Random balancing-like sequences

A balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.