function R generates the next character.