Sew the arms down (whip stitch is easiest) across the top and bottom. This should be enough to secure the arms on tightly.
Optionally (the period example doesn't seem to have it); you can stitch the sides down as well, where the pins were. It makes the arms less floppy, keeps the stuffing securely in the arms and secures the arms on better.
I use a back stitch as the longer stitch on the underside makes a more secure stitch through the body.