# Read logfile names with last position read yesterday
open $inf, "<", "loglist.txt" or die;
while (<$inf>) {
   chomp;
   my ($fname, $fpos) = split /\|/;
   $logs{$fname} = $fpos;
}
close $inf;

# Get new lines from each file
for my $fname (keys %logs) {
   open $ouf, '>>', $fname . ".cumulative" or die;
   open $inf, '<', $fname or die;
   if ($logs{$fname} < stat($inf)[7]) {
      # Continue from where we left off yesterday
      seek $inf, $logs{$fname}, SEEK_SET;
   }
   else {
      # start at beginning of file
   }
   while (<$inf>) {
      print $ouf $_;
   }
   $logs{$fname} = tell $inf;
   close $inf;
   close $ouf;
}

# Rewrite list of files and positions
open $ouf, ">", "loglist.txt" or die;
print $ouf join "\n", map { $_ . '|' . $logs{$_} } keys %logs;
close $ouf;