UPDATE 4-Blackstone CEO sees "more than green shoots"

* Can do transactions in the $3-$4 billion range

* Still nervous about strength of economic growth

* Has reduced or extended $10 bln of debt at portfolio cos

* Considering opening office in Middle East

* Blackstone shares up 4 percent (Recasts 1st paragraph, adds share rise)

By Megan Davies

DUBAI, Oct 14 (Reuters) - Private equity giant Blackstone Group's (BX.N) chief executive said the worst of the industry's problems had passed, with improved capital and equity markets finally providing an opportunity to do deals and sell existing investments through IPOs.

Stephen Schwarzman also said on Wednesday he was seeing "more than green shoots" of economic recovery, though the scale of growth through next year was still unclear.

Private equity firms have been hampered since the credit crisis shut off their ability to tap financing for leveraged buyouts; the financial turmoil has also damaged the health of their portfolio companies. Economic recovery and a rebound in financing markets are key for the industry.

"We do not expect the U.S. economy to slip back into recession but we do believe that weak consumer spending and continued constraints on bank lending will dampen the U.S. economic recovery in 2010 and 2011," Schwarzman said at the Super Return Middle East conference in Dubai.

While it would take several years before "freely flowing but responsible credit" was re-established, the private equity industry was in a "radically different place" than a year ago, given signs of life in the bank financing market, he said.

"We can certainly do transactions in the $3-$4 billion range at this stage in the cycle," he said on the sidelines of the conference. "And with low leverage involved, deals of that size can use in excess of $1 billion equity."

Blackstone, one of the world's biggest private equity firms, struck a deal earlier this month to buy Anheuser-Busch InBev's U.S. theme parks for up to $2.7 billion, adding to its amusement assets such as the Madame Tussauds wax museums, Legoland and the London Eye Ferris wheel.

Based on recent deals, Blackstone's implied investment rate is $4 billion to $5 billion a year, Schwarzman said.

He said right now is an excellent time to purchase stable businesses in developed markets, but added it was still too early for cyclical companies. He sees opportunities to buy growth companies in Asia.

Schwarzman said while he expects more deals ahead, Blackstone has been outbid by companies rather than private equity firms on several occasions recently.