Conn. Gen. Stat. § 51-72: Probate Court; stenographer may be called in by agreement.

Whenever, in any court of probate, the parties or their attorneys so agree in writing, the judge of the court may call in a competent and disinterested person who is capable to act as a stenographer to act as the official stenographer in the whole or in such portion of the cause or matter as may be agreed upon. The compensation of the stenographer shall be paid by the parties in such proportion as the judge of the court decides, except that the compensation shall not exceed that of the official court reporter of the Superior Court.