98 Sport - THEFT light on while driving

My dashboard anti-theft indicator has been staying on lately while
driving. When this started to happen a few years ago, other strange
things started to happen, like my key fob not working the locks
intermittently. Followed soon thereafter by some weak starts and
replacement of the battery. The service advisor at the dealer claimed
to know nothing about the anti-theft indicator having any connection to
a weak battery, and his handheld tester showed my battery to be OK.

I've been away from this newsgroup for a few years, so hopefully the
old-timers will remember if this topic has ever come up before.
Thanks.