Abstract: Let 0≤a₁≤a₂≤... be a sequence of fixed non-negative integers, the limit properties investment portfolio growth rate from a_n+1 to a_n+n are studied. By constructing a sequence of random variables with one parameter and bounded expectations, and using the Borel-Cantelli lemma, the limit theorems for the growth rate of any investment portfolio and the general market conditions are obtained, and a way to Markov inequality and the Borel- Cantelli lemma and other tools to the strong limit theorem is given.