Here's a summary of some research on hot and cold streaks. The summary is taken from Rob Neyer's column on ESPN.COM from Wed., Oct 13, 1999. I haven't seen the original reference or source material.

The following comes from my friend
Mitchel Lichtman, and stikes me as a fairly methodical look at the issue of
hot streaks and slumps, and whether they have any predictive qualities.

Mitchel studied the 1998 season, both major leagues. He broke down
the season into 12 two-week periods. For each two-week segment, he
made two lists: players who batted lower than .200, and players who
batted higher than .350 (minimum 40 at-bats for both lists). Presumably,
those are pretty fair definitions for "slump" and "streak."

Focusing on the "hot" group, those players combined for a .391 batting
average in their two-week hot stretches. That includes 14,294 at-bats, so
obviously the sample size is significant. That same group posted a .291
batting average including the hot streak and the previous season.
Essentially, we're talking about a giant .291 hitter who has hit .391 over
the last two weeks.

Now, if being hot is a predictor of future performance, then we would
expect this group, in the coming at-bats, to hit closer to .391 than .291,
right?

But of course, that's not what happened. The group actually hit .288 over
the next two weeks, practically identical to the previous season's average
(plus the hot streak). What's that? You think looking at the next two
weeks is too much? Well, Mitchel also looked at just the first day
following the two-week hot streak. Believe it or not, the hot hitters batted
.291 on that day ... exactly the same as their average in the previous
season. 

As Mitchel writes, "So there was not a significant difference between their
expected batting average and their actual batting average, strongly
suggesting that their two-week "hot" periods were nothing more than
random statistical variations. In more vernacular terms, when a batter
steps up to the plate, the only predictors of his likely performance for that
AB, or any subsequent AB's, are his lifetime statistics, adjusted for any
relevant factors, such as the opposing pitcher, weather, ballpark, age,
injury, et cetera."

Mitchel also ran the same study for "cold" hitters, the group that batted
less than .200 over a two-week period. His results were the same. No
effect. He also looked at hot and cold hitters over one-week periods.
Same results. No effect.

Wait, I know what you're thinking: "Well, maybe we don't find statistically
significant streaks if we're looking at the entire population, but couldn't
there be at least a few hitters who tend toward streakiness?"

Let's think about that. If some are indeed streaky, yet the overall numbers
are a wash, then that means some other hitters have to make up for it.
And does anybody really think that there are baseball players whose
chances of hitting safely go down with each hit they collect? It doesn't
compute, does it?

I know all this is counter-intuitive, and I feel bad about that, I really do.
But getting back to what brought all this on, Nomar Garciaparra isn't
more likely to hit a home run because he hit one 30 minutes ago, and
Nomar Garciaparra isn't more likely to get a hit against the Indians in
October because he hit .451 against the Indians during the regular
season. This failure to understand simple laws of probability might have
cost Cleveland a trip to the World Series.