Such funds, enabling investors to buy into the bonds or shares of countries looking to join the European Union or euro zone, were some of the hottest trends around until the first throes of the credit crisis five years ago.

The euro debt crisis, however, saw that convergence concept rapidly discarded. According to fund tracker Lipper, dedicated European convergence funds dropped from a peak of $9 billion in 2007 to $1 billion last year.

But as the euro zone's future has become a little clearer, investors have become more enthusiastic about buying emerging European assets. So convergence fund managers may be tempted to remove the mothballs.

There is as yet no great rush. Indeed, JP Morgan Asset Management switched its European convergence equity fund to a Turkey fund just last month.

But there could be money to be made out of remodeling the old convergence trade to fit the more upbeat mood.

"We will start to see the euro convergence story picking up steadily in the next two years," said Simon Quijano-Evans, central and eastern Europe economist at ING.

"The more positive signs we get out of the euro zone, the more of a story it will become again."

The trade is likely to be different this time around, with investors looking more closely at the local economies of converging countries, rather than seeing them purely as euro zone-lite as they did before.

Euro zone convergence will also be aided by a broader convergence trend in debt levels, credit ratings and fiscal discipline, investors say, with emerging markets generally going up, and developed markets going down.

This broad convergence path can be traced to the collapse of the Soviet Union in 1991 and the subsequent move to market economies, says Jan Dehn, co-head of research at Ashmore, and emerging European markets will be among those to benefit.

"We are part of a big global convergence trade that started 23 years ago, this is not going to stop."

COMING AND GOING

According to Lipper data, just over 100 funds have ever labeled themselves as convergence funds, in bonds and equities. About 60 percent of these were either liquidated, merged with other funds or were rebranded.

Of the remaining funds tracked by Lipper, only 15 total more than $10 million.

But for those that have survived, such as convergence bond funds run by Deka and Dexia, last year was a bumper one, with returns of more than 20 percent, according to Lipper. That compares with returns of 14 percent for Austrian or French bonds, and 8 percent for Spain.

Convergence may not be at the top of investors' minds just yet, with euro zone entry dates for EU members largely postponed. But emerging European countries have enjoyed increasing investment in the past few months because credit ratings are rising and default risks decreasing compared with many countries in the developed world.

Polish short-term debt yields, for example, have halved to 3 percent since the country joined the EU in 2004.

"Poland is considered a safe bet," said Koon Chow, emerging markets strategist at Barclays.

"It's working with debt levels that are lower than the euro zone, the government has presided over fiscal tightening, we have a central bank which is willing to cut but only gradually. All that has attracted flows from both emerging market and non-dedicated investors."

Joining the euro is not as popular as it once was, but euro zone or EU entry prospects are rising in some countries.

Latvia will make a formal euro zone entry application in February, for entry in 2014, and Poland's prime minister Donald Tusk last month raised the prospect of relaunching his country's bid to join the euro soon.

Croatia, meanwhile, is scheduled to join the EU in July and Serbia may get a date for EU entry talks by then.

These changes mean there could be another round of currencies entering the exchange rate mechanism (ERM), the waiting-room for euro zone entry.

For Helena Clijsters, who runs Dexia's convergence bond fund with more than $100 million under management, greater confidence about the euro after Euroepan Central Bank President Mario Draghi said last July he would do "whatever it takes" to preserve the single currency has helped bring in investors.

"As long as risk appetite remains, these markets will continue to perform," she said.

(Additional reporting by Joel Dimmock and Sujata Rao. Editing by Jeremy Gaunt.)

• A Permanent Economic Emergency