require "stat.pl";
...
for(;;) {
    open ML, "</var/log/local2" || die $!;
    &Stat(ML);
    $old_ML_size = $st_size;
    #seek(ML, 0, 2); # to EOF - Not, process from beginning

    for (;;){
        &Stat(ML);
        if ($old_ML_size > $st_size) {
            print "closing the file\n";
            last;  #file closed and new one started
        }else{
            $old_ML_size = $st_size;
        }
        while (<ML>){
            another_line();
        }
        sleep 5;
        seek(ML,  0, 1); # reset end-of-file error
    }
    close(ML);
}