Validation of Dollar cost averaging investment method

As a method of long term investment for private investor, the dollar cost averaging investment method is well known and seems to reduce the purchase cost because we purchase risk assets with same amount of monye every month, then we purchase many assets when the price of assets is low and few assets when the price is high. On the other hand, if the expectation of the return of the risk assets is positive, we have the maximum expectation of return when we purchase the risk assets with all of money to invest. To reduce the risk of investment, diversified investments are effective. However question whehter we use the dollar cost averaging investment method or investe money all at once to well-diversified risk assets remains. In this study, we validate the effecte of the ddollar cost averaging investment method by Monte Carlo simulation