At the end of the 12-week dose adjustment phase, at Visit 4, it will be determined whether meaningful tremor reduction has occurred at the target dose of 300 mg per day or at a lesser but tolerated dose. If not, the participant will discontinue from the study. If benefit has occurred, the subject will be invited to continue taking zonisamide for another 12 weeks, at the conclusion of which the participant will be seen in the final visit (Visit 5).